Light - The Physics of the Photon

LIGHT The Physics of the Photon

Ole KellerAalborg University, Denmark

LIGHT The Physics of the Photon

Cover image: Courtesy of Esben Hanefelt Kristensen, based on a painting entitled “A Wordless Statement.”

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Contents

Preface xiii

Acknowledgments xix

About the author xxi

I Classical optics in global vacuum 1

1 Heading for photon physics 3

2 Fundamentals of free electromagnetic fields 72.1 Maxwell equations and wave equations . . . . . . . . . . . . . . . . . . . . 72.2 Transverse and longitudinal vector fields . . . . . . . . . . . . . . . . . . . 82.3 Complex analytical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Monochromatic plane-wave expansion of the electromagnetic field . . . . . 132.5 Polarization of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Transformation of base vectors . . . . . . . . . . . . . . . . . . . . . 142.5.2 Geometrical picture of polarization states . . . . . . . . . . . . . . . 15

2.6 Wave packets as field modes . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Conservation of energy, moment of energy, momentum, and angular momen-

tum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Riemann–Silberstein formalism . . . . . . . . . . . . . . . . . . . . . . . . . 222.9 Propagation of analytical signal . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Optics in the special theory of relativity 273.1 Lorentz transformations and proper time . . . . . . . . . . . . . . . . . . . 273.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Four-vectors and -tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Manifest covariance of the free Maxwell equations . . . . . . . . . . . . . . 333.5 Lorentz transformation of the (transverse) electric and magnetic fields. Du-

ality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Lorentz transformation of Riemann–Silberstein vectors. Inner-product invari-

ance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II Light rays and geodesics. Maxwell theory in generalrelativity 39

4 The light-particle and wave pictures in classical physics 41

5 Eikonal theory and Fermat’s principle 455.1 Remarks on geometrical optics. Inhomogeneous vacuum . . . . . . . . . . . 455.2 Eikonal equation. Geometrical wave surfaces and rays . . . . . . . . . . . . 475.3 Geodetic line: Fermat’s principle . . . . . . . . . . . . . . . . . . . . . . . . 52

vii

viii Contents

6 Geodesics in general relativity 556.1 Metric tensor. Four-dimensional Riemann space . . . . . . . . . . . . . . . 556.2 Time-like metric geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 The Newtonian limit: Motion in a weak static gravitational field . . . . . . 596.4 Null geodesics and “light particles” . . . . . . . . . . . . . . . . . . . . . . 616.5 Gravitational redshift. Photon in free fall . . . . . . . . . . . . . . . . . . . 62

7 The space-time of general relativity 677.1 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.3 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.4 Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.5 Algebraic properties of the Riemann curvature tensor . . . . . . . . . . . . 737.6 Einstein field equations in general relativity . . . . . . . . . . . . . . . . . . 747.7 Metric compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.8 Geodesic deviation of light rays . . . . . . . . . . . . . . . . . . . . . . . . 76

8 Electromagnetic theory in curved space-time 798.1 Vacuum Maxwell equations in general relativity . . . . . . . . . . . . . . . 798.2 Covariant curl and divergence in Riemann space . . . . . . . . . . . . . . . 808.3 A uniform formulation of electrodynamics in curved and flat space-time . . 81

8.3.1 Maxwell equations with normal derivatives . . . . . . . . . . . . . . 818.3.2 Maxwell equations with E, B, D, and H fields . . . . . . . . . . . . 838.3.3 Microscopic Maxwell–Lorentz equations in curved space-time . . . . 848.3.4 Constitutive relations in curved space-time . . . . . . . . . . . . . . 858.3.5 Remarks on the constitutive relations in Minkowskian space . . . . . 878.3.6 Permittivity and permeability for static metrics . . . . . . . . . . . . 88

8.4 Permittivity and permeability in expanding universe . . . . . . . . . . . . . 898.5 Electrodynamics in potential description. Eikonal theory and null geodesics 918.6 Gauge-covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

III Photon wave mechanics 97

9 The elusive light particle 99

10 Wave mechanics based on transverse vector potential 10510.1 Gauge transformation. Covariant and noncovariant gauges . . . . . . . . . 10510.2 Tentative wave function and wave equation for transverse photons . . . . . 10710.3 Transverse photon as a spin-1 particle . . . . . . . . . . . . . . . . . . . . . 11010.4 Neutrino wave mechanics. Massive eigenstate neutrinos . . . . . . . . . . . 113

11 Longitudinal and scalar photons. Gauge and near-field light quanta 11911.1 L- and S-photons. Wave equations . . . . . . . . . . . . . . . . . . . . . . . 11911.2 L- and S-photon neutralization in free space . . . . . . . . . . . . . . . . . 12011.3 NF- and G-photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12211.4 Gauge transformation within the Lorenz gauge . . . . . . . . . . . . . . . . 124

Contents ix

12 Massive photon field 12712.1 Proca equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12712.2 Dynamical equations for E and A . . . . . . . . . . . . . . . . . . . . . . . 12912.3 Diamagnetic interaction: Transverse photon mass . . . . . . . . . . . . . . 13012.4 Massive vector boson (photon) field . . . . . . . . . . . . . . . . . . . . . . 13212.5 Massive photon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

13 Photon energy wave function formalism 14313.1 The Oppenheimer light quantum theory . . . . . . . . . . . . . . . . . . . . 14313.2 Interlude: From spherical to Cartesian representation . . . . . . . . . . . . 14613.3 Photons and antiphotons: Bispinor wave functions . . . . . . . . . . . . . . 15013.4 Four-momentum and spin of photon wave packet . . . . . . . . . . . . . . . 15313.5 Relativistic scalar product. Lorentz-invariant integration on the

energy shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

IV Single-photon quantum optics in Minkowskian space 159

14 The photon of the quantized electromagnetic field 161

15 Polychromatic photons 16515.1 Canonical quantization of the transverse electromagnetic field . . . . . . . 16515.2 Energy, momentum, and spin operators of the transverse field . . . . . . . 16815.3 Monochromatic plane-wave photons. Fock states . . . . . . . . . . . . . . . 17115.4 Single-photon wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 17315.5 New T-photon “mean” position state . . . . . . . . . . . . . . . . . . . . . 17715.6 T-photon wave function and related dynamical equation . . . . . . . . . . . 17915.7 The non-orthogonality of T-photon position states . . . . . . . . . . . . . . 181

16 Single-photon wave packet correlations 18316.1 Wave-packet basis for one-photon states . . . . . . . . . . . . . . . . . . . . 18316.2 Wave-packet photons related to a given t-matrix . . . . . . . . . . . . . . . 18416.3 Integral equation for the time evolution operator in the interaction

picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18616.4 Atomic and field correlation matrices . . . . . . . . . . . . . . . . . . . . . 18916.5 Single-photon correlation matrix: The wave function fingerprint . . . . . . 194

17 Interference phenomena with single-photon states 19717.1 Wave-packet mode interference . . . . . . . . . . . . . . . . . . . . . . . . . 19717.2 Young-type double-source interference . . . . . . . . . . . . . . . . . . . . . 19817.3 Interference between transition amplitudes . . . . . . . . . . . . . . . . . . 20117.4 Field correlations in photon mean position state . . . . . . . . . . . . . . . 201

17.4.1 Correlation supermatrix . . . . . . . . . . . . . . . . . . . . . . . . . 20217.4.2 Relation between the correlation supermatrix and the transverse pho-

ton propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

18 Free-field operators: Time evolution and commutation relations 20518.1 Maxwell operator equations. Quasi-classical states . . . . . . . . . . . . . . 20518.2 Generalized Landau–Peierls–Sudarshan equations . . . . . . . . . . . . . . 20718.3 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

18.3.1 Commutation relations at different times (τ 6= 0) . . . . . . . . . . . 20918.3.2 Equal-time commutation relations . . . . . . . . . . . . . . . . . . . 210

x Contents

V Photon embryo states 213

19 Attached photons in rim zones 215

20 Evanescent photon fields 22120.1 Four-potential description in the Lorenz gauge . . . . . . . . . . . . . . . . 22120.2 Sheet current density: T-, L-, and S-parts . . . . . . . . . . . . . . . . . . . 22320.3 Evanescent T-, L-, and S-potentials . . . . . . . . . . . . . . . . . . . . . . 22520.4 Four-potential photon wave mechanics . . . . . . . . . . . . . . . . . . . . . 22920.5 Field-quantized approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23120.6 Near-field photon: Heisenberg equation of motion and coherent state . . . . 234

21 Photon tunneling 23721.1 Near-field interaction. The photon measurement problem . . . . . . . . . . 23721.2 Scattering of a wave-packet band from a single current-density sheet . . . . 23821.3 Incident fields generating evanescent tunneling potentials . . . . . . . . . . 24321.4 Interlude: Scalar propagator in various domains . . . . . . . . . . . . . . . 24621.5 Incident polychromatic single-photon state . . . . . . . . . . . . . . . . . . 24721.6 Photon tunneling-coupled sheets . . . . . . . . . . . . . . . . . . . . . . . . 250

22 Near-field photon emission in 3D 25522.1 T-, L-, and S-potentials of a classical point-particle . . . . . . . . . . . . . 255

22.1.1 General considerations on source fields . . . . . . . . . . . . . . . . . 25522.1.2 Point-particle potentials . . . . . . . . . . . . . . . . . . . . . . . . . 257

22.2 Cerenkov shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26022.2.1 Four-potential of point-particle in uniform motion in vacuum . . . . 26022.2.2 Transverse and longitudinal response theory in matter . . . . . . . . 26322.2.3 The transverse Cerenkov phenomenon . . . . . . . . . . . . . . . . . 26622.2.4 Momenta associated to the transverse and longitudinal parts of the

Cerenkov field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26922.2.5 Screened canonical particle momentum . . . . . . . . . . . . . . . . . 272

VI Photon source domain and propagators 275

23 Super-confined T-photon sources 277

24 Transverse current density in nonrelativistic quantum mechanics 28324.1 Single-particle transition current density . . . . . . . . . . . . . . . . . . . 28324.2 The hydrogen 1s⇔ 2pz transition . . . . . . . . . . . . . . . . . . . . . . . 28624.3 Breathing mode: Hydrogen 1s⇔ 2s transition . . . . . . . . . . . . . . . . 28924.4 Two-level breathing mode dynamics . . . . . . . . . . . . . . . . . . . . . . 292

25 Spin-1/2 current density in relativistic quantum mechanics 29725.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29725.2 Covariant form of the Dirac equation. Minimal coupling. Four-current density 29925.3 Gordon decomposition of the Dirac four-current density . . . . . . . . . . . 30125.4 Weakly relativistic spin current density . . . . . . . . . . . . . . . . . . . . 30325.5 Continuity equations for spin and space four-current densities . . . . . . . 306

Contents xi

26 Massless photon propagators 30926.1 From the Huygens propagator to the transverse photon propagator . . . . 30926.2 T-photon time-ordered correlation of events . . . . . . . . . . . . . . . . . . 31126.3 Covariant correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 31326.4 Covariant quantization of the electromagnetic field: A brief review . . . . . 31426.5 The Feynman photon propagator . . . . . . . . . . . . . . . . . . . . . . . 31626.6 Longitudinal and scalar photon propagators . . . . . . . . . . . . . . . . . 318

VII Photon vacuum and quanta in Minkowskian space 321

27 Photons and observers 323

28 The inertial class of observers: Photon vacuum and quanta 32928.1 Transverse photon four-current density . . . . . . . . . . . . . . . . . . . . 32928.2 Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

28.2.1 Lorentz and Lorenz-gauge transformations of the four-potential . . . 33228.2.2 Plane-mode decomposition of the covariant potential . . . . . . . . . 33328.2.3 Mode functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

28.3 Physical (T-photon) vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . 337

29 The non-inertial class of observers: The nebulous particle concept 34529.1 Bogolubov transformation. Vacuum states . . . . . . . . . . . . . . . . . . 34529.2 The non-unique vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34829.3 The Unruh effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

29.3.1 Rindler space and observer . . . . . . . . . . . . . . . . . . . . . . . 35229.3.2 Rindler particles in Minkowski vacuum . . . . . . . . . . . . . . . . . 354

30 Photon mass and hidden gauge invariance 36330.1 Physical vacuum: Spontaneous symmetry breaking . . . . . . . . . . . . . . 36330.2 Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36630.3 The U(1) Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36830.4 Photon mass and vacuum screening current . . . . . . . . . . . . . . . . . . 37230.5 ’t Hooft gauge and propagator . . . . . . . . . . . . . . . . . . . . . . . . . 373

VIII Two-photon entanglement in space-time 377

31 The quantal photon gas 379

32 Quantum measurements 38532.1 Tensor product space (discrete case) . . . . . . . . . . . . . . . . . . . . . . 38532.2 Definition of an observable (discrete case) . . . . . . . . . . . . . . . . . . . 38632.3 Reduction of the wave packet (discrete case) . . . . . . . . . . . . . . . . . 38732.4 Measurements on only one part of a two-part physical system . . . . . . . 38732.5 Entangled photon polarization states . . . . . . . . . . . . . . . . . . . . . 390

33 Two-photon wave mechanics and correlation matrices 39333.1 Two-photon two times wave function . . . . . . . . . . . . . . . . . . . . . 39333.2 Two-photon Schrodinger equation in direct space . . . . . . . . . . . . . . 39633.3 Two-photon wave packet correlations . . . . . . . . . . . . . . . . . . . . . 397

33.3.1 First-order correlation matrix . . . . . . . . . . . . . . . . . . . . . . 39733.3.2 Second-order correlation matrix . . . . . . . . . . . . . . . . . . . . . 399

xii Contents

34 Spontaneous one- and two-photon emissions 40134.1 Two-level atom: Weisskopf–Wigner theory of spontaneous emission . . . . 401

34.1.1 Atom-field Hamiltonian in the electric-dipole approximation. RWA-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

34.1.2 Weisskopf–Wigner state vector . . . . . . . . . . . . . . . . . . . . . 40634.2 Two-level atom: Wave function of spontaneously emitted photon . . . . . . 409

34.2.1 Photon wave function in q-space . . . . . . . . . . . . . . . . . . . . 40934.2.2 The general photon wave function in r-space . . . . . . . . . . . . . 41034.2.3 Genuine transverse photon wave function . . . . . . . . . . . . . . . 41134.2.4 Spontaneous photon emission in the atomic rim zone . . . . . . . . . 413

34.3 Three-level atom: Spontaneous cascade emission . . . . . . . . . . . . . . . 41734.3.1 Two-photon state vector . . . . . . . . . . . . . . . . . . . . . . . . . 41734.3.2 Two-photon two-times wave function . . . . . . . . . . . . . . . . . . 42034.3.3 The structure of Φ2,T (r1, r2, t1, t2) . . . . . . . . . . . . . . . . . . . 422

34.3.4 Far-field part of Φ(1)2,T (r1, r2, t1, t2) . . . . . . . . . . . . . . . . . . . 425

Bibliography 429

Index 441

Preface

I have often been asked what is a photon? In order to attempt to answer this question,as communicating human beings, above all we must learn how to use the word is in anunambiguous manner. The learning process takes us on a journey into deep philosophicalquestions, and many of us end up being bewildered before we finally are snowed underwith philosophical thinking. In my understanding, the is-problem is like a Gordian knot.In physics we replace the word is by characterizes, although in everyday discussion amongphysicists we do not need to distinguish between is and characterizes, in general. So, I takethe liberty to replace the original question with what characterizes a photon? If someone asksyou who is this person you will “only” be able to answer by mentioning as many features,traits, etc., as you are aware of about the given person. In a sense, a good characterizationof a phenomenon in physics means to look at the phenomenon from various perspectives(through different windows). In the case of the photon, we approach the original questionwhat is a photon by looking at the phenomenon through as many windows as possible. Onlyin the never attainable limit, where the number (N) of windows [photon perspectives (PP)]approaches infinity, has one captured the photon phenomenon, at least in my understanding.Mathematically,

Observational possibility ≡N∑

i=1

(PP )i

⇒∞∑

i=1

(PP )i ≡ The photon phenomenon.

In this book I take a look at the photon phenomenon from a personal selection of a fewperspectives. The insight obtained by looking through some of the windows may already befamiliar to the reader.

Above I have made use of the word phenomenon, and replaced photon with photonphenomenon. The concept phenomenon was introduced in the physical literature by NielsBohr, and the definition he first formulated publicly at a meeting in Warsaw in 1938,arranged by the International Institute of Intellectual Co-operation of the League of Nations.Niels Bohr, one of the monumental figures in the establishment of quantum mechanics,throughout his life, with ever-increasing force of the argument, emphasized that we mustlearn to use the words of the common language in an unambiguous manner, because afterall, we as physicists essentially have only the common language when we discuss witheach other what we have learned in our field of study. According to Bohr, no elementaryquantum phenomenon is a phenomenon until it is a registered (observed) phenomenon. ForBohr quantum mechanics was a rational generalization of classical physics, and his definitionof the phenomenon concept made it possible to unite the seemingly incompatible particleand wave aspects of the photon phenomenon, e.g., the single- and double-slit experimentswith photons. Bohr’s phenomenon concept, as well as another of his central points, viz.,that the functioning of the measuring apparatus always must (and only can) be describedin the language of classical physics, will be important for us to remember. To Bohr, everyatomic phenomenon is closed in the sense that its observation is based on registrations

xiii

xiv Light—The Physics of the Photon

obtained by means of suitable macroscopic devices (with irreversible functioning). Bohrconsidered the closure of fundamental significance not only in quantum physics but in thewhole description of nature, and he often stressed in discussions that “reality” is a word inour language and that we must learn to use it correctly. Kalckar, in the 1967 book NielsBohr: His Life and Work as Seen by His Friends and Colleagues (edited by S. Rozental)quoted Bohr for the following statement: I am quite prepared to talk of the spiritual life ofan electronic computer, to say that it is considering or that it is in a bad mood. What reallymatters is the unambiguous description of its behaviour, which is what we observe. Thequestion as to whether the machine really feels, or whether it merely looks as though it did,is absolutely as meaningless as to ask whether light is “in reality” waves or particles. Wemust never forget that “reality” too is a human word just like “wave” or “consciousness.”Our task is to learn to use these words correctly − that is, unambiguously and consistently.It will be well to remember the fundamental (central) points of Niels Bohr throughout thereading of this book.

Notwithstanding the fact that field–matter interaction is needed for a photon to appearas a photon phenomenon, it is nevertheless indispensable to study the photon as a conceptbelonging to global vacuum (matter-free space). Although the photon of the vacuum is anabstraction of our mind, the photon concept must be firmly connected to the electromagneticfield concept in free space. The autonomy of the classical electromagnetic field in free space issolely connected with the vacuum speed of light (c): The classical electromagnetic field is anintermediary describing the delayed (with speed c) interaction between electrically chargedparticles in nonuniform motion. Although there is no room for accommodating the photonconcept in the framework of classical electrodynamics, it is of value to investigate how farone may proceed toward the introduction of a classical light particle concept in a classicalframework. The autonomy of the electromagnetic field increases in an essential manner withthe introduction of the quantum of action (Planck’s constant, h) in electrodynamics. Thephoton concept then flourishes, and the photon-free vacuum appears with its own autonomy.The modern era of the light particle (based on h) began when Einstein in 1905 concludedthat monochromatic (frequency: ν) radiation of low density (within the range of validity ofWien’s radiation formula) behaves thermodynamically as though it consisted of a number ofindependent energy quanta of magnitude hν.

In Part I, we prepare ourselves for photon physics by studying certain aspects of classicaloptics in a global vacuum on the basis of the free-space Maxwell equations. Since the photonin global vacuum (T-photon) is a transversely (T) polarized object belonging to the positive-frequency part of the electromagnetic spectrum, studies of transverse (longitudinal) vectorfields, complex analytical signals, and the various polarization states of light are central.With an eye to the point-like Einstein light particle we also describe how the electromagneticfield can be resolved into a complete set of wave-packet modes. Because the massless photonnecessarily is a relativistic object propagating with the vacuum speed of light, it is importantto consider the fields of classical optics from the perspective of special relativity. Our briefaccount of optics in special relativity culminates with a demonstration of the manifestcovariance of the Maxwell equations, and a discussion of the Lorentz transformation of thetransverse and longitudinal parts of the electromagnetic field.

In Part II, we study light rays and geodesics, and we also present a brief account of theMaxwell theory in general relativity. In the framework of classical electrodynamics thereis no hope for considering light as consisting of some sort of particles, in general. Thisis so because (wave) interference effects cannot occur in classical particle dynamics. In acorner of the classical field theory, known as geometrical optics, the wavelength (λ) of lightplays no role; however, in the short wavelengths limit and here (λ → 0) a geometrizationof the field description in the form of light rays appears. The eikonal equation is the basicequation of geometrical optics. A classical particle moves along a trajectory, and in the

Preface xv

framework of geometrical optics it makes sense to reflect on whether a kind of approximatelight particle theory can be established in which the particle follows a trajectory (light ray)according to the possibilities inherent in the eikonal theory. From a somewhat differentperspective a light ray appears as a geodetic line for particle motion. The equation forthe geodetic line is obtained from a variational principle which also gives one Fermat’sprinciple. Although it is not meaningless to consider a light ray as a particle trajectory,it is not possible to extend the formalism in such a manner that it describes the motionof a light particle which is spatially well-localized somewhere on the ray at a given time.The geodesic principle can be generalized to general relativity, and the “light particle” herepropagates along null geodesics. On the basis of the principle of equivalence, the geodesicapproach leads to the conclusion that the gravitational field may shift the frequency ofa locally monochromatic light beam along the geodetic line. This so-called gravitationalredshift can be understood from a somewhat different perspective that relates to quantumtheory, viz., as a monochromatic photon in free fall in a gravitational field. It is possibleto go beyond the geometrical optical approximation in general relativity, and establish anextension of the Maxwell–Lorentz theory to curved space-time. A beautiful reformulation ofthe basic theory allows one to present the Maxwell–Lorentz theory in general relativity in aform formally identical to that of macroscopic electrodynamics. Thus, the role of the metrictensor is reflected via effective permittivity and permeability tensors. In the quantum theoryof the photon the scalar and vector potentials play a central role, and for this reason alone itis important that the possibilities for establishing a potential description of electrodynamicsin curved space-time is presented to the reader.

In Part III, the theory of photon wave mechanics is discussed. The wave mechanicalpicture of light partly is based on a reinterpretation of the content of the free Maxwellequations. In this book, the properly normalized transverse part vector potential, a gauge-invariant quantity, is considered as the wave function of the free (transverse) photon. [Twotransverse photon types having orthogonal polarizations (e.g., opposite helicities) are neededto establish the general theory.] Starting from the wave equation for the transverse vectorpotential AT , a Schrodinger-like (Hamiltonian) wave equation for the analytical signal,

A(+)T , emerges. In the framework of classical electrodynamics there is no room for the quan-

tum of action, and only by brute force Planck’s constant can be attached to photon wavemechanics based on the reinterpretation of the Maxwell theory. The division of the vectorpotential into transverse and longitudinal parts is not Lorentz invariant. This fact is not initself a problem from the point of view that one finally always has to connect the abstractphoton concept to the photon phenomenon. This concept relates to what an observer canmeasure, and a given inertial observer cannot be in two different inertial frames at a giveninstant of time. A Lorentz invariant photon wave mechanical theory can be establishedif one is willing to introduce (in addition to the two transverse photon types) a longitu-dinal and a scalar photon. In free space there is no net physical effect of these photons,often called virtual photons. In the rim (near-field) zone of matter (source/detector) thiscanceling does not occur. It is possible however to replace the longitudinal and scalar pho-tons by two new ones, the so-called gauge and near-field photons. The gauge photon can beeliminated by a suitable gauge transformation within the Lorenz gauge, leaving us with thenear-field photon. As the name indicates this photon plays an important role in near-fieldelectrodynamics. Although the free photon in our present understanding (description) ofthe physical world is massless, it is interesting to reflect on the (hypothetical) situationwhere the transverse photon is endowed with a mass. The quantum mechanics of the mas-sive photon is governed by the Proca equation and the Lorenz condition of the potentials,which usually is a subsidiary condition, must be satisfied. In cases where the photon’s inter-action with matter is dominated by the diamagnetic coupling (as in a BCS superconductor,for example) the transverse photon may acquire an effective mass. This circumstance, in a

xvi Light—The Physics of the Photon

relativistic setting, leads to the conclusion that the interaction between a transverse pho-ton and a relativistic spinless boson particle under certain conditions makes the photonmassive, but still with the freedom of gauge invariance lost. Once the photon is made mas-sive, it is possible to make a Lorentz transformation to the photon’s rest frame. The newframe’s velocity equals the light particle’s group velocity in the original frame. Although themain emphasis in this book is devoted to a formalism in which photon wave mechanics isbased on the transverse vector potential, alternatives exist. Starting with the Oppenheimerlight quantum theory from 1930, I discuss the closely related photon energy wave functionformalism in some detail. In this connection remarks on antiphotons are given.

In Part IV, we turn toward the field-quantized description of the electromagnetic field,paying particular attention to single-photon quantum optics in Minkowskian space. In text-books the photon concept usually is connected with the elementary quantum excitations as-sociated to monochromatic plane waves, yet sometimes to monochromatic multipole waves.A single photon may be emitted when an atom makes a stimulated downward transitionfrom a stationary state |a〉 to a stationary state |b〉. From the Bohr relation Ea − Eb = ~ωit appears that the photon is monochromatic (angular frequency: ω). This result of theold quantum theory cannot be strictly correct in general since the decay time is finite. Forsingle photon emission from a general many-body transition the same conclusion holds: Thephoton is polychromatic. To qualify as a polychromatic single-photon state the eigenvalueof the global number operator must be 1. First, I develop and discuss the polychromaticone-photon theory in Hilbert space. Next, I introduce a (new) T-photon “mean” positionstate in the state space in order to introduce a polychromatic single-photon wave functionin direct space. Finally, I establish the dynamical (Schrodinger-like) wave equation for thephoton. Our choice of T-photon wave function is based on a mean position state, |R〉, in-troduced via the action of the negative-frequency part of the local vector-potential operator

A(−)T (r, t) on the global photon vacuum, |0〉. Hence |R〉(r, t) ≡ (2ǫ0c/~)

1/2A(−)T (r, t)|0〉.

This definition allows one to capture all observational photon phenomena, e.g., also thoserelated to the Aharonov–Bohm effect. It is shown that it is possible to form a polychromatic(wave packet) basis for one-photon states. Atomic and field correlation matrices allow oneto address the question: How can a single-photon phenomenon manifest itself? On the ba-sis of a single-photon correlation matrix interference phenomena with single-photon wavepackets are discussed.

In Part V, we concentrate on photon physics in the rim zone of matter, paying particularattention to photon emission processes. In the rim zone the “object” that ends up as a T-photon after the light source has stopped its activity is attached to matter. I have calledthe transverse part of the field state in the rim zone a photon embryo. As the T-embryopropagates outward from the source it gradually develops into a T-photon. Important insightinto photon physics in the rim zone can be obtained in the covariant field formalism. Thus,in this formalism the coupling of the T-photon to its source is described as an interactionwith longitudinal and scalar photons. I discuss basic aspects of the rim zone photon physicsvia studies of selected examples, viz., evanescent fields, photon tunneling, electric monopoledynamics, and Cerenkov shock waves. The chosen examples illustrate the first- and second-quantized versions of photon wave mechanics at work.

In Part VI, we take a closer look at the photon source domain, and the field propagatorsthat in a convenient manner describe the photon field propagation in the vicinity of and farfrom the electronic source domain. The source domain of a transverse photon is identical tothe domain occupied by the transverse part (JT ) of the electronic current density (J). Thecurrent density J is obtained via the relevant many-body (or single-particle) transition cur-rent densities. In most cases the related JT is algebraically confined for atomic transitions[distance dependencies from the nucleus: r−3 (ED-transitions), r−4 (MD+EQ-transitions),etc.] as we illustrate by a nonrelativistic study of the hydrogen 1s ⇔ 2pz transition. In a

Preface xvii

few exotic cases it turns out that JT = J. When this happens the source domain of theT-photon becomes exponentially confined. Such so-called super-confined T-photon sourcesappear in what I denote as breathing mode transitions. The breathing mode current densityoriginates in the diamagnetic part of the transition current density, a part that is needed forgauge invariance. I illustrate the breathing mode dynamics (and confinement) by a studyof the 1s ⇔ 2s transition in hydrogen. This transition is forbidden in all multipole ordersif the diamagnetic part of the current density is neglected. A pure spin-1/2 current densityalso may lead to exponential T-photon source confinement. Starting from the Gordon ex-pression for the spin part of the relativistic spin-1/2 current density (an expression that Idiscuss in some detail) it is shown that the spin current density in the weakly relativisticlimit is a transverse current density vector field. Part VI is closed with studies of masslessphoton propagators, such as the Huygens propagator, the transverse photon propagator,the Feynman photon propagator, and the longitudinal and scalar photon propagators. Theclose relation between the propagators and the related photon correlation matrices is em-phasized, and the connection between T-photon time-ordered correlation events (based onthe mean position state for transverse photons) and the transverse photon propagator isdetermined and discussed.

In Part VII, we study the photon vacuum and light quanta in Minkowskian space. In freespace a physical photon vacuum state, |0PHY S〉, is a state in which the number of transverse

photons is zero. When an arbitrary transverse photon annihilation operator †T acts on thephysical vacuum state one obtains

†T |0PHY S〉 = 0,

a definition of the T-photon vacuum state. It is important to understand that |0PHY S〉 isa state in “our physical world,” whereas the zero on the right side of the relation above is“outside this world.” In a sense one may say that the operator †T is the recipe for transfer-ring one to the state of Nirvana, 0 = NIRVANA. From this state no operation O can bringus back to the physical world. In Minkowskian space inertial observers have a privileged sta-tus. Although the physical photon vacuum state will be the same for all inertial observers,a Lorentz boost changes the number of scalar (S) and longitudinal (L) photons in |0PHY S〉.In free space there is no net physical effect of these photon types, and a given allowed ad-mixture of L- and S-photons can be removed by a suitable gauge transformation within theLorenz gauge. An observer that accelerates through the Minkowskian vacuum will observea spectrum of transverse photons. In the special case where the observer accelerates uni-formly, with a magnitude of the four-acceleration equal to a, she/he will measure a thermal(Planck) spectrum of T-photons corresponding to an absolute temperature T0 = a/(2πkB).The privileged status of inertial observers in special relativity makes the Minkowskian vac-uum the “natural” choice for the “correct” physical vacuum. In general relativity inertialobservers become free-falling observers, and in general detectors in different free falls willnot agree on a definition of “physical vacuum.” This fact raises deep unanswered questionsconcerning quantum electrodynamics in general relativity. If the photon vacuum in somesense is analogous to the ground state of an interacting many-body system, it is possiblethat the photon vacuum is degenerate (non-unique). Such a situation may lead one to amass of the T-photon in vacuum, and to the presence of vacuum screening currents in-volving a real Higgs field. Although we have no experimental indication of the existenceof a photon vacuum mass it is nevertheless of some interest to reflect on this topic. In aphysical vacuum with spontaneous symmetry breaking the photon can acquire a vacuummass without destroying the gauge invariance freedom.

A two-photon is not two photons, but a single entity one may call a biphoton. Thus,two-photon interference cannot be considered the interference of two photons. In Part VIII,we study the two-photon entanglement that is associated to the biphoton in space-time. In

xviii Light—The Physics of the Photon

the wake of a brief account of the general formalism for quantum measurements bearingon only one part of a two-part physical system, we turn to a description of the formalismfor two-photon wave mechanics. Afterward, the first- and second-order correlation matricesassociated with two-photon wave packet correlations are discussed. The general theory isillustrated via a treatment of the photon wave mechanical picture of the correlated spon-taneous photon cascade emission from a three-level atom. On the basis of the Weisskopf–Wigner theory for photon emission from a two-level atom I first determine the associatedspace-time photon wave function. My treatment extends previous studies by paying partic-ular attention to the spontaneous emission in the atomic rim zone. In this atomic near-fieldzone one finds an interesting interplay between the spatial photon localization problem andthe two-photon entanglement process.

Acknowledgments

On April 1, 2009, I was contacted by Dr. John Navas, senior acquisitions editor (physics)with Taylor & Francis, who invited me to discuss the idea of writing a theoretical book on“the nature of light.” Since for many years the physics of the photon had been a subjectof the greatest importance for me, it did not take me long to accept John’s proposal. Istarted writing the manuscript in December 2010, and thus it has taken me three yearsto accomplish this book project. In particular, I want to acknowledge my former physicsstudent, M.Sc. Dann S. Olesen, for the comprehensive work he has done converting myhandwritten manuscript into a professional LaTeX version. A special thank you goes toNiels Maribo Bache, currently a physics student at Aalborg University, who in the finalstage of the work has helped with the drawing of the figures.

xix

About the author

Ole Keller is professor emeritus of theoretical physics, Aalborg University, Denmark. Heearned his PhD degree in physics from the Danish Technical University in Copenhagen(1972), and the doctor of science degree from the University of Aarhus (1990). He is afellow of the Optical Society of America. In recent years he has worked on theoreticalresearch in fundamental photon physics, near-field quantum electrodynamics, mesoscopicphysical optics, and magnetic monopole photon wave mechanics.

Part I

Classical optics in globalvacuum

1

Heading for photon physics

Notwithstanding that it is possible to consider all electromagnetic fields as intermediaries,transmitting interactions between charged particles, it is fruitful to study the conceptsconnected to free fields, i.e., fields detached from the charges producing or absorbing them.To free electromagnetic fields, also referred to as radiation fields, one may associate manyof the properties we are so familiar with for matter, e.g., energy, momentum and angularmomentum. Even in the framework of classical physics, radiation fields take up a positionalmost on an equal footing with matter. In quantum physics, the autonomy of the radiationfield is fully developed through the emergence of the photon concept. The photon, theelementary excitation of the electromagnetic quantum field, appears as a particle just as“fundamental” as the massive elementary particles attached to other quantum fields.

The photons referring to the quantized free electromagnetic field are called transversephotons because the electric field of the radiation field, from which these photons emerge,is a divergence-free vector field. A divergence-free electric field in direct space (r-space) isperpendicular in the geometrical sense to the wave-vector (q) direction in reciprocal space(q-space), therefore the name transverse photon. Transverse photons are often referred toas physical (or real) photons, because of the (almost) autonomous status of the free field.However, it must be remembered that a physical photon is observable only when it inter-acts with matter (charged massive particles). In the photon-matter interaction photons arecreated or destroyed, so in a sense, one may say that a transverse photon manifests itselfonly during its birth or death process. After all, a real transverse photon is not very realleft alone in free space. Perhaps, the only fingerprint left of a free photon is the fact thata number, the speed of light in vacuum, is attached to its “propagation” from source todetector (delayed interparticle interaction). The words “real photon” thus at best refers tothe circumstance that one can establish an autonomous quantum theory of free electromag-netic fields. However, it must be remembered that a complete decoupling of the dynamicsof charged massive particles and transverse photons is impossible. On top of the discussionabove, we have learned from Niels Bohr that the words “real” and “reality” do not makemuch sense in physics unless they are attached to phenomena (in the Bohr sense) observableby human beings [207]. In the covariant theory of quantum electrodynamics (QED) so-calledlongitudinal and scalar photons are introduced. These types of photons are called virtualbecause they only play a physical role (“exist”) during the time where a given field-matterinteraction process takes place.

The virtual photons, which couple charged massive particles in so-called near-field con-tact, are active only in what I have named the rim (or Lorenz) zone of matter [122]. Therim zone is a vacuum domain in the sense that it is located outside what we refer to as amatter-filled region. In a quantum physical context “outside” means in a region where the(many-body) probability density distribution of the matter particle(s) effectively vanishes.The free transverse photon hence is an object related to those parts of vacuum regions thatdo not include rim zones.

By a certain reinterpretation of the Maxwell equations in free space, these appear asa first-quantized theory of the transverse photon, as we shall see later on. Because of this

3

4 Light—The Physics of the Photon

circumstance, it is important to study selected fundamental aspects of the classical electro-magnetic field theory in free space.

We start our journey into classical electromagnetics (optics) from the Maxwell equationsin free space, and the associated wave equations for the electric and magnetic fields (Sec.2.1). The magnetic field, B(r, t), is a divergence-free (transverse) field everywhere in spaceindependent of whether we are inside or outside of matter. The electric field, E(r, t), onthe other hand, is a transverse field (ET ) only in vacuum outside rim zones. In free space(global vacuum) both E and B are transverse fields. A vector field is called a transversevector field provided it is divergence-free in every space point within its domain of definition.The magnetic field thus is a genuine transverse vector field because its divergence vanishesin all space points. In the presence of field-matter interactions the electric field possessesa rotational-free (longitudinal) part inside matter and in the rim zone. In Sec. 2.2, weshow that up to a physically unimportant constant, a differentiable vector field is uniquelyseparable into divergence-free and rotational-free parts. Such a division for the electric fieldis of utmost importance in studies of field-matter interactions, in this book in particular inrelation to photon physics.

In relativistic wave mechanics wave packets constructed by superposition of plane wavesof positive frequencies relate to photons. Hence, it is important to introduce and discussthe complex analytical signal concept in classical optics. This is done in Sec. 2.3, where it isshown also that the real and imaginary parts of the analytical signal form a Hilbert trans-form pair (also called a conjugate pair). The analytic part of a signal is timely nonlocallyrelated to the signal itself, a fact which is thought-provoking in a photon perspective.

The photon concept most often is introduced starting from an expansion of the transversepart of the electromagnetic into monochromatic plane waves (Sec. 2.4). The individualphoton emerging from such an expansion attains energy E = ~ω and momentum p = ~q,where ω = c|q| and q are the angular frequency and wave-vector of the given (ω,q)-mode.Photons belonging to a selected monochromatic plane-wave mode appear in two differenthelicity eigenstates, which relate to right- and left-hand circular polarized field unit vectors.From these so-called positive- and negative-helicity states alternative sets of orthonormalbase vectors can be constructed. In Sec. 2.5, we analyze the linear transformation connectingdifferent sets of basis vectors and discuss the geometrical picture of the field polarizationstates.

Although it sometimes is claimed in the literature that the photons are synonymous withthe quanta associated to monochromatic plane-wave modes, certainly this need not be thecase. Thus, a wave packet composed of different (ω,q)-modes may represent a single photon.Throughout this book we shall often consider a photon as a wave packet. As shown in Sec.2.6, it is possible to expand a given transverse classical field after a set of orthonormalizedwave-packet modes. Upon quantization, transverse wave-packet photons emerge. The wave-packet modes satisfy a completeness theorem in the subspace of transverse vector fields.

In global vacuum, the energy, the moment of energy, the momentum, and the angularmomentum of the electromagnetic are conserved in time, as emphasized in Sec. 2.7. Theseconservation laws may be derived from Emmy Nother’s theorem [173] which provides uswith a relationship between the symmetry (invariance) properties and conservation laws ofa system [206]. The forms of the integrands appearing in the aforementioned quantities arenot universal. Thus, these forms are valid for observers at rest in the inertial frame in whichthe fields are specified.

By means of the complex Riemann–Silberstein (RS) vectors it is possible to write the setof free Maxwell equations in compact form (Sec. 2.8). The two RS-vectors relate to states inwhich the electromagnetic field is composed of positive- and negative-helicity species. Thedynamical equations for the positive-frequency parts of these vectors lead to photon wavemechanics based on the so-called energy wave function [16].

Heading for photon physics 5

Since the photon necessarily is a relativistic object, it is important to consider the fieldsof classical optics in the perspective of the Special Theory of Relativity (Chapt. 3). Abrief review of the Lorentz transformation and the proper time concept (Sec. 3.2), andthe important four-vector and four-tensor formalism (Sec. 3.3) is given before turning theattention to the free-space electromagnetic field. The set of microscopic Maxwell–Lorentzequations, constituting the foundation of classical electrodynamics, is form-invariant underLorentz transformations. This form-invariance, traditionally called covariance, necessarilyalso holds for the set of free Maxwell equations, and in Sec. 3.4 we rewrite this set in man-ifest covariant form. The virtual scalar and longitudinal photons appear in the wake of thecovariant formalism. The Lorentz transformation of the electric and magnetic fields (Sec.3.5) plays an important role in photon physics. The free-space electric and magnetic fieldsare transverse in all inertial frames, but a Lorentz transformation of the fields shows thatET and B have no independent “existence.” In the rim zone of matter, where the electricfield has both transverse and longitudinal (EL) components, a Lorentz transformation willchange EL in a manner which involves the charge current density of the particle (system).The limitations on the localization of a transverse photon in space is linked to the spatialextension of the rim zone [123], and since this zone essentially is determined by the longitu-dinal field distribution, the spatial photon localization does not appear to be the same fordifferent inertial observers.

2

Fundamentals of free electromagnetic fields

2.1 Maxwell equations and wave equations

Classical electromagnetics is summed up in the Maxwell–Lorentz equations [56, 57, 133],and in the absence of charges the electric and magnetic fields, E(r, t) and B(r, t), satisfythe equations

∇×E(r, t) = − ∂

∂tB(r, t), (2.1)

∇×B(r, t) = c−2 ∂

∂tE(r, t), (2.2)

∇ · E(r, t) = 0, (2.3)

∇ ·B(r, t) = 0, (2.4)

in space (r)-time (t). Eqs. (2.3) and (2.4) specify that both E and B are divergence-free(solenoidal) fields in matter-free regions of space. The magnetic field remains divergence-freein matter-filled domains, and this is so because our present theory is based on the fact thatthere is no experimental evidence for the existence of magnetic charges or monopoles. Sinceelectric charges do exist, the electric field will not be divergence-free in matter-filled regions,and Eq. (2.3) thus must be modified in such regions. Whether a region can be characterizedas matter-filled in the context of classical electromagnetics requires some remarks. In themacroscopic Maxwell theory matter is conceived as a continuum and the characterizationcomplies with this. In the microscopic Maxwell–Lorentz theory all relevant charged particles(electrons, protons, ions) are treated as point-like entities. In consequence matter is presentonly in discrete points, and in these the charge density is infinite. In the covering theory ofclassical electrodynamics, named semiclassical electrodynamics [206], the dynamics of thecharged elementary particles (electrons, etc.) is treated on the basis of quantum mechanics.Although we think of these particles as point-like entities, quantum theory does not allowone to determine (at a given time) a particle’s position precisely. The probabilistic nature ofquantum mechanics in a way leads us back to a continuum view of matter, yet in a quantumstatistical sense to be described later on.

It appears from Eqs. (2.1) and (2.2) that the electric and magnetic fields are coupled,and a transformation of our description from one inertial frame to another shows that Eand B have no independent existence, as we shall realize in Sec. 3.5.

It can be shown from Eqs. (2.1)-(2.4) that the electric and magnetic fields satisfy form-identical wave equations, viz.,

E(r, t) = 0, (2.5)

B(r, t) = 0, (2.6)

where

= ∇2 − c−2 ∂2

∂t2(2.7)

7


is the d’Alembertian operator. The constant c is the speed of light, a universal quantitywhich is the same in all inertial systems. The wave equations in Eqs. (2.5) and (2.6) suggestthe existence of electromagnetic waves that propagate through vacuum domains with speedc, a statement we shall put into the perspective of photon physics in Secs. 2.2-2.4. When weturn to the particle description of electrodynamics it will be seen that all photons propagatewith speed c.

2.2 Transverse and longitudinal vector fields

In the following we take up a topic of utmost importance when we later discuss how photonsare created and destroyed in space-time in their interaction with matter. The subject wetouch upon here also is of relevance for the epistemology related to the photon concept(the photon measurement problem), and for a basic understanding of optical diffraction inregions near matter.

The electric and magnetic fields we deal with in classical electrodynamics always aregenerated by sources occupying a finite domain in space-time. In consequence, these fieldsvanish infinitely far away from their sources. In the Maxwell–Lorentz theory the fields aredifferentiable functions of the space coordinates except at the locations of the point-particles.Here the fields diverge. The fields are differential functions of time. In semiclassical elec-trodynamics, where the inherent probabilistic interpretation of quantum mechanics smearsevery singular behavior, the fields become differentiable everywhere in space-time.

Let W be a generic name for E and B, and let us for brevity omit the reference to thetime from the notation. Starting from the vector function

F(r) =

∫ ∞

−∞

W(r′)4π|r− r′|d

3r′, (2.8)

which is a solution of the vectorial Poisson equation

W(r) = −∇2F(r) = ∇× (∇× F(r)) −∇∇ ·F(r), (2.9)

it is possible to prove that

W(r) = ∇×∫ ∞

−∞

∇′ ×W(r′)4π|r− r′| d3r′ −∇

∫ ∞

−∞

∇′ ·W(r′)4π|r− r′| d

3r′, (2.10)

because W(r) vanishes at infinity. The result in Eq. (2.10) shows that the vector field W(r)is uniquely separable into a divergence-free part,

WT (r) = ∇×∫ ∞

−∞

∇′ ×W(r′)4π|r− r′| d3r′, (2.11)

and a rotational-free part,

WL(r) = −∇∫ ∞

−∞

∇′ ·W(r′)4π|r− r′| d

3r′, . (2.12)

Fundamentals of free electromagnetic fields 9

This property,

W(r) = WT (r) +WL(r), (2.13)

is called Helmholtz’s theorem [163]. A field, WT , which is divergence-free in direct (r) spaceis in reciprocal (q) space perpendicular to q, i.e., q ·WT (q) = 0, and a field, WL, which isrotational-free in r-space is parallel to q in q-space, that is q×WL(q) = 0. This geometricalsignificance in reciprocal space is the reason that we shall use also the names transverse(with subscript T ) and longitudinal (subscript L) for such fields in the remaining part ofthis book.

It is important to emphasize that a field, W(r), by definition, only qualifies as a trans-verse vector field if its divergence vanishes in every space point, i.e.,

∇ ·W(r) = 0, ∀r. (2.14)

In accordance with this, a combination of Eqs. (2.12) and (2.14) gives WL(r) = 0 for all r,and thus W(r) = WT (r). In analogy, a field W(r) is a longitudinal vector field only if

∇×W(r) = 0, ∀r, (2.15)

in agreement with the fact that Eqs. (2.11) and (2.15) leads to WT (r) = 0 for all r, andhence W(r) = WL(r).

Returning now to the free-space Maxwell equations given in (2.1)-(2.4), it appears thatif no charges were present in the universe both the electric and magnetic field would be(genuine) transverse vector fields, i.e.,

E(r, t) = ET (r, t), (2.16)

B(r, t) = BT (r, t). (2.17)

Because of the absence of magnetic monopoles, the magnetic field will still qualify as atransverse vector field in the presence of matter. This circumstance makes it superfluous toadd the subscript T to B. The presence of a charge density distribution, ρ(r, t), changes Eq.(2.3) to ∇·E(r, t) = ρ(r, t)/ǫ0, where ǫ0 is the vacuum permittivity. With W(r, t) = E(r, t),Eqs. (2.11)-(2.13) then give

E(r, t) = ET (r, t)−1

4πǫ0∇∫ ∞

−∞

ρ(r′, t)|r− r′|d

3r′. (2.18)

It appears from Eq. (2.18) that the electric field is not a transverse vector field when acharge density exists in a region of space. The E-field has a longitudinal part EL(r, t)not only inside the charge distribution but also in the vacuum in a usually narrow zonesurrounding matter; see Fig. 2.1. This zone, called the rim zone, is part of the sourcedomain for transverse photons, as we shall understand later on. The rim zone concept playsa central role, e.g., in studies of evanescent fields (Chapt. 20), photon tunneling (Chapt.21), the Cerenkov effect (Chapt. 22), and photon emission from atoms (Chapt. 24).


FIGURE 2.1The black region indicates a domain in space where the quantum mechanical charge densityof a system of (elementary) particles (atoms, molecules, a solid, etc.) is nonvanishing. Al-though transverse (T) photons can be generated by (certain) many-body (or single-body)transitions in the charge system, one cannot claim that such photons for sure are born withinthe charge distribution if one wants to uphold the criterion that T-photons propagate withthe vacuum speed of light everywhere in space outside matter-filled domains. To maintainso-called Einsteinian causality one must admit that a T-photon in a quantum statisticalsense also can be emitted (born) from every point within a larger so-called rim zone ofmatter. Schematically, the rim zone of the black charge density distribution is shown as agrey-toned domain. The fading out of the grey-toning away from the charge density regionis meant to indicate that there is no sharp boundary between the rim zone domain and thesurrounding vacuum.

2.3 Complex analytical signals

In free space relativistic wave equations have two main types of wave packet solutions, viz.,those built from plane waves of positive frequencies, corresponding to particles, and thosebuilt from negative frequencies, relating to antiparticles [88, 209]. For photon physics itis therefore of interest to study the positive-frequency solutions to the free-space Maxwellequations.

Since the real and nonsingular vector field W(r, t)[= E(r, t)orB(r, t)] always has finitesupport in time it may be represented as a Fourier integral

W(r, t) =1

2π

∫ ∞

−∞W(r;ω)e−iωtdω. (2.19)

Below we are interested only in the time dependence ofW, and for brevity, we therefore leaveout the reference to r. Because W(t) is real, the (generally complex) Fourier amplitudes


W(ω) obey the relation

W(−ω) = W∗(ω). (2.20)

It appears from Eq. (2.20) that the negative frequency components (ω < 0) do notcontain any information not already carried by the positive frequency part of the spectrum.In a broader perspective this implies that the photon and antiphoton are identical.

The complex analytical signal [75, 155], denoted by W(+)(r, t)[≡ W(+)(t) below] isobtained from the Fourier integral in Eq. (2.19) by suppressing the negative frequencycomponents:

W(+)(t) =1

2π

∫ ∞

0

W(ω)e−iωtdω. (2.21)

Since W(t) is real, the negative frequency part of Eq. (2.19),

W(−)(t) =1

2π

∫ 0

−∞W(ω)e−iωtdω, (2.22)

is the complex conjugate of the analytical signal, i.e.,

W(−)(t) = (W(+)(t))∗, (2.23)

as the reader may verify by a direct calculation involving use of Eq. (2.20). For what follows,it is useful to write the analytical signal as an integral over all ω:

W(+)(t) =1

2π

∫ ∞

−∞W(+)(ω)e−iωtdω. (2.24)

where

W(+)(ω) =

W(ω) for ω ≥ 0,0 for ω < 0.

(2.25)

The Fourier amplitude W(+)(ω) thus is given by an integral

W(+)(ω) =

∫ ∞

−∞W(+)(t)eiωtdt. (2.26)

which is zero for ω < 0.Let us next, albeit in a not quite rigorous manner, examine the possibility for extending

the definition of the analytic signal given in Eq. (2.24) to complex valued arguments τ =t+ is. The demand that W(+)(ω) is zero for negative values of ω implies that W(+)(τ) isan analytic function in the lower half of the complex τ -plane. To see this, we consider thecontour integral

I(ω) =

∮

C

W(+)(τ)eiωτdτ, (2.27)

and choose as the closed contour C a portion −T < t < T of the real axis plus a semi-circle(of radius T ) in the lower half-plane. It may be deduced that the integral along the semi-circle is zero in the limit T → ∞ [155] for ω < 0. In the limit, the integral along the realaxis is just W(+)(ω). For ω < 0, we thus must have

I(ω;T → ∞) = W(+)(ω) = 0. (2.28)

If we require that W(+)(τ) is analytic in the lower half-plane, Cauchy’s theorem ensuresthat I(ω;T → ∞) is zero for ω < 0.


The analyticity ofW(+)(τ) for s ≤ 0 allows one to make use of Cauchy’s integral formula,and thus obtain

∮

C

W(+)(τ)

τ − τ0dτ = πiW(+)(τ0) (2.29)

in the case where τ0 lies on the boundary curve. The integral must be interpreted as theCauchy principal (P ) value, and the curve C (located in the domain s ≤ 0) is circulatedin the counterclockwise sense. Let us now take for C the same contour as used in relationto Eq. (2.27), and let τ0 = t be a point on the real axis. Since the contribution from thesemi-circle again vanishes in the limit where the radius becomes infinite, we obtain

P

∫ ∞

−∞

W(+)(t′)t′ − t

dt′ = −πiW(+)(t). (2.30)

It appears from this integral identity that the real (R) and imaginary (I) parts of thecomplex analytic signal form a Hilbert transform pair, i.e.,

IW(+)(r, t) =1

πP

∫ ∞

−∞

RW(+)(r, t′)t′ − t

dt′ (2.31)

RW(+)(r, t) = − 1

πP

∫ ∞

−∞

IW(+)(r, t′)t′ − t

dt′ (2.32)

in a notation where the reference to the space coordinate has been reinserted. Utilizing thatW(t) = 2RW(+)(t), it is easy to show that the analytical part of a signal is related to thesignal itself as follows:

W(+)(r, t) =1

2

(

W(r, t) +i

πP

∫ ∞

−∞

W(r, t′)t′ − t

dt′)

. (2.33)

The fact that the relation between W(+) and W is nonlocal in time (yet local in space) isthought-provoking from the perspective of photon physics. Thus, as we shall see (Part III),it is possible in photon wave mechanics to associate the wave function of a transverse photon

to a combination of the complex analytical fields E(+)T (r, t) and B(+)(r, t) (Chapt. 13), or

alternatively, to the positive-frequency part of the transverse vector potential A(+)T (r, t)

(Chapt. 10). For an electromagnetic field (ET ,B) of finite support in time, say from t = 0to t = T0, the associated photon wave function will be nonvanishing also outside the interval(0|T0). It must be remembered, however, that also a transverse antiphoton is associated tothe given field. Together the photon and antiphoton have no net effect outside the (0|T0)-interval.

Since W(r, t) satisfies the wave equation W(r, t) = 0, cf. Eqs. (2.5) and (2.6), itfollows from the Fourier integral representation in Eq. (2.19) that the Fourier amplitudeobeys the Helmholtz equation [∇2 + (ω/c)2]W(r;ω) = 0. An integration of this equationover all positive frequencies shows that the analytical part of the signal satisfies the samewave equation as the signal itself. The analytical parts of the free electric and magneticfields hence obey wave equations of the usual form:

E(+)T (r, t) = 0, (2.34)

B(+)(r, t) = 0. (2.35)

In Sec. 2.9, we shall see that the complex analytical signal also satisfies a certain type ofintegro-differential equation. Because this equation is of first-order in time, it is possible todetermine the values of W(+)(r, t) for all r and t from a knowledge if W(+)(r, t0) at anyparticular time t0. This first-order equation in time helps us to obtain a unified view of thewave mechanics of massless and massive particles.


2.4 Monochromatic plane-wave expansion of the electromagneticfield

A Fourier integral representation of the spatial part of W(r, t), together with Eq. (2.19),lead to the monochromatic plane-wave expansion

W(r, t) = (2π)−4

∫ ∞

−∞W(q, ω)ei(q·r−ωt)d3qdω. (2.36)

Since W(r, t) must satisfy the free-space wave equation W(r, t) = 0, the (angular) fre-quency (ω) and wave number (q = |q|) are connected by the two-branch dispersion relation

ω = ±cq. (2.37)

In the context of wave mechanics ω = +cq(> 0) and ω = −cq(< 0), upon multiplicationby Planck’s constant divided by 2π, give us the energy-momentum relation for plane-wavephotons and antiphotons, respectively, as we shall see later on. The constraints in Eqs.(2.37), imply that the Fourier amplitude W(q, ω) can be written in the form

W(q, ω) = 2π (W(q, cq)δ(ω − cq) +W(q,−cq)δ(ω + cq)) , (2.38)

where δ is the Dirac delta function. A combination of Eqs. (2.36) and (2.38) then splitsW(r, t) into its positive- and negative-frequency parts:

W(r, t) = W(+)(r, t) +W(−)(r, t), (2.39)

where

W(+)(r, t) =

∫ ∞

−∞W(q, cq)ei(q·r−cqt) d3q

(2π)3, (2.40)

and

W(−)(r, t) =

∫ ∞

−∞W(q,−cq)ei(q·r+cqt) d3q

(2π)3. (2.41)

Because (W(−)(r, t))∗ = W(+)(r, t) the Fourier amplitudes satisfy the relation

W(−q,−cq) = W∗(q, cq), (2.42)

as the reader may verify by complex conjugation of Eq. (2.41), followed by a variable changeq ⇒ −q.

By inserting the expansion in Eq. (2.36) into the Maxwell equations in (2.1)-(2.4) itappears that the Fourier amplitudes satisfy the algebraic equations

q×ET (q, ω) = ωB(q, ω), (2.43)

− c2q×B(q, ω) = ωET (q, ω), (2.44)

q · ET (q, ω) = 0, (2.45)

q ·B(q, ω) = 0, (2.46)

where ω = ±cq. Instead of E(q, ω) we have written ET (q, ω) to emphasize that the electricfield in a completely free space is a transverse vector field. In the photon wave mechanical


description, to follow in Part III, the analytical parts of the fields play a prominent role.With the use of the dispersion relation ω = cq(> 0) inserted one obtains from Eqs. (2.43)-(2.46)

κ×ET (q, cq) = cB(q, cq), (2.47)

− cκ×B(q, cq) = ET (q, cq), (2.48)

κ · ET (q, cq) = 0, (2.49)

κ ·B(q, cq) = 0, (2.50)

where κ = q/q is a unit vector in the direction of the wave vector q. The algebraic setof equations satisfied by the negative-frequency [ω = −cq(< 0)] components of the fields[ET (q,−cq),B(q,−cq)] is readily obtained from Eqs. (2.47)-(2.50) utilizing the relation inEq. (2.42).

2.5 Polarization of light

2.5.1 Transformation of base vectors

It appears from Eq. (2.45) that the electric field vector ET (q, ω) always lies in a planeperpendicular to the wave vector q. To characterize the state of the field we resolve thevector ET (q, ω) into two orthogonal components by selecting a pair of generally complexorthonormal base vectors ε1(κ) and ε2(κ) that obey the following conditions:

κ · εs(κ) = 0, s = 1, 2 (2.51)

ε∗s(κ) · εs′(κ) = δss′ , s, s′ = 1, 2 (2.52)

where δss′ is the Kronecker symbol. Thus,

ET (q, ω) =∑

s=1,2

ET,s(q, ω)εs(κ). (2.53)

The projections of ET on the complex conjugates two basis vectors give the field componentsin the chosen basis. i.e.,

ET,s(q, ω) = ε∗s(κ) · ET (q, ω). (2.54)

The conditions in Eqs. (2.51) and (2.52) do not determine the basis vectors uniquely,and this is convenient because in a given application a particular set of basis vectors maybe more useful than the others. Starting from a given set of OLD basis vectors, NEW setscan be constructed via a linear transformation

(

εNEW1

εNEW2

)

= T

(

εOLD1

εOLD2

)

, (2.55)

where

T =

(

a bc d

)

(2.56)

is a 2× 2 transformation matrix. In order that also the new set (εNEW1 , εNEW

2 ) satisfies theconditions in Eq. (2.52), the components of T must be related as follows:

aa∗ + bb∗ = cc∗ + dd∗ = 1, (2.57)

a∗c+ b∗d = 0, (2.58)


as one readily may show. The constraints among the components are obeyed if the trans-formation matrix is unitary, i.e.,

T−1 = T†, (2.59)

where

T−1 = D−1

(

d −b−c a

)

, (2.60)

and

T† =

(

a∗ c∗

b∗ d∗

)

, (2.61)

are the inverse and Hermitian conjugate of T, respectively. The quantity D = ad− bc is thedeterminant of T. Written in terms of the components unitarity is expressed by

a = d∗D, b = −c∗D, c = −b∗D, d = a∗D. (2.62)

The reader may verify to himself that these relations lead to

DD∗ = 1, (2.63)

and the fulfilment of Eqs. (2.57) and (2.58). It follows from Eq. (2.63) that the modulus ofD is equal to one, and one may therefore write the determinant in the form D = exp(iδ),where δ is a real phase parameter. By now, one may express the transformation matrix inthe form

T =

(

a b−b∗eiδ a∗eiδ

)

. (2.64)

Remembering that |a|2 + |b|2 = 1, it appears that T contains four free parameters. One ofthese is δ. With a view to the geometrical analysis of the polarization states of light, givenin Subsec. 2.5.2, it is useful to take

a =(

1 + |∆|2)− 1

2 eiα, (2.65)

b =(

1 + |∆|2)− 1

2 ∆, (2.66)

with ∆ = ∆R+ i∆I . Expressed in terms of the four real free parameters δ, α, ∆R, and ∆I ,the transformation is

T =1

(1 + |∆|2) 1

2

(

eiα ∆

−∆∗eiδ ei(δ−α)

)

. (2.67)

For what follows it is sufficient to employ only three free parameters, and it turns out to beconvenient to make the choice α = 0. Thus, the reduced transformation matrix

T(∆, δ) =1

(1 + |∆|2) 1

2

(

1 ∆−∆∗eiδ eiδ

)

(2.68)

will serve as the starting point for the subsequent study of the various polarization statesof the electric field associated with a given monochromatic plane wave.

2.5.2 Geometrical picture of polarization states

The role of the phase factor exp(iδ) becomes clear if one considers the vectorial product ofthe base vectors. Hence, one obtains from the unitary transformation in Eq. (2.55), withuse of Eq. (2.68),

εNEW1 × εNEW

2 = eiδεOLD1 × εOLD

2 . (2.69)


With the choice δ = 0, the vectorial product ε1(κ)× ε2(κ) therefore is the same for all setsof basis vectors. The vector product always gives a vector parallel or antiparallel to κ. Withthe choice

ε1(κ)× ε2(κ) = κ, δ = 0, (2.70)

the vectors (ε1(κ), ε2(κ),κ) form a right-handed orthonormal set [in a generalized sensesince ε1(κ) and ε2(κ) may be complex]. Below, we shall keep the phase factor in the analysis.

Classical electrodynamics is a deterministic theory. This implies that the end point ofthe electric field vector, at a fixed point in space, with increasing time describes a smoothcurve. In general, the form of this curve is extremely complicated, and the curves are verydifferent at the various points in space. For a monochromatic field, the curve is never morecomplicated than what results from a linear superposition of two ellipses. Below, we shallprove this assertion for a plane-wave field. The generalization of the proof from plane-wavefields to more complicated monochromatic fields is easy.

To examine the polarization state of the (transverse) electric field belonging to a givenplane wave, one must analyze the expression

R[

ET (q, ω)ei(q·r−ωt)

]

=∑

s=1,2

R[

ET,s(q, ω)εs(κ)ei(q·r−ωt)

]

. (2.71)

If one writes the complex amplitude ET,s(q, ω) in the polar form ET,s(q, ω) =|ET,s(q, ω)| exp[iφs(q, ω)], one obtains

R[


]

=∑

s=1,2

|ET,s(q, ω)|R[

εs(κ)ei(q·r−ωt+φs(q,ω))

]

, (2.72)

a form which is convenient for the subsequent analysis. Let us assume now that the old basisvectors are real (superscript R): (εOLD

1 , εOLD2 ) = (εR1 , ε

R2 ). From Eqs. (2.55) and (2.68), we

then find that the new basis vectors (εNEW1 , εNEW

2 ) = (ε1, ε2) are given by

εs(κ) = (ps + iqs) exp(iδδs2), s = 1, 2, (2.73)

where

p1 = K(εR1 +∆RεR2 ), (2.74)

q1 = K∆IεR2 , (2.75)

and

p2 = K(−∆RεR1 + εR2 ), (2.76)

q2 = K∆IεR1 , (2.77)

with K = (1 + |∆|2)−1/2. The four vectors (ps,qs)[s = 1, 2] are real, and their geometricalsignificance will soon become clear. With the abbreviation

Φs = q · r+ φs(q, ω) + δδs2, (2.78)

we obtain by combining Eqs. (2.72) and (2.73)

R[


]

=∑

s=1,2

|ET,s| [ps cos(ωt− Φs) + qs sin(ωt− Φs)] . (2.79)

As a function of time the expression in the bracket describes (in general) for the given s anellipse, and ps and qs are a pair of so-called conjugate semi-diameters for the ellipse. Thestate of polarization of the field associated with the monochromatic plane wave (q, ω), thus


appears as a superposition of two elliptical polarization states with weights |ET,s|, s = 1, 2.Since

p1 × q1 · κ = −p2 × q2 · κ = K2∆I , (2.80)

the end points of the two vectors describing the ellipses traverse these in opposite directions.For ∆I > 0, we say that the polarization is right-handed for s = 1, and left-handed for s = 2(see Fig. 2.2). For ∆I < 0, the classification is opposite. The corresponding conjugate semi-diameters for the two ellipses are orthogonal, i.e., p1 · p2 = q1 · q2 = 0. When ∆I = 0,

e

D >

R

q

p

x 1

2

eR

p

q

1

1

2

2

I0

e

D >

R

q

p

1

2

eR

p

q

1

1

2

2

I0

FIGURE 2.2Sketch of the vector sets (ps,qs), s = 1, 2, related to a general transformation from anOLD set of real and orthonormalized polarization basis vectors, (εR1 , ε

R2 ), to a NEW set of

generally complex polarization vectors [see Eqs. (2.73)–(2.77)]. For ∆I > 0, the polarizationis right-handed (with respect to the wave-vector direction κ) for s = 1, and left-handed fors = 2. For ∆I < 0, the classification is the opposite. The real quantity ∆I is one of threeparameters characterizing the employed transformation matrix T (Eq. (2.68)).


we have q1 = q2 = 0, and the two fundamental states now are linearly polarized. Inrelation to photon physics, the states characterized by ∆R = 0 and ∆I = +1 [and henceK = 1/

√2] are of special importance. Since, now p1 = q2 = εR1 /

√2 and p2 = q1 = εR2 /

√2,

the fundamental states are circularly polarized: right-handed for s = 1 and left-handedfor s = 2. If one makes the choice ∆I = −1, instead, the two circles are traversed in theopposite sense as before.

The reader may convince herself that an expression of the form R [(p+ iq) exp(iωt)]describes an ellipse: set p+ iq = (a+ ib) exp(iη), and choose the real phase parameter η sothat the real vectors a and b become mutually orthogonal. With tan(2η) = 2p ·q/(p2− q2)we find a ⊥ b. The connection R [(p+ iq) exp(−iωt)] = a cos(ωt − η) + b sin(ωt − η) inturn evidently shows that the original expression represents an ellipse, with semi-axes |a|and |b|.

2.6 Wave packets as field modes

In Sec. 2.4, we made an expansion of the electromagnetic field in monochromatic plane-wavemodes, and in Sec. 2.5 an analysis of the polarization states associated with the individualmodes was undertaken. Notwithstanding the extreme importance of the monochromaticplane-wave expansion in both classical and quantum optics, not least for technical mathe-matical reasons, it is from a conceptual point of view interesting to investigate the possibilityfor expanding the classical free field in wave-packet modes. If we follow Einstein’s originalidea [60] that light might consist of quanta of energy with a point-like structure, it is natu-ral, if one starts from classical optics, to seek to localize the electromagnetic field in narrowwave packets in space-time.

Without loss of generality, let us focus the attention on the positive-frequency part of the

transverse electric field, i.e., E(+)T (r, t). In Sec. 2.4 the field was expanded over infinite space.

In the following we replace this continuous mode representation by an expansion over a finitecubic volume, V = L3. Here, this is done for mathematical simplicity but we note that theexpansion over a finite volume, not necessarily cubic, is of physical importance in studiesof for instance atom-field interaction in cavities [211, 158] and field propagation in confinedstructures [225, 101, 91]. In going from infinite-space to finite-space mode expansion, Fourierintegrals involving a continuum of wave vectors q, are replaced by Fourier series (discretesums) with discrete wave vectors qα, only. In expressions involving bilinear combinationsof fields (e.g., field energy, momentum, and angular momentum) the correspondence is asfollows:

∫ ∞

−∞(· · · ) d3q

(2π)3⇔ 1

L3

∑

qα

(· · · ). (2.81)

If, for brevity , we let the index i (or j) stand for the combination (qα, sβ) of wave vectorand polarization (β = 1, 2) indices, the correspondence for the positive-frequency electricfield takes the form

E(+)T (r, t) =

∑

s

∫ ∞

−∞ET,s(q, cq)εs(κ)e

i(q·r−cqt) d3q

(2π)3

⇔ 1

L3

2

∑

i

ET,iεiei(qi·r−ωit), (2.82)


with ωi = c|qi|. The discrete (ET,i) and continuous (ET,s(q, cq)) amplitudes do not havethe same dimension, but the precise relation between them is not needed in the treatmentbelow.

Let us thus consider wave packets formed by linear combination of monochromatic plane-wave modes, viz.,

w(+)m (r, t) =

1

L3

2

∑

j

tmjεjei(qj ·r−ωjt), (2.83)

where the coefficients tmj are elements of a certain type of square matrix, as we shall seesoon. To each m belongs a wave-packet mode composed of only positive-frequency plane

waves. With an appropriate t-matrix, the general field E(+)T (r, t) can be resolved in terms

of a complete orthonormal set of w(+)m (r, t)-modes. Utilizing Eq. (2.83) one obtains

∫

V

[w(+)m (r, t)]∗ ·w(+)

n (r, t)d3r

=∑

i,j

t∗mitnjε∗i · εjei(ωi−ωj)t

[

1

L3

∫

V

ei(qj−qi)·rd3r

]

=∑

i,j

t∗mitnjε∗i · εjδqiqj

. (2.84)

The presence of the Kronecker delta δqiqjimplies that the base vectors εi and εj belong

to the same wave-vector direction. Use of the orthonormality condition in Eq. (2.52) thenleads to the result

∫

V

[w(+)m (r, t)]∗ ·w(+)

n (r, t)d3r = δmn (2.85)

provided the t-matrix satisfies the condition

∑

i

t∗mitni = δmn. (2.86)

If Eq. (2.85) is obeyed the wave-packet modes are orthonormalized. The condition in Eq.(2.86) will be met if t is a unitary matrix:

tt† = U = t†t, (2.87)

or in component form

∑

m

timt∗jm = δij =

∑

m

t∗mitmj , (2.88)

since per definition t†mj = t∗jm (t†im = t∗mi). [The first member of Eq. (2.88) leads to Eq.(2.86) upon the following renaming of indices: i→ n, j → m, m → i.] The second memberof Eq. (2.88) allows one to rewrite the expression for the positive-frequency electric field inthe form

E(+)T (r, t) = L− 3

2

∑

i,j

ET,iδijεjei(qj ·r−ωjt)

= L− 3

2

∑

i,j,m

ET,it∗mitmjεje

i(qj ·r−ωjt). (2.89)


By means of Eq. (2.83) one finally obtains

E(+)T (r, t) =

∑

m

ET,mw(+)m (r, t), (2.90)

where

ET,m =∑

i

ET,it∗mi. (2.91)

Eq. (2.90) shows that an arbitrary transverse electric field, E(+)T (r, t), can be expanded

after an orthonormalized set of wave-packet modes, w(+)m (r, t), the expansion coefficient

being ET,m [Eq. (2.91)] for the mth mode. Such an expansion is important in photon physics,because it enables one to associate a photon concept also with non-monochromatic localizedfield distributions.

The wave-packet modes satisfy a completeness (closure) theorem in the subspace of

transverse vector fields. To prove this, one inserts the plane-wave expansion of w(+)m [Eq.

(2.83)] in the relevant sum of dyadic products. Hence,

∑

m

[w(+)m (r′, t)]∗wn(r, t) =

1

L3

∑

m,i,j

t∗mitmjε∗i εje

i(qj·r−ωjt)e−i(qi·r′−ωit). (2.92)

The sum over m gives δij , since t is unitary ([Eq. (2.88)], last member), and the presenceof the Kronecker delta immediately reduces the remaining double summation to a singlesummation. Therefore,

∑

m

[w(+)m (r′, t)]∗w(+)

m (r, t) =1

L3

∑

i

ε∗i εieiqi·(r−r′)

⇔∫ ∞

−∞

[

∑

s

ε∗s(κ)εs(κ)

]

eiq·(r−r′) d3q

(2π)3. (2.93)

If one multiplies Eq. (2.52) (with the factors in the scalar product interchanged) by εs(κ),and thereafter makes a summation over s, one obtains

εs′(κ) ·[

∑

s

ε∗s(κ)εs(κ)

]

=∑

s

δss′εs(κ) = εs′(κ). (2.94)

The sum in the square bracket hence must be the 2 × 2 unit tensor in the subspace of thetransverse vector fields. If U denotes the unit tensor (3 × 3) in the full vector field space,we have the dyadic relation

∑

s

ε∗s(κ)εs(κ) = U− κκ. (2.95)

When Eq. (2.95) is inserted into the integral expression in Eq. (2.93), one obtains a com-pleteness theorem of the form

∑

m

[w(+)m (r′, t)]∗w(+)

m (r, t) = δT (r− r′), (2.96)

where δT (r−r′) is the transverse delta function, a dyadic quantity, which in the descriptioninvolving a continuum of wave vectors is given by

δT (r− r′) =∫ ∞

−∞(U − κκ)eiq·(r−r′) d3q

(2π)3. (2.97)


The transverse delta function is not zero for r 6= r′, but decays as |r− r′|−3. The fact thatthe left-hand side of Eq. (2.96) is appreciably different from zero for space points with so-called near-field separation has important consequences for the spatial localization problemfor transverse photons, as we shall see in later chapters. Since δT (r − r′) formally is theFourier transform of a function, U−κκ, which does not tend to zero for |q| → ∞, δT (r−r′)has a singularity at r′ = r which one must regularize by a procedure relating in the propermanner to the physics over short distances.

The photon localization problem relates to the impossibility of creating an electromag-netic field which is different from zero only in a single space point (at a given time). Afield with only delta function support is physically untenable for several reasons, as we shalldiscuss later on. From a mathematical point of view, all purely transverse electric fields,E(r, t) = ET (r, t), ∀r, must obey the identity

ET (r, t) = ∇×∫ ∞

−∞

∇′ ×ET (r′, t)

4π|r− r′| d3r′, (2.98)

cf. the Helmholtz theorem (in particular Eq. (2.11)). The reader may readily convinceherself that a postulated field of the form ET (r, t) = A(t)δ(r) cannot satisfy Eq. (2.98).Furthermore, such a field is not a genuine transverse vector field because its divergence doesnot vanish at r = 0 (the subscript T on the postulated field thus is misleading!).

2.7 Conservation of energy, moment of energy, momentum, andangular momentum

For a free electromagnetic field the total energy

HT =ǫ02

∫ ∞

−∞

[

ET (r, t) · ET (r, t) + c2B(r, t) ·B(r, t)]

d3r, (2.99)

the total moment of energy

KT =ǫ02

∫ ∞

−∞r[

ET (r, t) ·ET (r, t) + c2B(r, t) ·B(r, t)]

d3r, (2.100)

the total momentum

PT = ǫ0

∫ ∞

−∞ET (r, t)×B(r, t)d3r, (2.101)

and the total angular momentum

JT = ǫ0

∫ ∞

−∞r× [ET (r, t)×B(r, t)] d3r, (2.102)

all are constants, that is, independent of time. In order to prove that

dHT

dt= 0,

dPT

dt=

dJT

dt=

dKT

dt= 0, (2.103)

one first differentiates the various integrands of Eq. (2.99)-(2.102) with respect to t, andthereafter the time derivatives ∂ET (r, t)/∂t and ∂B(r, t)/∂t are eliminated by means of


the Maxwell equations given in Eqs. (2.1) and (2.2). Finally, certain integrals, viz., thosecontaining integrands of the types ∇ · f and ∇f , are transformed into surface integrals.Since all free electromagnetic fields are generated (emitted) by sources with finite supportin space-time, the detached fields are for all finite times contained in a certain (finite)volume of space. If the surfaces mentioned above are placed outside this volume, the surfaceintegrals vanish, and the results in Eqs. (2.103) follow.

In relativistic quantum theory, the ten generators of the Poincare group are identifiedwith the operators HT , KT , PT , and JT associated with the classical quantities in Eq.(2.99)-(2.102). These operators generate infinitesimal time (HT )- and space (PT )- transla-

tions, rotations (JT ), and boosts (KT ). Physical states are labelled by the eigenvalues ofthose operators which are conserved, i.e., that commute with the energy operator. Sincethe commutator [KT , HT ] 6= 0, the eigenvalues of the boost operator are not used to labela free photon state [209, 242].

The quantities WT (r, t) ≡ (ǫ0/2)[E2T (r, t) + c2B2(r, t)] and c2PT (r, t) = ST (r, t) ≡

µ−10 ET (r, t) × B(r, t) are known as the electromagnetic energy density and the Poynting

vector (or energy flux density). It may be shown, that these designations are meaningfulonly relative to an observer at rest in the frame in which the fields are specified [206, 101].

2.8 Riemann–Silberstein formalism

The information contained in the set of free-space Maxwell equations, given in Eqs. (2.1)-(2.4), can be written in compact form by introduction of the complex so-called Riemann–Silberstein vector

F+(r, t) =

√

ǫ02(ET (r, t) + icB(r, t)) . (2.104)

Thus, by multiplying Eq. (2.2) by ic, and adding hereafter the resulting equation and Eq.(2.1), it appears that F+(r, t) satisfies the dynamical equation

i

c

∂

∂tF+(r, t) = ∇× F+(r, t). (2.105)

By taking the divergence of this equation, and interchanging the ∂/∂t- and ∇-operators itfollows that ∇ · F+(r, t) must equal a time independent constant, which possibly may bespace dependent. However, the combination of the Maxwell Eqs. (2.3) and (2.4) shows thatF+(r, t) is a divergence-free complex vector field, i.e.,

∇ · F+(r, t) = 0. (2.106)

Together, Eqs. (2.105) and (2.106) represent a complex version of the free-space Maxwellequations. Applications in the framework of classical electrodynamics of this so-calledRiemann–Silberstein version, which seems to date back to Riemann [240], were given ahundred years ago first by Silberstein [219, 220, 221], and shortly afterward by Bateman[11]. In the presence of charges, the electric field E(r, t) is no longer a transverse vector field,and in the original Riemann–Silberstein description of classical electrodynamics E(r, t) en-ters the definition of F+(r, t) instead of ET .

In recent time F+(r, t), and its complex conjugate

F−(r, t) =

√

ǫ02(ET (r, t)− icB(r, t)) , (2.107)


have played an important role in photon wave mechanics, because the positive-frequencyparts of these vectors relate to the so-called energy wave function of the photon [see Chapt.13]. Because only the transverse field dynamics is quantized in this approach it is useful hereto use ET (r, t) in the definition of F+(r, t) and F−(r, t). It is easy to show that F−(r, t)satisfy the dynamical equation

i

c

∂

∂tF−(r, t) = −∇× F−(r, t), (2.108)

and, of course

∇ ·F−(r, t) = 0. (2.109)

It is convenient for later use in photon wave mechanics and quantum optics, to write theequations for F+(r, t) and F−(r, t) in the compact notation

i∂

∂tF±(r, t) = ±c∇× F±(r, t), (2.110)

∇ ·F±(r, t) = 0. (2.111)

It appears from Eq. (2.40) that the positive-frequency parts of the Riemann–Silbersteinvectors have the plane-wave expansions

F(+)± (r, t) =

∫ ∞

−∞F±(q, cq)e

i(q·r−cqt) d3q

(2π)3, (2.112)

and by combining Eqs. (2.110) and (2.112) it follows that the Fourier amplitudes mustsatisfy the algebraic equations

F±(q, cq) = ±iκ× F±(q, cq). (2.113)

To determine the polarization states of the Riemann–Silberstein vectors one writes these inthe form

F±(q, cq) = F±(q, cq)e±(κ), (2.114)

where e±(κ) = p±+ iq± are the relevant unit polarization vectors. Since these vectors obeythe relations

e±(κ) = ±iκ× e±(κ), (2.115)

their real (p±) and imaginary (q±) parts must be connected by

q± = ±κ× p±, (2.116)

and |p±| = |q±| = 1/√2. From the triple products

p± × q± · κ = ±1

2(2.117)

it then appears that the Riemann–Silberstein F+(q, cq) and F−(q, cq) relate to states whichare right (e+)- and left (e−)-hand circular polarized, respectively. Expressed in terms of aset of real basis vectors, and with a phase parameter δ included, one has

e+(κ) =1√2

(

εR1 (κ) + iεR2 (κ))

, (2.118)

e−(κ) =eiδ√2

(

−εR1 (κ) + iεR2 (κ))

. (2.119)


In photon wave mechanics, F(+)+ (r, t) and F

(+)− (r, t) may be used to describe single-photon

wave packets composed of positive (+) and negative (−) helicity species. The helicity con-cept will be introduced in Sec. 10.3 and employed throughout this book.

Since the two Riemann–Silberstein vectors F+ and F−, given in Eqs. (2.104) and (2.107),and now divided into their positive- and negative-frequency parts, i.e.,

F±(r, t) = F(+)± (r, t) + F

(−)± (r, t), (2.120)

are each other’s complex conjugate, the following relations must be satisfied:

F(−)± (r, t) =

(

F(+)∓ (r, t)

)∗. (2.121)

In Sec. 2.7, expressions were given for the energy, moment of energy, momentum, andangular momentum of the transverse electromagnetic field. The reader may prove to himselfthat these conserved quantities can be given in forms of integrals involving only one of theRiemann–Silberstein vectors. Thus, for instance

HT =

∫ ∞

−∞F∗

+(r, t) ·F+(r, t)d3r, (2.122)

KT =

∫ ∞

−∞rF∗

+(r, t) ·F+(r, t)d3r, (2.123)

PT =1

ic

∫ ∞

−∞F∗

+(r, t)× F+(r, t)d3r, (2.124)

and

JT =1

ic

∫ ∞

−∞r×

(

F∗+(r, t) × F+(r, t)

)

d3r. (2.125)

It turns out that these classical field quantities lead directly to proper quantum mechanicalmean values of the photon energy wave function [16, 123]; see also Secs. 13.3-13.5.

2.9 Propagation of analytical signal

If a vector field W(r, t) satisfies the free-space wave equation W(r, t) = 0, we have seenthat the analytical signal W(+)(r, t) may be given in terms of the integral representation inEq. (2.40). What kind of propagation equation does the analytical signal satisfy? To answerthis question, it is useful first to operate with −∇2 on Eq. (2.40). This gives one the formula

−∇2W(r, t) =

∫ ∞

−∞q2W(q, cq)ei(q·r−cqt) d3q

(2π)3. (2.126)

This suggests that we use the symbolic notation√−∇2W(+)(r, t) to denote the integral

that appears if q2 is replaced by q on the right-hand side of Eq. (2.126):

√

−∇2W(+)(r, t) ≡∫ ∞

−∞qW(q, cq)ei(q·r−cqt) d3q

(2π)3. (2.127)


Now, if we differentiate Eq. (2.40) with respect to time and use the definition in Eq. (2.127),we obtain the following equation of propagation for the complex analytical signal:

i∂

∂tW(+)(r, t) = c

√

−∇2W(+)(r, t). (2.128)

The equation for W(+)(r, t) is of first order in time whereas the one for W(r, t) is ofsecond order. The price paid for this simplification appears when the reader realizes thatEq. (2.128) is nonlocal in space. To reach this conclusion one notes that the right-hand sideof Eq. (2.127) formally is the Fourier-integral representation of a folding integral of twofunctions which Fourier amplitudes are q and W(q, cq) exp(−icqt). The propagation of theanalytical signal thus is governed by the integro-differential equation

i∂

∂tW(+)(r, t) =

∫ ∞

−∞h(r− r′)W(+)(r′, t)d3r′, (2.129)

with the singular kernel

h(R) =

∫ ∞

−∞cqeiq·R

d3q

(2π)3. (2.130)

If one multiplies Eq. (2.129) by ~ (Planck’s constant divided by 2π), the resulting equationappears like a quantum mechanical wave equation written in Hamiltonian form and witha Hamilton operator density ~h(R). In q-space, the Hamilton operator then is c~q, i.e.,precisely that of a plane-wave photon. An equation of the form given in Eq. (2.128) wasoriginally established by Landau and Peierls [141] in their search for a wave mechanicalequation for the photon [126]. Much later, an equivalent equation was rederived by Su-darshan [226] in connection to his studies of the propagation of second-order correlationfunctions [155].

Because Eq. (2.128) is of first order in time, it is possible to determine W(+)(r, t) inall space-time points from a knowledge of W(+)(r′, t′) in the entire space at any particularinstance of time (t′). In order to express this statement explicitly, we take the Fourier inverseof Eq. (2.40). This gives

W(q, cq) = eicqt′

∫ ∞

−∞W(+)(r′, t′)e−iq·r′d3r′, (2.131)

and when this expression for W(q, cq) is reinserted in Eq. (2.40), we obtain

W(+)(r, t) =

∫ ∞

−∞K(r− r′, t− t′)W(+)(r′, t′)d3r′, (2.132)

where

K(R, τ) =

∫ ∞

−∞ei(q·R−cqτ) d3q

(2π)3. (2.133)

The kernel (scalar Green function) K(r− r′, t− t′) thus describes how the analytical signalevolves in space and time.

In relation to free-space electromagnetic fields one cannot assume that the evolution“starts” (at t = t′) from a completely localized distribution, i.e., a distribution with delta

function support, because the vector field is transverse W(+) = W(+)T (= E

(+)T or B(+)).

The reader may remember that this circumstance was briefly discussed in Sec. 2.6 [see Eq.(2.98) and the text below this equation].


Although the integral in Eq. (2.133) can be expressed in a nice manner in terms ofthe positive [δ(+)(R− cτ)] and negative [δ(−)(R+ cτ)] wave-number parts of the associatedDirac delta functions, it is perhaps more useful for the reader to calculate here the frequencyFourier transform of the kernel, viz.,

K(R;ω) =

∫ ∞

−∞K(R, τ)eiωτdτ. (2.134)

By inserting the expression given in Eq. (2.133) for K(R, τ) into Eq.(2.134), the integrationover τ gives 2πδ(ω − cq), and the remaining integral over q-space, i.e.,

K(R;ω) =

∫ ∞

−∞δ(ω − cq)eiq·R

d3q

(2π)3, (2.135)

is easily done in spherical coordinates (place the polar axis in theR-direction, for simplicity).With the abbreviation ω/c = q0 one obtains

K(|R|;ω) = q20πc

sin(q0R)

q0R, ω > 0. (2.136)

For ω < 0, the integral in Eq. (2.135) is zero, in agreement of course with the fact that ouranalysis deals with analytic signals. As expressed explicitly in Eq. (2.136), the kernel K isa function only of the magnitude of R, a fact which is obvious already from the integralform in Eq. (2.133).

3

Optics in the special theory of relativity

In this chapter a brief account is given of those aspects of the Special Theory of Relativitywhich are of particular relevance for photon physics. For a comprehensive treatment ofSpecial Relativity the reader may consult, e.g. [161, 229, 185, 204]; see also [101].

3.1 Lorentz transformations and proper time

The Principle of Special Relativity states that the laws of nature are invariant under aparticular group of space-time coordinate transformations [61, 62, 148], the so-called Lorentztransformations. A Lorentz transformation is a linear transformation from one system ofspace-time coordinates xα to another system x′α, i.e.,

x′α = Λαβx

β + aα. (3.1)

In the following we use a notation where Greek letters α, β, γ, . . ., always run over fourvalues 0, 1, 2, 3. For the contravariant space-time position four-vector this means that

x ≡ xα = (x0, x1, x2, x3) = (x0,x) = (ct,x), (3.2)

where x = xi = (x1, x2, x3) is the space position vector, and, as indicated, x0 the timemultiplied by the vacuum speed of light. Wherever needed, the Cartesian components ofusual three-vectors are denoted by Latin letters, i, j, k, . . .. The repeated index β in Eq.(3.1) implies a summation over β = 0 − 3, and henceforth this summation convention forrepeated indices (one as subscript, the other as superscript) is employed, unless otherwisestated. The fundamental property that characterizes Lorentz transformations is that theseleave invariant the proper time differential dτ , defined by

c2dτ2 ≡ c2dt2 − dx · dx = −ηαβdxαdxβ , (3.3)

where ηαβ are the αβ-element of the metric tensor η for the flat space-time of specialrelativity. This tensor is diagonal and its nonvanishing elements are given by

η00 = −1, η11 = η22 = η33 = +1. (3.4)

The invariance of the proper time (dτ ′ = dτ) restricts the elements of the matrix Λ by theconditions

ηαβΛαγΛ

βδ = ηγδ. (3.5)

It appears from Eq. (3.1) that the coordinate differentials in the two systems are related by

dx′α = Λαβdx

β , (3.6)

27


and with the help of this relation Eq. (3.5) can be proved. Thus,

c2dτ ′2 = −ηαβdx′αdx′β = −ηαβΛαγΛ

βδdx

γdxδ = −ηγδdxγdxδ = c2dτ2, (3.7)

provided the conditions in Eq. (3.5) are satisfied.Because of the minus sign in the second member of Eq. (3.3) the geometry specified

for the four-dimensional space is non-Euclidean. It is called pseudo-Euclidean space orMinkowski space. Neighboring points x and x + dx are said to be timelike separated ifdτ2 > 0. Particles with nonzero rest mass always move along timelike world lines in classicalphysics. When dτ = 0, the points are said to be null separated or lightlike separated. Thelast term associates to the fact that a light wave front (or a photon) will have |dx/dt| = c,and thus dτ = 0. Other zero rest mass particles also must move along null separated worldlines. When dτ < 0, the points are said to be spacelike separated. The hypothetical entitiescalled tachyons would always move with a speed larger than the speed of light and thus havespacelike world lines. Events that are spacelike separated do play a role when we go beyondclassical physics to relativistic quantum mechanics for massive particles, but only a lengthscale comparable to or smaller than the Compton wavelength λC = h/(mc) [h is Planck’sconstant, and m is the particle’s rest mass]. In photon physics, spacelike separations enterthe discussion of the photon localization problem, as we shall discuss later on (Parts V andVII).

Provided x′(x) and x(x′) are differentiable functions, so that the matrix ∂x′α/∂xβand its well-defined inverse ∂xβ/∂x′α are nonsingular, the Lorentz transformation in Eq.(3.1) is the only coordinate transformation that leaves dτ2 invariant. In order to show thiswe use the differential relation

dx′α =∂x′α

∂xβdxβ (3.8)

to relate dτ ′2 to dτ2. Thus,

c2dτ ′2 = −ηαβdx′αdx′β = −ηαβ∂x′α

∂xγ∂x′β

∂xδdxγdxδ. (3.9)

The demand dτ ′2 = dτ2 then gives

ηγδ = ηαβ∂x′α

∂xγ∂x′β

∂xδ, (3.10)

and upon differentiation with respect to xε

0 = ηαβ

[

∂2x′α

∂xγ∂xε∂x′β

∂xδ+∂x′α

∂xγ∂2x′β

∂xδ∂xε

]

. (3.11)

By adding to this equation the same with γ and ε interchanged, and subtract the same withε and δ interchanged, one obtains

0 = 2ηαβ∂2x′α

∂xγ∂xε∂x′β

∂xδ, (3.12)

and because ηαβ and ∂x′β/∂xδ are nonsingular matrices we can conclude that

∂2x′α

∂xγ∂xε= 0. (3.13)

The general solution to Eq. (3.13) is precisely a linear function of the type given in Eq.

Optics in the special theory of relativity 29

(3.1). By inserting Eq. (3.1) into Eq. (3.10), it becomes clear that the elements of Λ satisfythe condition in Eq. (3.5).

The 16 real elements (parameters) Λαβ are restricted by the 10 independent conditions

in Eq. (3.5), and this leaves us with a (6 + 4)-parameter family of transformations. Of thesix parameters in Λα

β three, Λij ≡ Λij = −Λji, describe space rotations. The remaining

three linearly independent parameters can be given by Λi0 (or Λ0

i) and they specify therelative velocity of the two inertial frames. The four parameters in aα describe relativetranslations of the coordinate systems in space, ai, and time, a0. The set of all Lorentztransformations of the form in Eq. (3.1) form a group, the so-called Poincare group (orthe inhomogeneous Lorentz group). The subset with aα = 0 is called the homogeneousLorentz group. The difference between the Galileo group and the Poincare group arises onlyin those transformations (called boosts) that change the velocity of the frame.

Let us now take a brief look at the homogeneous Lorentz transformation. Suppose thatone observer O sees a particle at rest (dx = 0), and a second observer O′ sees it movingwith velocity v. From Eq. (3.1) we obtain dx′i = cΛi

0dt and cdt′ = cΛ0

0dt, and hence

vi =dx′i

dt′= c

Λi0

Λ00

. (3.14)

If one sets γ = δ = 0 in Eq. (3.5) we have second relation between Λi0 and Λ0

0, viz.,

3∑

i=1

(Λi0)

2 − (Λ00)

2 = −1. (3.15)

By combining Eqs. (3.14) and (3.15) the solution for the Λα0 ’s, given in terms of the velocity

v, is readily obtained. Thus,

Λi0 =

vicγ, Λ0

0 = γ, (3.16)

where

γ =

[

1−(v

c

)2]− 1

2

. (3.17)

The remaining Λij ’s are not uniquely determined. A convenient choice, which corresponds

to the case where the axes of the two frames are parallel, is

Λij = δij +

vivjv2

(γ − 1), Λ0j =

vjcγ. (3.18)

The most general homogeneous Lorentz transformation (without inversion) is obtained byrotating the axes of only one of the two frames, say the primed frame, S′. If we denote therotation matrix by R, we have

t′ = γ(

t+v · xc2

)

(3.19)

R · x′ = x+ γvt+ (γ − 1)vv · xv2

= γ (x+ vt) + (γ − 1)v × (v × x)

v2. (3.20)

In the nonrelativistic limit, v/c→ 0, Eqs. (3.19) and (3.20) reduce to a homogeneous Galileotransformation, here written in the form R · x′ = x+ vt and t′ = t.


3.2 Tensors

Before we go on to study the free electromagnetic field in a relativistic perspective, it isconvenient to outline a tensorial notation that makes the Lorentz transformation proper-ties of physical quantities transparent in a broad context. We begin by summarizing someelements of tensor analysis in a vector space which might be non-Euclidean. Although thefirst part of the analysis holds for quite general vector spaces, it is useful here to think ofthe space-time continuum defined in terms of a four-dimensional space with (Cartesian)coordinates xα (α = 0, 1, 2, 3). Suppose that a set of new coordinates x′α is determinedby a nonsingular transformation

x′α = x′α(x0, x1, x2, x3), α = 0− 3. (3.21)

For the moment the transformation law is not specified, but soon we shall identify it withthe Lorentz transformation.

Tensors (of all ranks) are associated with each space-time point x, and they are definedby their transformation properties under the transformation x→ x′. A tensor of rank zero,also called a scalar, is a single quantity whose value is not changed by the transformation. Atensor of rank one, which also is called a vector, has four components, and two kinds mustbe distinguished, The elements V α of a contravariant vector V α are denoted by upperindices (superscripts), and they transform according to the rule

V α → V ′α =∂x′α

∂xβV β . (3.22)

A covariant vector Wα has elements Wα, denoted by lower indices (subscripts), thattransform as

Wα →W ′α =

∂xβ

∂x′αWβ . (3.23)

In the literature on general relativity contravariant vectors are often just called vectors,and covariant vectors are referred to as dual vectors. Associated with the vector space onedefines a dual vector space as the space of all linear maps from the original vector space tothe real numbers. As we shall learn in Sec. 3.4, the word covariant is used also in a quitedifferent context in electrodynamics. The inner or scalar product of two vectors is definedas the following product of a covariant and contravariant vector:

WV (= W · V ) ≡WαVα = V αWα (3.24)

The scalar product is an invariant (scalar) since

W ′V ′ =∂xβ

∂x′α∂x′α

∂xγWβV

γ =∂xβ

∂xγWβV

γ = δβγWβVγ =WV. (3.25)

Above, δβγ is the Kronecker symbol. The differential elements dxα form a contravariantvector since

dx′α =∂x′α

∂xβdxβ , (3.26)

and the rule of implicit differentiation, viz.,

∂

∂x′α=∂xβ

∂x′α∂

∂xβ, (3.27)


shows that the gradient operator ∂/∂xα is a covariant vector. The invariance of thescalar product shows that the so-called four-divergence of a contravariant vector V α, i.e.,∂V α/∂xα, is a scalar.

A contravariant tensor of rank two Tαβ consists of sixteen elements that transformaccording to

Tαβ → T ′αβ =∂x′α

∂xγ∂x′β

∂xδT γδ, (3.28)

and the elements of the covariant tensor of rank two Rαβ transform as

Rαβ → R′αβ =

∂xγ

∂x′α∂xδ

∂x′βRγδ. (3.29)

Finally, we have a mixed second-rank tensor Sαβ with the transformation rule

Sαβ → S′α

β =∂x′α

∂xγ∂xδ

∂x′βSγ

δ. (3.30)

The transformation properties introduced above for first- and second-rank tensors shouldmake the generalization to contravariant, covariant, and mixed tensors of arbitrary rankobvious to the reader.

Speaking in general terms, the fundamental usefulness of the tensor concept for physicsis associated to the fact that if two tensors, with the same combination of upper and lowerindices, are equal in one coordinate system, then they are equal in any other coordinatesystem. A tensor which is zero in one coordinate system thus is zero in all other systems. Theimportance of tensor analysis culminates in the field of general relativity. Below, we shalllearn the usefulness of tensor analysis in the framework of the Special Theory of Relativity,where the transformation in Eq. (3.21) is the Lorentz transformation.

3.3 Four-vectors and -tensors

In Sec. 3.2, we discussed the tensor concept without specifying the law of transformation inour four-dimensional space continuum. In Special Relativity it is the Lorentz transformation(Eq. (3.1)) which defines the transformation from one system of space-time coordinates toanother, as we have seen. Thus,

∂x′α

∂xβ= Λα

β , (3.31)

and, since the Lorentz transformation is nonsingular

∂xβ

∂x′α= Λ β

α , (3.32)

where Λ βα is the inverse matrix of Λα

β, that is

Λ βα Λα

γ = δβγ . (3.33)

Vectors and tensors which transform according to the Lorentz transformation are calledfour-vectors and four-tensors. The elements of the contravariant four-vector V α hence havethe transformation rule

V α → V ′α = ΛαβV

β , (3.34)


and those of the covariant (dual) four-vector transform as

Wα →W ′α = Λ β

α Wβ . (3.35)

The invariance of the scalar product dx′αdxα, together with the invariance of the proper

time, expressed in the form given in Eq. (3.3), show that the elements of the differentialfour-vector dxα are connected to the elements of its dual four-vector dxα via

dxα = ηαβdxβ . (3.36)

The coordinate independence of the metric tensor in the Special Theory of Relativity impliesthat Eq. (3.36) can be integrated to give xα = ηαβx

β . To every contravariant four-vectorthere hence corresponds a covariant four-vector, that is,

Vα = ηαβVβ , (3.37)

and to every covariant vector there corresponds a contravariant vector with

Wα = ηαβWβ . (3.38)

The elements of the matrix ηαβ are the same as those of ηαβ, i.e., ηαβ = ηαβ .To the contravariant four-vector

V α = (V 0,V) (3.39)

there correspond in special relativity a covariant four-vector

Vα = (−V 0,V) = (V0,V) (3.40)

for the signs of the flat-space metric defined as in Eq. (3.4) [or equivalently Eq. (3.3)]. Withthis choice the scalar product takes the explicit forms

WαVα = W ·V −W 0V 0 = W ·V +W0V

0 =W0V0 +WiV

i, (3.41)

where, in the last member of Eq. (3.41) the summation over i runs over i = 1− 3. It is clearfrom the foregoing analysis that the gradient [cf. Eq. (3.27)]

∂α ≡ ∂

∂xα = (

1

c

∂

∂t,∇) (3.42)

must be a covariant four-vector, a fact underlined by the notation ∂α. The correspondingcontravariant four-vector, with elements determined by

∂α ≡ ∂

∂xα= ηαβ

∂

∂xβ= ηαβ∂β , (3.43)

thus is

∂α = (−1

c

∂

∂t,∇). (3.44)

The scalar product

∂α∂α = ∇2 − 1

c2∂2

∂t2≡ (3.45)


shows that the d’Alembertian operator is invariant against Lorentz transformations. Thesame holds, as we already know, for the divergence of a four-vector

∂αVα = ∂αVα = ∇ ·V +

1

c

∂

∂tV 0 = ∇ ·V − 1

c

∂

∂tV0. (3.46)

On the basis of Eqs. (3.31) and (3.32) it is easy to write down the Lorentz transformationsfor the second-rank tensors in Eqs. (3.28)-(3.30), and for higher-rank tensors. As an example,the transformation rule for a mixed second-rank tensor will be

Sαβ → S′α

β = ΛαγΛ

δβ S

γδ. (3.47)

With the help of Eq. (3.33), the Lorentz transformation restriction in Eq. (3.5) may bewritten in the form

ηαβ = Λ γα Λ δ

β ηγδ, (3.48)

as the reader may verify to herself. Eq. (3.48) shows that ηαβ is a covariant tensor. From

Λ βα = ηαγη

βδΛγδ, (3.49)

it is not difficult to prove that ηαβ is a contravariant tensor. Since a combination of Eqs.(3.37) and (3.38) gives

ηαβηβγ = δαγ , (3.50)

it appears that ηαβ = ηαβ , as already mentioned. The metric tensor, also called theMinkowski tensor, therefore is both a covariant and contravariant tensor.

The product of the components of two tensors (called the direct product) yields a newtensor. For instance,

∂αTβγ ≡ T βγ

α . (3.51)

Setting an upper and lower index of a tensor equal and using the convention of summationover repeated indices (a procedure called contraction), yields a tensor two ranks lower. Asan example take the third-rank mixed tensor in Eq. (3.51), written in the form ∂αT

γβ, andthen set γ = α. The transformation for this new object becomes

∂αTαβ → ∂′αT

′αβ = Λ γα Λα

δΛβε∂γT

δε = δγδΛβε∂γT

δε = Λβε∂γT

γε, (3.52)

having made use of Eq. (3.33). The result in Eq. (3.52) demonstrates that ∂αTαβ ≡ V β is

a first-rank contravariant tensor (contravariant four-vector).

3.4 Manifest covariance of the free Maxwell equations

The form-invariance of the microscopic Maxwell–Lorentz equations under Lorentz trans-formations was known before the formulation of the Special Theory of Relativity [236,145, 146, 147, 237, 180]. This invariance in form has traditionally been called covariance.Although the covariance of electrodynamics holds in the presence of charged matter, weshall below confine ourselves to an examination of the covariance of the set of free Maxwellequations. Later on when we study the light-matter interaction the covariance of the fullMaxwell–Lorentz theory is proved.


To express the free Maxwell equations given in Eqs. (2.1)-(2.4) in a form which is man-ifestly covariant, we introduce the antisymmetric contravariant field tensor

Fαβ =1

c

0 E1 E2 E3

−E1 0 cB3 −cB2

−E2 −cB3 0 cB1

−E3 cB2 −cB1 0

, (3.53)

where the subscript set (1, 2, 3) = (x, y, z) denotes the Cartesian coordinates. The rows andcolumns of the matrix are numbered α = 0− 3, and β = 0− 3. The definition in Eq. (3.53)is general, but in free space the electric field only has a transverse part, i.e., E = ET . Therelated field tensor Fαβ

T thus is

FαβT = Fαβ(E = ET ). (3.54)

The content of the matrix equation in Eq. (3.53) may conveniently be expressed as (i, j =1, 2, 3)

−F i0 = F 0i =1

cEi (3.55)

F ij = εijkBk, (3.56)

where εijk is the three-dimensional Levi–Civita symbol, given by

εijk =

+1 if ijk even permutation of 123−1 if ijk odd permutation of 1230 otherwise

. (3.57)

In terms of the elements FαβT of the free-space field tensor, the Maxwell equations in

Eqs. (2.2) and (2.3) [E = ET ] can be written as

∂αFαβT = 0. (3.58)

The reader is urged to verify to herself that with β = 0, the Maxwell equation ∇ · ET = 0follows, and for β = 1, 2, and 3, the three Cartesian components of ∇ × B = c−2∂ET/∂temerge. In the presence of charges and currents precisely these two Maxwell equations aremodified, and Eq. (3.58) then is replaced by

∂αFαβ = −µ0J

β , (3.59)

where Jβ is the so-called contravariant four-current density (see Secs. 8.3 and 12.1). Thisis a tensor, and therefore also the left-hand side of Eq. (3.59) must be a tensor. Since ∂αis a tensor, Fαβ is a tensor. The expression ∂αF

αβ is just an example of the contraction

given in Eq. (3.52). The free-space field tensor FαβT is also a genuine tensor because the

left-hand side of Eq. (3.58) is the Zero Tensor (of rank one).In order to rewrite the two remaining free-space Maxwell equations in (2.1) and (2.4) in

a manifest covariant form, we first introduce a covariant field tensor Fαβ by

Fαβ = ηαγηβδFγδ, (3.60)

in analogy with the correspondence for first-rank tensors; see Eq. (3.37). In matrix notationone has

Fαβ =1

c

0 −E1 −E2 −E3

E1 0 cB3 −cB2

E2 −cB3 0 cB1

E3 cB2 −cB1 0

. (3.61)


The covariant form of the field tensor can be obtained from the contravariant form by thereplacement E ⇒ −E, that is

Fαβ(E,B) = Fαβ(−E,B). (3.62)

In free space Fαβ is replaced by FT,αβ.By introduction of the four-dimensional Levi–Civita symbol εαβγδ, defined by

εαβγδ =

+1 if αβγδ even permutation of 0123−1 if αβγδ odd permutation of 01230 otherwise

, (3.63)

the free-space Maxwell equations given in Eqs. (2.1) and (2.4) [E = ET] can be written inthe compact form

εαβγδ∂βFT,γδ = 0. (3.64)

This form also holds in the presence of charges and currents. We just need to replace FT,γδ

by Fγδ, that is, in general

εαβγδ∂βFγδ = 0. (3.65)

The form in Eq. (3.65) [Eq. (3.64)] is clearly Lorentz invariant because it consists of a sumof third-rank mixed tensors, ∂βFγδ [∂βFT,γδ]. Again, I encourage the reader to show thatα = 0 gives the Maxwell equation ∇ ·B = 0, and that α = 1, 2, and 3 lead to the x, y, and zcomponents of ∇×E = −∂B/∂t. There is a useful alternative form of Eq. (3.65) in whichone avoids the Levi–Civita symbol, namely,

∂αFβγ + ∂βFγα + ∂γFαβ = 0, (3.66)

and, of course, an analogous one for Eq. (3.64)[Fαβ ⇒ FT,αβ ]. If two of the indices in Eq.(3.66) are equal, Eq. (3.66) is an identity [0=0] because the field tensor is antisymmetric.With one of the indices in Eq. (3.66) equal to zero, one reproduces the Cartesian componentsof ∇×E = −∂B/∂t, and if no index is zero one arrives at ∇ ·B = 0.

The manifest covariant form of the free-space Maxwell equations thus is given by Eqs.(3.58) and (3.64), and in the presence of matter these are replaced by Eqs. (3.59) and (3.65).

3.5 Lorentz transformation of the (transverse) electric and mag-netic fields. Duality

We have realized in the previous section that the object Fαβ is a tensor (covariant andantisymmetric). This means that its component values in one inertial system O′ can beexpressed in terms of the values in another inertial system O according to the generaltransformation formula for contravariant rank two tensors [combine Eqs. (3.28) and (3.31)].Thus,

F ′αβ = ΛαγΛ

βδF

γδ. (3.67)

Since the field tensor is build from the electric and magnetic fields in the manner given inEq. (3.53), Eq. (3.67) contains the transformation formulae for E and B. For the purpose


of this book we do not need the most general transformation formulas, and we thereforelimit ourselves to the specific case where the coordinate axes in O and O′ are parallel.Furthermore, it is assumed that O′ moves with a velocity v seen from O. The specificLorentz transformation for the coordinates here is a bit different from the one written downin Sec. 3.1 [Eqs. (3.16) and (3.18)]. However, it is easy to obtain the new one from the oldone. We just need to take the following two steps: (i) Interchange O and O′. This givest = t(t′,x′) and x = x(t′,x′). (ii) Take the inverse transformation. This is simply done bythe replacement v → −v. Altogether, one just needs the replacement v → −v in Eqs. (3.19)and (3.20) (with the rotation matrix equal to the unity matrix). Hence, with

Λi0 = Λ0

i = −vicγ, Λ0

0 = γ, (3.68)

and, as before,

Λij = δij +

vivjv2

(γ − 1) (3.69)

inserted into Eq. (3.67), one finds that the transformation formulas for the fields can bewritten as

E′ = γ (E+ cβ ×B)− γ2

1 + γββ · E, (3.70)

B′ = γ

(

B− 1

cβ ×E

)

− γ2

1 + γββ ·B, (3.71)

where

β =v

c. (3.72)

The result in Eqs. (3.70) and (3.71) show that the electric and magnetic fields have noindependent “existence.” A pure electric field in O, for instance appears as a mixture E′ =γ[U−γββ/(1+γ)] ·E, and B′ = −γβ×E/c in O′, and a pure magnetic field in O becomesa mixture in O′.

For the specific choice adopted above (parallel O and O′ axes) the results in Eqs. (3.70)and (3.71) are general in the sense that they hold for electric fields having both transverseand longitudinal parts, i.e., E = ET + EL. In photon physics, the transverse part of theelectric field is associated with transverse photons, and the longitudinal electric field relatesto the longitudinal and scalar photons (in the framework based on the scalar and vectorpotentials). It is therefore important to have a qualitative understanding of the transforma-tion properties of ET and EL. For this purpose it is convenient to study the transformationin the four-dimensional Fourier space [see Eq. (2.36)]. If one writes the phase Φ = q · r−ωtin the form

Φ = qµxµ, (3.73)

where

qµ =(

−ωc,q)

, (3.74)

it appears that qµ must be a covariant four-vector, the so-called four-wave vector. This isso because the phase Φ is a Lorentz invariant quantity, and xµ is the position four-vector.With the replacements ct → ω/c (ct′ → ω′/c), x → q (x′ → q′), and v → −v [see the


remarks above Eq. (3.68)] in Eqs. (3.19) and (3.20) one obtains the following transformationfor the four-wave vector:

ω′

c= γ

(ω

c− βq‖

)

, (3.75)

q′ = q+ eβ

[

(γ − 1)q‖ − γβω

c

]

, (3.76)

where eβ = β/β and q‖ = eβ ·q. Below only the transformation law for the three-wave vector(q) is needed, but in passing the reader will note that Eq. (3.75) is the well-known relativisticDoppler shift formula (for parallel O and O′ axes). Remembering that E′ = E′(r′, t′) andE = E(r, t) [B = B(r, t)] in Eq. (3.70), one obtains the relation

E′(q′, ω′) = γ (E(q, ω) + cβ ×B(q, ω)) + (1− γ)eβeβ ·E(q, ω) (3.77)

among corresponding Fourier components in O and O′ [E′(q(q′, ω′), ω(q′, ω′)) ≡ E′(q′, ω′)].By combining Eqs. (3.76) and (3.77) one may show that the scalar products between thewave vector and the electric field in the two frames are related by

q′ ·E′(q′, ω′) = γq · E(q, ω)− γβ ·(ω

cE(q, ω) + cq×B(q, ω)

)

. (3.78)

This result can be simplified utilizing the Maxwell–Lorentz equation iq × B(q, ω) =µ0J(q, ω)− (iω/c2)E(q, ω) [Eq. (8.52) transformed to the wave-vector−frequency domain].Thus,

q′ ·E′(q′, ω′) = γ (q ·E(q, ω) + iµ0cβ · J(q, ω)) , (3.79)

where J(q, ω) is the charge current density of the particle system in the frequency-wavevector domain. It appears from Eq. (3.79) that the transformation of the longitudinal partof the electric field only depends on the current density. This conclusion holds for the mostgeneral Lorentz transformation. Thus, if the charge density is denoted by ρ, it is knownthat Jµ = (cρ,J) is a four-vector [see Secs. 8.3 and 12.1]. A Lorentz transformation ofthe Maxwell–Lorentz equation iq′ · E′(q′, ω′) = ρ′(q′, ω′)/ǫ0 back from O′ to O thereforeinvolves only a combination of ρ(q, ω) = iǫ0q ·E(q, ω) [Eq. (8.51) in the (q, ω)-domain] andJ(q, ω). Once the transformation of the longitudinal electric field, viz.,

E′L(q

′, ω′) = (q′)−2q′q′ ·E′(q′, ω′)

=γ

q+ eβ[

(γ − 1)q‖ − γβ ωc

]

(q · EL(q, ω) + iµ0cβ · J(q, ω))q2⊥ + γ2

(

q‖ − β ωc

)2 , (3.80)

has been determined, the transverse field in the O′ frame, E′T (q

′, ω′) = E′(q′, ω′) −E′

L(q′, ω′), can be obtained subtracting the results in Eqs. (3.77) and (3.80) from each

other. In the denominator of Eq. (3.80), q⊥ = (q2 − q2‖)1/2 denotes the magnitude of the

component of the wave vector perpendicular to the eβ-direction. In free space, where onlytransverse fields can exist, the Lorentz transformation for the ET and B fields (with parallelO and O′ axes) reads

E′T (r

′, t′) = γ (ET (r, t) + cβ ×B(r, t))− γ2

1 + γββ · ET (r, t), (3.81)

B′(r′, t′) = γ

(

B(r, t) − 1

cβ ×ET (r, t)

)

− γ2

1 + γββ ·B(r, t), (3.82)


with E′T (r

′, t′) ≡ E′T (r(r

′, t′), t(r′, t′)) and an analogous definition of B′(r′, t′).Remembering that E(r, t) = ET (r, t) in the free Maxwell equations, given in Eqs. (2.1)-

(2.4), it is easy to show that this set is form-invariant under the combined transformation

ET ⇒ cB , B ⇒ −1

cET . (3.83)

The transformation above is the simplest example of a so-called duality transformation[101]. A duality transformation of Eq. (3.81) leads to Eq. (3.82), as expected.

3.6 Lorentz transformation of Riemann–Silberstein vectors. Inner-product invariance

For the specific choice of coordinate systems O and O′ which leads to the transformationformulas for the electric and magnetic fields given in Eqs. (3.70) and (3.71) one readily maydetermine the transformation equations for the positive- (Eq. (2.104)) and negative- (Eq.(2.107)) helicity Riemann–Silberstein vectors. Hence,

F′±(r

′, t′) = γ

(

F±(r, t)∓ iβ × F±(r, t)−γ

1 + γββ · F±(r, t)

)

. (3.84)

How does the inner product of the Riemann–Silberstein vectors change under a generalLorentz transformation? To answer this question we examine the real and imaginary partsof

F± · F± =ǫ02(E2

T − c2B2)± iǫ0cET ·B, (3.85)

where, for notation simplicity, E2T ≡ ET · ET and B2 ≡ B ·B. To determine the transfor-

mation of E2T − c2B2, we start from the result

LF ≡ − ǫ0c2

4FαβF

αβ =ǫ02

(

E2 − c2B2)

. (3.86)

The quantity LF is the density of the field Lagrangian [88, 101]. The last member of Eq.(3.86) follows immediately by combination of Eqs. (3.53) and (3.61). The quantity FαβF

αβ

is a contraction (summation over α and β) of the covariant and contravariant field tensors,and therefore FαβF

αβ is a Lorentz invariant scalar. In free space E = ET , so one mayconclude that E2

T − c2B2 is an invariant. From Eqs. (3.70) and (3.71) the reader may showby a direct calculation that E′ · B′ = E · B. The scalar product of E and B hence is thesame in the O and O′ frames which here have parallel axes. Since the scalar product of twothree-vectors is invariant against spatial rotations, it follows that E · B is a pseudoscalarinvariant also if the axes of the two inertial systems are not parallel. In free space ET · Bhence is a Lorentz invariant pseudoscalar. On the basis of the considerations above we thushave come to the conclusion that the norms of the two Riemann–Silberstein vectors arepreserved, i.e.,

F′±(r

′, t′) · F′±(r

′, t′) = F±(r, t) · F±(r, t) (3.87)

for the squared norms. The duality transformation in (3.83) implies that

F± ⇒ (∓i)F±, (3.88)

corresponding to phase shifts ∓π/2.

Part II

Light rays and geodesics.Maxwell theory in general

relativity

4

The light-particle and wave pictures in classical

physics

The Greek philosophers of antiquity speculated about the nature of light, and they werefamiliar with the rectilinear propagation of light, and the reflection and refraction phenom-ena of light rays at an interface. The law of reflection was known to the Greeks, but the lawof refraction was discovered experimentally only in 1621 by Snell [38]. Shortly afterwardDescartes derived the laws of reflection and refraction by comparison with the trajectory ofa ball [59, 177]. The observations of the rectilinear propagation of light made it plausiblefor Newton and others to consider a light ray as a stream of particles [177, 169]. AlthoughNewton devoted himself to the development of the corpuscular model of light, his views onthe nature of light were as ambivalent as that nature itself, as emphasized by Nussenzveig[177]. The first observations of interference, and the detection of the presence of light in thegeometrical shadow of a material body (diffraction) [87] made it more likely that light wasa wave phenomenon, a view first advocated by Hooke, [99] and much later by Young, whoformulated the basic principle of interference [255]. The earliest known example of frustratedtotal reflection is due to Newton [170] who observed the phenomenon now known as New-ton’s rings [38]. Although the visual appearance of the rings suggests that light is a wavephenomenon, Newton invoked his corpuscular light theory [177] to model total reflection.

In our time, we consider frustrated total reflection as an optical tunneling process. Todescribe the phenomenon, quantum physics, with its wave-particle duality picture of light, isneeded, as well as an understanding of the light-source concept in quantum electrodynamics.It is perhaps not surprising that even Newton did not commit himself completely to thecorpuscular model of light in relation to total reflection. In “Principia” [177, 169] he thusstates: “Therefore because of the analogy there is between the propagation of the rays oflight and the motion of bodies, I thought it not amiss to add the following Propositions foroptical uses; not at all considering the nature of the rays of light or inquiring whether theyare bodies or not; but only determining the curves of bodies which are extremely like thecurves of rays.”

All experiences in classical electrodynamics are summed up in the microscopic Maxwell–Lorentz equations, relating the electromagnetic field to the microscopic charge and currentdensities of massive point-like particles [56, 57, 133]. Although light is considered as a wavephenomenon in these equations, one may ask whether there is any indirect trace pointingtoward a possible particle character (behavior) of light. Since the distinction between par-ticles and fields is sharp in classical physics, one must look for an approximation to theset of Maxwell–Lorentz equations in which the (always finite) wavelength (λ) of the fieldis negligible in order to keep the hope of finding a fingerprint of a possible existence of a“light particle.” Such an approximation (λ→ 0) does exist, and the related theory is knownas geometrical optics [38]. The fundamental quantity in geometrical optics is the so-calledeikonal, S(r), a space (r) dependent scalar quantity, which satisfies the eikonal equation [38].The orthogonal trajectories to the geometrical wave surfaces, given by S(r) = C, where Cis a parameter, determine the (geometrical) light rays (Sec. 5.2). A classical particle followsa trajectory, and in geometrical optics the field energy flows (on the average) along the light

41


ray. In the absence of matter the rays are straight lines, like the trajectories for a massive freeparticle in Newtonian mechanics. In general, the trajectory of a classical particle will bendif the particle is subjected to an external force. “External forces” in geometrical optics stemfrom the presence of electrically charged massive particles in the field. In the framework ofgeometrical optics the light ray-particle interaction is described in an approximation wherethe (microscopic) susceptibility is assumed to be spatially local and frequency independent.Certainly, it is not an easy matter to establish precise physical criteria for validity of suchan approximation (Sec. 5.1). An ideal medium in which the ray-particle interaction can bedescribed via a susceptibility χ = χ(r) I name an inhomogeneous vacuum. In the spirit ofa light particle concept, one might imagine that a light ray consists of a stream of lightparticles. The idea that there might be a link between geometrical optics and classical par-ticle mechanics is strengthened by rewriting Newton’s second law in the Hamilton–Jacobiform [38, 228, 140, 9, 80]. In Newtonian mechanics the dynamical state of a point-particleat time t is characterized by the particle’s position, r = r(t), and its momentum p = p(t).In the eikonal limit it is tempting to associate a local momentum

pLight(r) = ~q0∇S(r) (4.1)

to a light particle with an underlying monochromatic field of wave number q0. To reachEq. (4.1) one must add a piece of de Broglie quantum physics. To the movement of a (free)particle with four-momentum pµ = (E/c,p) we thus associate the propagation of a locallymonochromatic phase wave with four-vector qµ = (q0,q) in such a manner that

pµ = ~qµ, µ = 0− 3, (4.2)

where h = 2π~ is Planck’s constant [126, 55]. Although it makes a certain sense to associateat the space point r a local momentum to our hypothetical light particle, it turns out thatit is impossible to localize the light particle at r (at a given time).

Newton’s and Hooke’s contemporary Huygens was the first to propose an explicit wavetheory of light. His classic treatise Traite de La Lumiere [100], formulated twelve yearsbefore its publication in 1690 [253], became a landmark in the history of optics. Actually,Huygens’ theory dealt with (spherical) light pulses, excluding the idea of periodicity [177].Thus, in a sense the wave picture of Huygens is not that far from the corpuscular model oflight.

In 1657 Fermat enunciated the celebrated Principle of Least Time [71]. According to histeleological principle, light follows that path which brings it to destination in the shortesttime. Fermat’s principle can be seen in a perspective which is extremely useful when theanalysis of light rays in flat space-time is extended to the curvilinear space-time of generalrelativity. Hence, if the metric tensor of the so-called inhomogeneous vacuum is defined bygij(r) = n2(r)U, where n(r) is the local refractive index, and U the unit tensor, theequation for the geodetic line is obtained from a conventional variational principle: Thegeodetic line between two given points in space is a line drawn in such a manner that thelength of the line obtains a stationary value. From the Euler–Lagrange equation associatedto the problem the differential equation for the light ray emerges. The analysis leads to anextension of Fermat’s Principle of Least Time to a Principle of Stationary Time (Sec. 5.3).

In general relativity light rays are associated to null geodesics, i.e., geodesics subjectedto the constraint that the (infinitesimal) proper time in the curved Riemann space is zero[241]. Using a suitable affine parameter σ as independent variable (instead of the propertime) the geodesic equation for the µth component of the position four-vector xµ takesthe form

d2xµ

dσ2+ Γµ

αβ(x)dxα

dσ

dxβ

dσ= 0, µ = 0− 3, (4.3)

The light-particle and wave pictures in classical physics 43

where Γµαβ(x) is the Christoffel symbol. Despite its name, x = xµ cannot be considered

as the space-time position of a classical light particle.To understand the physics of light in classical general relativity it is necessary to study

also the behavior of massive particles. This is so because light only manifests itself in itsinteraction with matter: The fingerprints of light are found in the light-induced motion ofcharged massive particles. The motion of a free massive test particle in curved space-timeis governed by a time-like geodesic equation (Sec. 6.2). The quantity

FµG(x) = −

Γµαβ(x)

dxα

dσ

dxβ

dσ

(4.4)

expresses the gravitational field strength on a test particle freely falling in a gravitationalfield (τ is the proper time). The Newtonian equation of motion is obtained in the limit wherethe gravitational field is weak and (essentially) static. In Sec. 6.5 we derive a well-knownexpression for the gravitational redshift of light using the Principle of Equivalence, and weindicate how this shift can be related to de Broglie quantum mechanics and to the lessonof Bohr concerning the epistemology of quantum physics. In a weak static gravitationalpotential, Φ, it turns out that the sum of the “kinetic” energy, hν, and “potential” energy,hνΦ/c2, of a monochromatic (frequency ν) light particle, viz.,

E = hν

(

1 +Φ

c2

)

, (4.5)

is a conserved quantity (Sec. 6.5). In some sense one may say that the result in Eq. (4.5)relates to a light particle freely falling in the gravitational field.

In Chapt. 2 we discussed the framework of classical electrodynamics in Minkowskianspace, paying particular attention to those aspects of the Maxwell theory in free spacewhich points toward photon physics. In Chapt. 8 the Maxwell–Lorentz theory is extendedto curved space-time. Such an extension is based on the Principle of General Covariance[241], which states that the physical laws in Special Relativity preserve their forms undergeneral coordinate transformations. To prepare ourselves for a generalization of the electro-magnetic field theory, we devote Chapt. 7 to a brief discussion of the space-time structureof general relativity. After having introduced the tensor field concept in a four-dimensionalRiemannian space with indefinite metric in Sec. 7.1, we turn our attention toward the con-cept named covariant derivative. The Maxwell–Lorentz equations in Minkowskian space(Eqs. (3.59) and (3.66)) involves first-order derivatives of the field tensor, and when the po-tential description of electrodynamics is formulated second-order derivatives of the potentialvector appear. The usual derivative of a vector (tensor) is not a tensor in general (it is ofcourse in Minkowski space). The problem thus consists of defining the derivative of a vector(tensor) so that it is a tensor. The covariant derivative solves this problem. In order that thetransformation between the normal (∂µ) and covariant (µ) derivatives is compatible withthe geodesic equation it is required that the connection between these derivatives is givenby the Christoffel symbol, provided one wants the connection to be linear and symmetric.After having discussed the covariant derivative we turn our attention toward the paralleltransport of a tensor along the path xµ(σ) (directional covariant derivative), and the Rie-mann curvature tensor in Secs. 7.3-7.5. Although the Einstein field equations do not playa role as such in our description of photon physics, these equations are briefly discussed inSec. 7.6. In the presence of matter the (here) so-called energy-momentum (or stress) tensorappears in the Einstein equations. This tensor, defined in terms of the density and currentof the energy-momentum four-vector of the particle system, in itself does play an importantrole in photon physics [206, 101, 188], but there is no room for discussing it in this book.We end our brief introduction to the space-time of general relativity studying the so-called


geodesic deviation of light rays, and it is concluded that initially parallel geodesics in emptyspace will not remain parallel unless the space-time curvature vanishes.

The Maxwell–Lorentz equations are extended to curved space-time by (i) requiring thatthe contravariant field tensor, Fµν, and the contravariant current density four-vector,Jµ, behave as tensors under general coordinate transformations, and that they reduce totheir original forms in locally inertial Minkowskian coordinates, and (ii) replacing all normalderivatives by covariant derivatives (∂µ ⇒ µ); see Sec. 8.1. For many purposes, it isuseful to rewrite the Maxwell–Lorentz equations in general relativity in such a mannerthat only normal derivatives occur. To achieve this goal the concepts covariant curl anddivergence are examined. It is easy to see that the covariant curl of a vector, V µ, isidentical to the ordinary curl in four dimensions. The covariant divergence of a contravariantvector turns out to be given by an expression which involves the determinant g of the metrictensor, gµν. Hence,

√−gµ Vµ = ∂µ

(√−gV µ)

, (4.6)

as shown in Sec. 8.2. The Maxwell–Lorentz equations in curved space-time are rewritten interms of normal derivatives in Subsec. 8.3.1.

The antisymmetric covariant field tensor, Fµν, has six independent components, andone can label these by names which correspond to those of the electric (E) and magnetic (B)fields in flat space-time. The antisymmetric contravariant field tensor, Fµν, is obtainedfrom Fµν with the help of the metric tensor of curved space-time, gµν, used twice.The six independent components of

√−gFµν we label by names known from macroscopicelectrodynamics, viz., D and H. With the help of the tensor relation

Fµν(x;D,H) = gµα(x)gνβ(x)Fαβ(x;E,B), (4.7)

the Maxwell–Lorentz equations in general relativity attain a form which looks exactly as themacroscopic Maxwell equations in Minkowski space; see Subsecs. 8.3.2 and 8.3.3. In flat andempty space, one obtains D = ǫ0E and B = µ0H, as expected. The connection in Eq. (4.7)expresses what one may call constitutive relations for curved space-time (Subsec. 8.3.4). Weexemplify this by calculating the relative permittivity and permeability for static metrics(Subsec. 8.3.6), and for the flat Robertson–Walker metric of an expanding homogeneousand isotropic universe (Sec. 8.4).

In this book photon wave mechanics based on (i) four-potentials and (ii) covariant quan-tum electrodynamics play central roles, and therefore it is fruitful to study the fundamentalsof the potential description of classical electrodynamics. We undertake an examination ofthe potential formulation in general relativity in Sec. 8.5. With the elements of covariantfield tensor given by Fµν(x) = µAν(x) −νAµ(x), where Aµ(x) is the covariant four-potential, one obtains in the family of Lorenz gauges, where ∂µAµ(x) = 0, the followingfield equation:

ν ν Aµ(x)−Rνµ(x)Aν (x) = −µ0Jµ(x), (4.8)

where Rνµ(x) is the Ricci tensor (contracted Riemann curvature tensor), and Jµ(x) is the

suitably extended Minkowski current density four-vector. Starting from Eq. (4.8) we studythe eikonal theory and null geodesics. In the limit of geometrical optics our field analysisalso suggests that light travel on null geodesics in free space.

The Principle of General Covariance governs the effect of gravitational fields, and theprinciple is often characterized as a dynamic symmetry [241]. In electrodynamics anotherdynamic symmetry called local gauge invariance appears. We study this dynamic symmetry,and the related so-called gauge-covariant derivative operator, in Sec. 8.6, and exemplify thegauge-invariance concept in relation to the Dirac equation.

5

Eikonal theory and Fermat’s principle

5.1 Remarks on geometrical optics. Inhomogeneous vacuum

In the perspective of classical theory it is the ability of light to interfere and diffract whichabove all prevents us from believing that light consists of a sort of classical particles, perhapspointlike. Even if we took the standpoint that propagating light fields in free space didconsist of streams of pointlike entities, these can certainly not be robust in their interactionwith material particles, although this was suggested by the scientist G. N. Lewis who in 1926coined the name photon for Einstein’s light quanta [144]. Despite the fact that the idea ofthe existence of uncreatible and indestructible “light atoms” has turned out to be untenable,the theory of classical optics shows reminiscences of particle behavior in situations wherethe finite wavelength of light is negligible.

The branch of optics which deals with phenomena where the wavelength of light playsno role is known as geometrical optics. In this section I shall briefly discuss the foundationof geometrical optics, but in an approach which deviates slightly from the traditional one.I do this because the aim here is to guide the reader toward the particle aspects of lightwithin a purely classical context. A good account of the basics of geometrical optics, andthis theory’s many applications can be found in the book by Born and Wolf [38].

To appreciate the importance and beauty of geometrical optics one must go beyond thefree-space Maxwell equations, given in Eqs. (2.1)-(2.4), and introduce the interaction of theelectromagnetic field with matter (charged massive particles). Although this interaction, inthe context of geometrical optics, usually is treated on the basis of macroscopic electrody-namics, we here seek a more microscopic approach for later convenience in photon physics.In the presence of charged particles, the forms of Eqs. (2.2) and (2.3) are changed to

∇×B(r, t) = µ0J(r, t) + c−2 ∂

∂tE(r, t), (5.1)

and

∇ · E(r, t) = ǫ−10 ρ(r, t), (5.2)

where the charge (ρ) and current density (J) together constitute a four-current density,Jα = (cρ,J), as already indicated in Sec. 3.4, where Eqs. (5.1) and (5.2) were written incovariant form [see Eq. (3.59)]. Together, Eqs. (2.1), (2.4), (5.1), and (5.2) constitute theso-called microscopic Maxwell–Lorentz equations. Without loss of generality, we introducea microscopic polarization field, P(r, t), by the definition

J(r, t) ≡ ∂

∂tP(r, t). (5.3)

The charge density can be expressed in terms of the polarization as follows:

ρ(r, t) = −∇ ·P(r, t). (5.4)

45


The connection in Eq. (5.4) holds up to a time-independent constant, which plays no role inelectrodynamics. The reader may prove Eq. (5.4) by combining the equation of continuityfor charge, viz.,

∇ · J(r, t) + ∂

∂tρ(r, t) = 0, (5.5)

and Eq. (5.3).In line with the standard approach to geometrical optics, we assume that the polarization

is linearly related to the selfconsistent electric field. This approximation (simplification),which in the present context is harmless, implies that the (now) most general constitutiverelation between P and E takes the form

P(r, t) = ǫ0

∫ ∞

−∞χ(r, r′, t, t′) ·E(r′, t′)dt′d3r′, (5.6)

where χ(r, r′, t, t′) is the so-called microscopic susceptibility. This tensorial quantity con-nects P and E in a nonlocal manner in both space and time. Although such a space-timenonlocality can be of significant importance in both classical and quantum electrodynam-ics, as the reader may appreciate by consulting, e.g., my book on the quantum theory ofnear-field electrodynamics [127], we now take three steps more down the ladder of simplifi-cations: (i) we assume that the properties of the matter system are time invariant, so thatχ(r, r′, t, t′) ⇒ χ(r, r′, t− t′), (ii) we assume that the spatial connection between polariza-tion and field essentially is local, which means that χ(r, r′, t− t′) ⇒ χ(r, t− t′)δ(r−r′), and(iii) we assume that the susceptibility tensor is isotropic, that is χ(r, t− t′) ⇒ χ(r, t− t′)U.Altogether, we end up with a constitutive relation

P(r, t) = ǫ0

∫ ∞

−∞χ(r, t− t′)E(r, t′)dt′. (5.7)

The fact that our new susceptibility χ(r, t − t′) only depends on the time difference t − t′

makes the right side of Eq. (5.7) a folding integral. A transformation to the space-frequencydomain therefore gives

P(r;ω) = ǫ0χ(r;ω)E(r;ω). (5.8)

When the susceptibility χ(r;ω) has an essential frequency dependence, it must neces-sarily be a complex quantity. In a strict sense this can be proved for a (slightly) modifiedsusceptibility χext(r;ω) which relates the polarization P(r;ω) to the externally impressedfield Eext(r;ω) needed in the first place to excite the particle system. The causal connectionbetween Eext and P in the time domain [χext(r, t− t′) = 0 for t < t′] implies that the realand imaginary parts of χext(r, t) form a Hilbert transform pairs [cf. Eqs. (2.31) and (2.32)];see also the analysis in, e.g., Refs. [155, 127]. If one neglects the frequency dispersion ofχext there is no longer any physically ACTIVE interaction between light and matter: Noabsorption and scattering processes exist any longer. Also the χ relating the polarization tothe selfconsistent electric field necessarily must lose its dependence on ω when active field-matter interaction is forbidden. In the standard theory of geometrical optics it is assumedthat the susceptibility is frequency independent, i.e., χ = χ(r). In certain macroscopic ap-plications based on geometrical optics a frequency dependence of the refractive index isincluded after the basic (eikonal) equation of the theory has been established.

Here, in the perspective of photon physics, I introduce the concept INHOMOGENEOUSVACUUM for a matter-filled space where the linear interaction between light and matter isdescribed in terms of a frequency independent susceptibility of the form χ = χ(r). Notwith-standing the fact that the concept relates to an idealized situation it may serve as an

Eikonal theory and Fermat’s principle 47

interesting and fruitful link to the study of the behavior of light rays in general relativity,as we shall see later on. The concept also helps us to understand how far we may go inconsidering classical light from the, albeit peculiar, particle point of view.

With J(r, t) and ρ(r, t) given by

J(r, t) = ǫ0χ(r)∂

∂tE(r, t), (5.9)

ρ(r, t) = −ǫ0∇ · (χ(r)E(r, t)) , (5.10)

Eqs. (5.1) and (5.2) can be written as

∇×B(r, t) =n2(r)

c2∂

∂tE(r, t), (5.11)

∇ ·(

n2(r)E(r, t))

= 0, (5.12)

where

n2(r) = 1 + χ(r). (5.13)

The Eqs. (2.1), (2.4), (5.11), and (5.12) thus are the basic set for the description of electro-dynamics in an inhomogeneous vacuum (with flat space-time). If one compares Eqs. (2.2)and (5.11) it appears that the local phase velocity of light is given by

v(r) =c

n(r)(5.14)

in inhomogeneous vacuum. The quantity n(r) may be called the refractive index. It appearsfrom Eq. (5.12) that the electric field is not a divergence-free vector field when the refractiveindex is space dependent.

5.2 Eikonal equation. Geometrical wave surfaces and rays

In inhomogeneous vacuum, the basic field equations are given by

∇×E(r;ω) = iωB(r;ω), (5.15)

∇ ×B(r;ω) = − iωc2n2(r)E(r;ω), (5.16)

∇ ·(

n2(r)E(r;ω))

= 0, (5.17)

∇ ·B(r;ω) = 0, (5.18)

in the space-frequency representation. From a light-particle point of view a special type offield is of central importance, viz., those of the form

E(r;ω) = E0(r)eiq0S(r), (5.19)

B(r;ω) = B0(r)eiq0S(r), (5.20)

where q0 = ω/c is the µ = 0 component of the contravariant four-wave vector. Note, thatin Eqs. (5.19) and (5.20) we do not allow E0, B0, and S to depend on frequency.


Let us now investigate under what conditions the forms in Eqs. (5.19) and (5.20) possiblymay satisfy Eqs. (5.15)-(5.18). Using well-known vector relations we obtain

iq0(

∇S ×E0 − cB0)

= −∇×E0, (5.21)

iq0(

n2

cE0 +∇S ×B0

)

= −∇×B0, (5.22)

iq0∇S ·E0 = −(

∇n2 +∇)

·E0, (5.23)

iq0∇S ·B0 = −∇ ·B0. (5.24)

Since the right-hand sides of these equations are independent of ω, the factors to iq0 on theleft sides must vanish. In turn, if these factors are zero, one must take the limit q0 → ∞in order to uphold nonvanishing right-hand sides. We are thus led to the (preliminary)conclusion: For sufficiently small wavelengths (asymptotically) the ansatz given by Eqs.(5.19) and (5.20) satisfies the fundamental field equations of the inhomogeneous vacuumprovided

∇S(r)× E0(r) = cB0(r), (5.25)

∇S(r)×B0(r) = −1

cn2(r)E0(r). (5.26)

The result in Eqs. (5.25) and (5.26) emerge from Eqs. (5.21) and (5.22). The last twoconditions E0 · ∇S = 0 and B0 ·∇S = 0 [arising from Eqs. (5.23) and (5.24)] also followfrom Eqs. (5.25) and (5.26) upon scalar multiplication of these equations with ∇S. Thefunction S = S(r) is called the eikonal. By eliminating B0 from Eqs. (5.25) and (5.26), oneobtains

∇S ×(

∇S ×E0)

= −n2E0, (5.27)

and since E0 ·∇S = 0, it appears that Eqs. (5.25) and (5.26) only have nontrivial solutionsprovided

∇S(r) ·∇S(r) = n2(r). (5.28)

Eq. (5.28) is known as the eikonal equation. It is the basic equation of geometrical optics,and as such it has numerous applications in situations where the optical wavelength isnegligible [38, 215, 134]. The eikonal is a real scalar function, but E0 and B0 in general arecomplex vector fields. These field vectors are everywhere in space perpendicular to ∇S(r),and they also are mutually orthogonal, in such a manner that E0 ×B0 ‖ ∇S. The surfaces

S(r) = C, (5.29)

where C is a constant (parameter) are called the geometrical wave surfaces or geometricalwavefronts. The orthogonal trajectories to the geometrical wave surfaces are called thegeometrical light rays (or just the light rays).

It appears from the eikonal equation that

t(r) =∇S(r)

n(r)=

∇S(r)

|∇S(r)| (5.30)

is a local unit vector. This fact may be put into a geometrical context: Let P be a point ona light ray (LR). The position of P can be characterized by the arc length s measured from


a point P0 [s = 0]. If, in a (Cartesian) coordinate system O, r = r(s) denotes the positionvector of P , then

t =dr

ds(5.31)

is the tangential unit vector to the LR at s. The reader may prove that t is in the directionof the time-averaged (〈· · · 〉) Poynting vector

〈S〉(r) = 1

2µ0R[

E0(r)×(

B0(r))∗]

, (5.32)

so for an observer at rest in O, the average energy flow in the field is along the light ray.Note that 〈S〉 has been characterized as a field-energy flow although E(r) is not a transversevector field in our inhomogeneous vacuum; see also [127, 53]. In a sense, this is meaningfulbecause the frequency independence of n = n(r) implies that an inhomogeneous vacuum isa passive medium, which cannot absorb nor release energy. A somewhat similar situationoccurs when one considers light rays in general relativity. Here, curved space-time may beconsidered as a passive medium, as we shall see in Chapt. 8; consult in particular Sec. 8.5.Since the differential change in the eikonal between neighboring geometrical wave surfacesis given by

dS = dr ·∇S = dst ·∇S, (5.33)

so that

dS

ds= n, (5.34)

it appears that the distance ds is inversely proportional to the refractive index, n. Theintegral

S(P2)− S(P1) =

∫

LR

nds, (5.35)

taken along a light ray from P1 to P2 is known as the optical length of the curve (P1 → P2).Because

nds = cds

v= cdt, (5.36)

where dt is the time needed for light (with local speed v(r)) to travel the distance ds alongthe ray in inhomogeneous vacuum, it follows that the optical path length is equal to thevacuum speed of light multiplied by the time needed for light to travel from P1 to P2, thatis

S(P2)− S(P1) = c

∫

LR

dt. (5.37)

At this stage in our classical description of light it is fruitful to reflect on the relationbetween the eikonal theory and the particle picture of light. A classical particle has nowave properties, and in geometrical optics (λ0 → 0) the fundamental eikonal equation hasno reference to wavelength. In its motion a classical particle follows a trajectory, and inthe eikonal limit the field energy flows, on the average, along the light ray. If one allowsanisotropy in inhomogeneous vacuum the considerations must be a bit modified because thegroup and phase velocity of light are not collinear in general, but the trajectory description


still holds in a slightly generalized sense. In his revolutionary article from 1905, Einsteinsuggested that light in its interaction with matter thermodynamically behaves as though itconsisted of independent energy (light) quanta [60]. He came to this proposal by analyzingthe high-frequency (λ0 → 0) Wien limit [249] of Planck’s radiation law [189, 190]. A bril-liant description of the interplay between the ideas of Planck and Einstein has been givenby Pais [180]. From a quantum physical perspective the particle aspect of the wave-particleduality tends to dominate precisely at high frequencies, as we shall see later on. The ideathat there is a link between geometrical optics and classical particle mechanics is strength-ened by the observation that Newton’s second law, when written in the Hamilton–Jacobiformulation, becomes form-identical to the eikonal equation, as the reader may learn bystudying, e.g., [38, 228, 140, 9, 80]. Thus, if Hamilton’s characteristic function is denotedby W , the Hamilton–Jacobi equation takes in Cartesian coordinates the form

∇W (r) ·∇W (r) = 2m(E − V (r)) (5.38)

for a particle of mass m and potential energy V (r). Since E is the total energy of theparticle, the quantity

p = ∇W (r) (5.39)

appears as the particle’s momentum at r. Since a monochromatic field with vacuum wavenumber q0 has a local wave number q(r) = q0n(r) in inhomogeneous vacuum, a forward leapto quantum mechanics (Planck’s constant h = 2π~) seems to indicate that a light particlein the eikonal limit might have the local momentum

pLight = ~q0∇S(r), (5.40)

when the classical electromagnetic field is monochromatic. The association in Eq. (5.40)at least brings the eikonal equation in the form pLight · pLight = (~q)2. In order to makea tight link to a particle description one must be able to localize the particle at a giventime somewhere on the ray, but such a demand runs into difficulties. In order to seek tolocalize the field (particle) we make a Fourier integral transformation of the electric field inits eikonal form [Eq. (5.19)], viz.,

E0(t− c−1S(r)) =

∫ ∞

−∞E0(r;ω)e−iω(t−c−1S(r)) dω

2π. (5.41)

where the amplitude weight factor, E0(r;ω), necessarily must be a function of the frequency,because only (numerically) high frequencies, |ω|, are allowed in the framework of the eikonaltheory. The field E0(t− c−1S(r)) propagates with local speed ds/dt = v(r) along the givenlight ray. Only by assuming (incorrectly) that the eikonal theory holds for all frequenciesare we allowed to choose frequency independent amplitudes, E0(r), in the superposition,and thus obtain a field

E0(t− c−1S(r)) = E0(r)δ(t− c−1S(r)), (5.42)

which is completely localized along the ray. Even if Eq. (5.42) could be satisfied, one wouldnot be able to localize (at time t) the field completely in the wave-front plane S(r) = ct. Aschematic illustration of this fact is shown in Fig. 5.1.

A differential equation for the ray r = r(s), which involves only n(r), can be obtainedstarting from the relation

n(r)dr(s)

ds= ∇S(r), (5.43)


v

P

"Light particle"

Geodetic line

Light

v

v

v

FIGURE 5.1Schematic illustration of a classical “light particle” propagating along a given geodesicline. Such a line can be identified with a light ray in geometrical optics. The light particleshown (as a grey-toned domain) is here assumed to be strongly localized in the directionof the ray tube. In consequence, the particle is weakly localized parallel to the wave fronts.Light particles exhibiting extreme spatial localization in one direction sometimes are calledwavefront light particles.

which follows directly from Eqs. (5.30) and (5.31). Differentiating Eq. (5.43) with respectto s one gets

d

ds

(

n(r)dr(s)

ds

)

= ∇n(r), (5.44)

since d∇S/ds = ∇(dS/ds) = ∇n. The last member of this equation is correct becauseof Eq. (5.34). When the refractive index is independent of r, the differential equation forthe ray reduces to d2r(s)/ds2 = 0. In a homogeneous vacuum, the light rays whence arestraight lines, i.e.,

r(s) = st+ a, (5.45)

where a is a constant vector, specifying that the given ray passes through the point r(0) = a.The vector multiplied on s is the here constant tangential unit vector to the LR, see Eq.(5.31). Setting n(r) = n0, Eq. (5.30) gives the eikonal

S(r) = n0t · r+ S0, (5.46)

where S0 is a constant. In a homogeneous vacuum the wavefronts, belonging to a t fixed tothe same value in every space point, are plane, and characterized by the unit normal vectort. Note that the monochromatic plane-wave field

E(r, t) = E0 exp[

i(

n0q0t · r− ωt

)]

, (5.47)

with wave vector n0q0t is a solution to the general set of Maxwell equations (for constant

n), and not only to the set obtained in the geometrical optical approximation.


5.3 Geodetic line: Fermat’s principle

Let us now consider the light ray and optical length concepts from a somewhat differentperspective, which when extended leads us to a beautiful description of light rays in thecurvilinear space-time of general relativity, and much more in photon physics.

Let

dsR =(

gij(r)dxidxj

)1

2 , i, j = 1− 3 (5.48)

be the infinitesimal line element in a three-dimensional Riemannian (R) space, here writtenin Cartesian coordinates. The quantity gij(r) ≡ gij

(

xi)

is the ij-component of the metrictensor gij, and as before summation over repeated indices is implicit. The line elementin Euclidean space, viz.,

ds =(

dx2 + dy2 + dz2)

1

2 , (5.49)

is obtained with gij = δij . A geodetic line between the points P1 and P2 in space is a line

drawn in such a manner that∫ P2

P1

dsR is stationary. The equation for the geodetic line isobtained from the variational principle

δ

∫ P2

P1

(

gijdxidxj

)1

2 = 0. (5.50)

To proceed from here one has to know the metric tensor for an inhomogeneous vacuum.Since the vacuum is assumed to be isotropic, the metric tensor must be diagonal, and thethree elements identical. Physically, the length of the Riemannian line element dsR can onlybe determined by the wavelength of light, λ(r) = λ0/n(r). Thus, the line element must beproportional to the refractive index, that is

dsR(r) = n(r)(

dx2 + dy2 + dz2)

1

2 , (5.51)

corresponding to a metric tensor with components

gij(r) = n2(r)δij . (5.52)

Next, we return to the variational principle in Eq. (5.50). In order to calculate the lineintegral along a given path we introduce a parameter description. Let σ be the parametercharacterizing the position on the arc, i.e., s = s(σ). The parameter σ can of course bechosen in (infinitely) many ways. A simple choice is σ = s. Could time be chosen as theparameter? For a massive particle moving along a given trajectory such a parameter choiceis possible not only here, but also in an extended relativistic space-time formulation of thevariational principle. For relativistic massive particles the frame-independent proper timeis a natural choice for σ. In photon physics such a choice is not possible because the propertime “seen” by the photon is zero.

Parameterized by an appropriate σ, the variational problem in Eq. (5.50) takes the form

δ

∫ P2

P1

(

gijdxi

dσ

dxj

dσ

)1

2

dσ = 0. (5.53)

With the notation

L = L

(

xi, dxi

dσ)

=

[

gij(

xi) dxi

dσ

dxj

dσ

]1

2

, (5.54)


the variational problem reads

δ

∫ P2

P1

Ldσ =

∫ P2

P1

δLdσ = 0, (5.55)

with the variation in L given by

δL =∂L

∂xiδxi +

∂L

∂(dxi/dσ)δ

(

dxi

dσ

)

, (5.56)

with the summation over i kept implicit. Since

δ

(

dxi

dσ

)

=d

dσδxi, (5.57)

Eq. (5.55) now becomes

δ

∫ P2

P1

Ldσ =

∫ P2

P1

[

∂L

∂xi+

∂L

∂(dxi/dσ)

d

dσ

]

δxidσ = 0. (5.58)

Remembering that the variations δxi vanish at the endpoint (P1 and P2) of the path, apartial integration of the last term of the integrand results in

∫ P2

P1

∂L

∂xi− d

dσ

[

∂L

∂(dxi/dσ)

]

δxidσ = 0. (5.59)

Since Eq. (5.59) must hold for all infinitesimal variations, one must have

∂L

∂xi− d

dσ

[

∂L

∂(dxi/dσ)

]

= 0 (5.60)

for i = 1, 2, 3. Eq. (5.60) is known as the Euler–Lagrange equation.By now, we are prepared to examine where the variational principle (the Euler–Lagrange

equations), with the metric tensor in Eq. (5.52), brings us. By inserting Eq. (5.52) into Eq.(5.54) one obtains

L = n(r)

[

∑

i

(

dxi

dσ

)2]

1

2

= n(r)ds

dσ, (5.61)

where ds is the Euclidean line element [Eq. (5.49)]. Then,

∂L

∂xi=

∂n

∂xids

dσ, (5.62)

and

d

dσ

(

∂L

∂(dxi/dσ)

)

=d

dσ

(

ndxi

dσdsdσ

)

=d

dσ

(

ndxi

ds

)

=ds

dσ

d

ds

(

ndxi

ds

)

. (5.63)

By combining Eqs. (5.60), (5.62), and (5.63), the Euler–Lagrange equation, after divisionby ds/dσ, gives

d

ds

(

ndxi(s)

ds

)

=∂n

∂xi, i = 1, 2, 3. (5.64)

This equation is just the ith component of the differential equation for the light ray; seeEq. (5.44). The result in Eq. (5.64) is the (extended) principle of Fermat which asserts that:The actual light ray is distinguished from other curves lying in a regular neighborhood bya stationary value of the optical length (between points P1 and P2).

6

Geodesics in general relativity

6.1 Metric tensor. Four-dimensional Riemann space

In Special Relativity, the four-dimensional space-time continuum (Minkowski space) is inrectilinear coordinates, xµ, characterized by the metric tensor η = ηµν, which is thesame in all space-time points, and given by the diagonal form

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

; (6.1)

see Eq. (3.4). The metric relates to the “infinitesimal squared distance,” ds2, associatedwith infinitesimal separated events in Minkowski space as follows:

ds2 = ηµνdxµdxν . (6.2)

The metric is called indefinite because ds2 can take on both positive and negative values, asis evident from Eq. (3.3). The lowering of indices by means of the covariant metric tensor,ηµν, is supplemented by a raising of indices with the help of the contravariant metrictensor, ηµν, as indicated for four-vectors in Eqs. (3.37) and (3.38). The two tensors arerelated by Eq. (3.50), and ηµν is numerically identical with ηµν : They are in matrix notationinverse matrices.

The generalization from flat space (Minkowski space [159, 160], also called pseudo-Euclidean space) to curved (four-dimensional (pseudo-)Riemann space [202]) is obtainedby replacing ηµν by the metric tensor gµν(x). The notion of an invariant “infinitesimalsquared distance” in curved space now is given by

ds2 = gµν(x)dxµdxν , (6.3)

where, as indicated, the metric tensor gµν(x) in general depends on the space-time po-sition, given by the curvilinear coordinates x ≡ xµ, or any set of reference coordinates.The metric tensor is required to be symmetric, gµν(x) = gνµ(x), and it has real elements. Asuitable choice of local coordinates therefore allows one to diagonalize the metric tensor ata given position, xµ. Different sets of local coordinates are needed to diagonalize gµν(x)at the various space-time points. The metric gµν(x) is indefinite, and we again speak ofspace-like (ds2 > 0) and time-like (ds2 < 0) separation of neighboring points xµ andxµ +dxµ. If ds2 = 0, the distance is called null. One usually, though not always, assumesthat the metric tensor is nondegenerate, meaning that the determinant, g = g(x), does notvanish:

g(x) = |gµν(x)| 6= 0, ∀x. (6.4)

At given point P it is possible to find coordinates such that the metric tensor (i) equals

55


the Minkowski metric tensor, and (ii) the first derivatives of the metric tensor vanish. Acoordinate system that satisfies these two conditions is called a local inertial frame (at P ) ingeneral relativity. Since the determinant of the covariant metric tensor, gµν(x), is differentfrom zero, it is possible to define an inverse contravariant metric tensor, gµν(x), via

gµαgαν = δµν ; (6.5)

cf. Eq. (3.50), which relates the covariant and contravariant forms of the metric tensor inflat space. In the tensor formalism of general relativity (Sec. 7.1), lowering and raising ofindices can be done by means of the covariant and contravariant forms of the metric tensor.

6.2 Time-like metric geodesics

Let us now study how free test particles move through the curved space-time of generalrelativity. By a test particle we mean a particle which has a mass so small that it producesno significant space-time curvature by itself. The test particle concept may be familiar tothe reader from electrodynamics. Here, an electrically charged test particle is a body whosemotion in an external electromagnetic field can be described without taking into accountthe field produced by the body itself. The question of the measurability of electromagneticfield quantities by means of charged test bodies raises deep questions in quantum electrody-namics. In classical electrodynamics fields should be measurable by means of point-like testcharges. In two landmark papers Bohr and Rosenfeld showed that in studying the measur-ability of field components in quantum electrodynamics one must use as test bodies chargeand current density distributions of linear extensions large compared to atomic dimensions[35, 36]. This conclusion emphasizes Bohr’s point of view that measurements always needto be described within the framework of classical physical ideas. In this section we considerthe motion of a particle with finite rest mass. Such a particle always moves along a time-likeworld line. In a notation where we for brevity set the speed of light to unity (c = 1 units),the (positive) infinitesimal proper time is given by

dτ = + (−gµν(x)dxµdxν)1

2 . (6.6)

The motion of our particle is governed by the so-called metric geodesic equation, and thisequation can be established from a variational principle that reads as follows: The worldline of a free test particle between two time-like separated points (P and Q) extremizes theproper time between them, that is

δ

∫ Q

P

dτ = 0. (6.7)

At this point, I remind the reader of Fermat’s principle for a light ray in an inhomogeneousvacuum (Sec. 5.3). A parameterization

τ = τ(u), (6.8)

where one without loss of generality lets the parameter u vary between u = 0 (at P ) andu = 1 (at Q), allows one to rewrite the variational principle in Eq. (6.7) in the form

δ

∫ 1

0

Ldu =

∫ 1

0

δLdu = 0, (6.9)

Geodesics in general relativity 57

where

L

(

xµ,

dxµ

du

)

=dτ

du=

(

−gµν(x)dxµ

du

dxν

du

)1

2

(6.10)

may be called the Lagrangian for the variational problem. As the reader may have antic-ipated by looking back at the variational problem described in Sec. 5.3, Eq. (6.9) leadsto Euler–Lagrange equations for the four space-time coordinates. These equations togetherconstitute what we call the time-like metric geodesic equation. Let me now give the detailsof the calculation. In the expression for the variation of the Lagrangian, i.e.,

δL =∂L

∂xµδxµ +

∂L

∂(dxµ/du)δ

(

dxµ

du

)

, (6.11)

we make use of the result

δ

(

dxµ

du

)

=d

duδxµ. (6.12)

An essentially analogous relation was given in Eq. (5.57) without proof. The proof of Eq.(6.12) runs as follows:

δ

(

dxµ

du

)

= δ lim∆u→0

(

∆xµ

∆u

)

= lim∆u→0

δ

[

xµ(u+∆u)− xµ(u)

∆u

]

= lim∆u→0

δxµ(u +∆u)− δxµ(u)

∆u

= lim∆u→0

δxµ(u) + ∆u dduδx

µ(u) +O(

(∆u)2)

− δxµ(u)

∆u

=d

duδxµ(u), (6.13)

q.e.d. Inserting Eq. (6.12) into Eq. (6.11) the variational problem in Eq. (6.9) takes theform

∫ 1

0

[

∂L

∂xµδxµ +

∂L

∂(dxµ/du)

d

duδxµ

]

du = 0. (6.14)

Remembering that δxµ vanishes at the endpoint of the integral, a partial integration of thelast term in Eq. (6.14) now gives

∫ 1

0

∂L

∂xµ− d

du

[

∂L

∂(dxµ/du)

]

δxµdu = 0, (6.15)

and since this equation must hold for all infinitesimally small variations around the worldline of the free test particle one is led to the four Euler–Lagrange equations

d

du

(

∂L

∂(dxµ/du)

)

− ∂L

∂xµ= 0, u = 0− 3 (6.16)

with L given by Eq. (6.10). In the next step one needs the partial derivatives of L withrespect to xµ and dxµ/du. With a replacement of the dummy indices (µ, ν) in Eq. (6.10)by (α, β), one obtains

∂L

∂xµ= − 1

2L

∂gαβ∂xµ

dxα

du

dxβ

du= −L

2

∂gαβ∂xµ

dxα

dτ

dxβ

dτ, (6.17)


where the last member follows from the fact that

d

dτ=

du

dτ

d

du=

1

L

d

du. (6.18)

Furthermore, since the metric tensor only depends on xµ,

∂L

∂(dxµ/du)= −gαβ

2L

∂

∂(dxµ/du)

(

dxα

du

dxβ

du

)

= − 1

2L

(

gµβdxβ

du+ gαµ

dxα

du

)

, (6.19)

and, in view of Eq. (6.18), thus

∂L

∂(dxµ/du)= −1

2

(

gµβdxβ

dτ+ gαµ

dxα

dτ

)

. (6.20)

By dividing Eq. (6.16) by L, and inserting the expressions in Eqs. (6.17) and (6.20) onegets, remembering the operator relation in Eq. (6.18), the intermediate result

d

dτ

(

gµβdxβ

dτ+ gαµ

dxα

dτ

)

− ∂gαβ∂xµ

dxα

dτ

dxβ

dτ= 0. (6.21)

Because the metric is (assumed to be) symmetric, the expression behind the d/dτ -operatorcan be written in the compact form

gµβdxβ

dτ+ gαµ

dxα

dτ= 2gµα

dxα

dτ, (6.22)

after having replaced the dummy index β by α. Hence, we first reach

d

dτ

(

gµαdxα

dτ

)

− 1

2

∂gαβ∂xµ

dxα

dτ

dxβ

dτ= 0, (6.23)

and then

gµαd2xα

dτ2+

dgµαdτ

dxα

dτ− 1

2

∂gαβ∂xµ

dxα

dτ

dxβ

dτ= 0. (6.24)

Since

dgµαdτ

=∂gµα∂xλ

dxλ

dτ, (6.25)

it appears that the second and third terms (sum) in Eq. (6.24) have the same structuralform, and one may write Eq. (6.24) as follows:

gµαd2xα

dτ2+

1

2

(

2∂gµα∂xλ

dxλ

dτ

dxα

dτ− ∂gαβ

∂xµdxα

dτ

dxβ

dτ

)

= 0. (6.26)

After some changes in the notation of certain dummy variables the reader may convinceherself that

2∂gµα∂xλ

dxλ

dτ

dxα

dτ=

(

∂gµβ∂xα

+∂gµα∂xβ

)

dxα

dτ

dxβ

dτ. (6.27)


Inserting this relation into Eq. (6.26) we find

gµαd2xα

dτ2+

1

2

[

∂gµβ∂xα

+∂gµα∂xβ

− ∂gαβ∂xµ

]

dxα

dτ

dxβ

dτ= 0. (6.28)

In order to remove the metric tensor in front of the acceleration term d2xα/dτ2, one mul-tiplies Eq. (6.28) by gγµ (summation over µ), and make use of Eq. (6.5). This gives

d2xγ

dτ2+

1

2gγµ

[

∂gµβ∂xα

+∂gµα∂xβ

− ∂gαβ∂xµ

]

dxα

dτ

dxβ

dτ= 0. (6.29)

The index relabelling γ → µ and µ → λ finally has led us from the variational principle inEq. (6.7) to the following equation of motion

d2xµ

dτ2+ Γµ

αβ

dxα

dτ

dxβ

dτ= 0, (6.30)

where

Γµαβ =

1

2gµλ (∂αgλβ + ∂βgλα − ∂λgαβ) (6.31)

is the so-called the Christoffel symbol (of the second kind) [remember ∂α = ∂/∂xα]. Notethat Γµ

αβ = Γµβα. Eq. (6.30) is the time-like geodesic equation for a massive free test particle

moving in a Riemann space with the metric given by Eq. (6.3). Together, µ = 0 − 3, thetime-like geodesics constitute the equation of motion for a test particle freely falling in agravitational field. In Minkowski space, where Γµ

αβ = 0, one regains the usual equation ofmotion for a free particle, viz.,

d2xµdτ2

= 0. (6.32)

The quantity

FµG(x) = −

Γµαβ

dxα

dτ

dxβ

dτ

(6.33)

expresses the gravitational field strength in terms of the Christoffel symbol, which involvesno derivatives of the metric tensor higher than the first. This circumstance gives the metrictensor the role as a gravitational potential in space-time.

6.3 The Newtonian limit: Motion in a weak static gravitationalfield

The equation of motion for a massive particle in a Newtonian gravitational field can bederived from the time-like geodesic equation by three requirements: (i) The particles aremoving slowly with respect to the speed of light, (ii) the field is so weak that one can adopta nearly Cartesian coordinate system in which the metric is close to that of flat space, and(iii) the field is (essentially) static.

Moving slowly in Minkowski space, where

(

dt

dτ

)2

− 1

c2dx

dτ· dxdτ

= 1, (6.34)


cf. Eq. (3.3), means that

1

c

∣

∣

∣

∣

dxi

dτ

∣

∣

∣

∣

≪∣

∣

∣

∣

dt

dτ

∣

∣

∣

∣

≈ 1, i = 1− 3. (6.35)

The condition |dxi/dτ | ≪ |dt/dτ | [in c = 1 units] thus is our “slow motion” criterion alsowhen the metric deviates slightly from the Minkowski metric in Eq. (6.1). In the slow-motionapproximation the geodesic equation in (6.30) is reduced to

d2xµ

dτ2+ Γµ

00

(

dt

dτ

)2

= 0, (6.36)

where, from Eq. (6.31),

Γµ00 =

1

2gµλ (2∂0gλ0 − ∂λg00) , (6.37)

remembering that the metric tensor is symmetric. With the assumption that the gravita-tional field is static (∂0gλ0 = 0), the relevant Christoffel symbol becomes

Γµ00 = −1

2gµλ∂λg00. (6.38)

The weakness of the field allows us to take

gµν = ηµν + hµν , |hµν | ≪ 1 (6.39)

where hµν is the small perturbation to the covariant Minkowski metric. From Eq. (6.5) wefind that the contravariant metric to first order in h is given by

gµν = ηµν − hµν , (6.40)

where hµν = ηµαηνβhαβ . To first order in hµν we thus have

Γµ00 = −1

2ηµν∂νh00. (6.41)

The static-field assumption, here ∂0h00 = 0, tells us that Γ000 = 0, and the µ = 0 component

of the geodesic equation in Eq. (6.36) therefore gives d2t/dτ2 = 0. In agreement with theapproximation in (6.35), dt/dτ hence is a constant (≈ 1). Since Γi

00 = − 12∂ih00, i = 1− 3,

the spatial components of the geodesic equation satisfy

d2xi

dτ2− 1

2∂ih00

(

dt

dτ

)2

= 0, i = 1− 3. (6.42)

After multiplication by (dτ/dt)2, and when written in three-vector form, the geodesic equa-

tion becomes

d2x

dt2=

1

2∇h00. (6.43)

Eq. (6.43) has precisely the Newtonian equation of motion form, i.e.,

d2x

dt2= −∇Φ ≡ g. (6.44)


Considering gravitational fields generated exclusively in a finite region of space, and settingthe Newtonian gravitational potential Φ to zero at infinity, we have the identification

h00 = −2Φ, (6.45)

and hence

g00 = − (1 + 2Φ) . (6.46)

Under the appropriate approximations the geodesic equation for a massive (electricallyneutral) particle therefore leads back to Newtonian mechanics.

6.4 Null geodesics and “light particles”

In Sec. 5.2, we reflected on the relation between the eikonal theory and the particle pictureof light, and we realized that the ith component of the differential equation for a light raysatisfies

d2xi(s)

ds2= 0 (6.47)

in empty (n = 1) flat space; cf. Eq. (5.44). Instead of using the Euclidean length s alongthe ray as parameter we may take another (affine) parameter σ, related to s by σ = as+ b,as independent variable. With this, the differential equation in Eq. (6.47) turns into

d2xi(σ)

dσ2= 0. (6.48)

For a light ray one cannot use the proper time, τ , as independent variable because dτ = 0,cf. Eq. (3.3). According to the Principle of Equivalence [241], there exists locally, for aparticle moving freely under the influence of purely gravitational forces, a freely fallingcoordinate system, ζα, in which the equation of motion of the particle is such that theparticle propagates along a straight line:

d2ζα(σ)

dσ2= 0. (6.49)

For a “light particle” the proper time is zero, that is,

−ηαβdζα

dσ

dζβ

dσ= 0. (6.50)

Starting from Eqs. (6.49) and (6.50) a transformation to curved-space coordinates withmetric gµν(x) leads to the geodesic equation [241]

d2xµ

dσ2+ Γµ

αβ

dxα

dσ

dxβ

dσ= 0, µ = 0− 3 (6.51)

Γµαβ being the Christoffel symbol [Eq. (6.31)], and to the constraint

−gµνdxµ

dσ

dxν

dσ= 0. (6.52)


In some sense, Eqs. (6.51) may be considered as the equation of motion for a masslessparticle [photon, neutrino (approximately),...] in general relativity. From Eq. (6.52) onefinds that the time dt it takes for a massless particle to travel a distance dx is given by

dt =1

g00

−gi0dxi −[

(gi0gj0 − gijg00) dxidxj

]1

2

, (6.53)

as the reader may show to herself. The time required for the massless particle to travelalong an arbitrary path between given points may be calculated by integrating dt along thepath.

When the considerations above are applied to the photon (light particle) one must notforget that this particle is only weakly localizable in space-time. In a classical framework, onemust ultimatively discuss the geometrical optical (particle) approach in general relativity,starting from a generalization of the Maxwell equations to curved space-time; see Chapt. 8.Curves that satisfy Eq. (6.51) with the vanishing proper time constraint in Eq. (6.52) arecalled null geodesics.

6.5 Gravitational redshift. Photon in free fall

The Principle of Equivalence [180, 241, 66] rests on the equality of gravitational and inertialmass, and the principle may be formulated as the following statement: At every space-timepoint in an arbitrary gravitational field it is possible to choose a locally inertial (freelyfalling) system, such that all laws of nature, within a sufficiently small region of the pointin question, take the same form as in an unaccelerated coordinate system in the absence ofgravitation. By “form in an unaccelerated coordinate system” we mean the form given tothe laws of nature in Special Relativity.

The Principle of Equivalence allows one to obtain an expression for the so-called gravi-tational redshift of light [241, 63, 65]. To illustrate the phenomenon let us consider the lightemitted (at the Bohr frequency) by a particular atomic transition from an atom located atthe fixed position P1 in space. Let the coordinate time separation between successive wavecrests measured by an observer at P1 be ∆t1, with corresponding proper time ∆τ1. FromEq. (6.6) it follows that

∆t1 = (−g00(x1))−1

2 ∆τ1. (6.54)

In a local (at P1) freely falling (FF) coordinate system (coordinates denoted by xµFF ) theproper time interval is calculated from

∆τ1 = (−ηµνdxµFF dxνFF )

1

2 . (6.55)

Let us next observe the light coming from P1 at another fixed position P2, and let us assumethat the gravitational field is stationary. The time t1→2 it takes for a given wave crest topropagate from P1 to P2 is given by the integral of Eq. (6.53) over the path, i.e.,

t1→2 =

∫ P2

P1

dt. (6.56)

Since the gravitational field is stationary, the time of travel of successive wave crests willalso be t1→2. The coordinate time separation between successive crests, coming from theemitter atom located at P1, and arriving at P2 will therefore be given by Eq. (6.54). If


now an identical atom is placed at P2 and the same atomic transition is observed at P2 thecoordinate time interval between the light wave crests (at the Bohr frequency) will be

∆t2 = (−g00(x2))−1

2 ∆τ2. (6.57)

The proper times ∆τ1 and ∆τ2 relating to free falls at P1 and P2 will not be the same forthe two identical atoms but we must have ∆t1 = ∆t2. Because of this identity the ratio ofthe Bohr frequencies (measured by clocks at P1 and P2) ν1 = 1/∆τ1 and ν2 = 1/∆τ2 hencebecomes

ν2ν1

=

(

g00(x1)

g00(x2)

)1

2

. (6.58)

In the weak-field limit [Φ(x)/c2 ≪ 1]

g00(x) = −(

1 + 2Φ(x)

c2

)

, (6.59)

in a notation where the speed of light has been reinserted. [With the replacement Γµ00 →

Γµ00c

2 in Eq. (6.36), Eq. (6.46) is turned into Eq. (6.59)]. By combining Eqs. (6.58) and(6.59), the now small relative frequency shift [|ν2 − ν1| ≪ ν1 ≈ ν2] is given by

ν2 − ν1ν1

=1

c2[Φ(x1)− Φ(x2)] , (6.60)

the well known expression for the gravitational redshift. [The frequencies are compared atP2, and if Φ(x1) < Φ(x2), the light coming from P1 has a higher frequency (ν1) than thatcoming from P2 (ν2)]. A schematical illustration of the gravitational redshift is shown inFig. 6.1.

The gravitational redshift can be understood from a somewhat different perspectivewhich relates to quantum theory, and in particular the photon measurement problem (seeChapt. 32). Before giving details it is important to recall the following two central pointsof Niels Bohr [32, 33, 34]: (i) I advocate the application of the word phenomenon exclusivelyto refer to observations obtained under specified circumstances, including an account ofthe whole experimental arrangement. (ii) However far the phenomena transcend the scopeof classical physical explanation, the account of all evidence must be expressed in classicalterms. A free photon is an abstraction in the sense that it cannot be observed, only itscreation and absorption during field-matter interaction processes leave a fingerprint.

When a photon is generated at P1 by an atom necessarily coupled to a heavy non-relativistic apparatus, an observer in a local inertial coordinate system moving with theapparatus will see its rest energy change from M1c

2 to (M1 +∆M1)c2, where

∆M1c2 = −hν1, (6.61)

ν1 being the frequency of the emitted photon. When the photon later is annihilated at P2

by an identical atom coupled to a second heavy nonrelativistic apparatus, an observer ina freely falling frame related to P2 will see the rest energy of this apparatus change fromM2c

2 to (M2 +∆M2)c2, where

∆M2c2 = hν2, (6.62)

ν2 being the photon frequency the second observer sees. Now, the total energy of the twopieces of apparatus must be the same before the photon was emitted and after it has been


∆t1

∆t1 = ∆t2

v2

v1

Ф1

Ф2

FIGURE 6.1Gravitational redshift. Two identical atoms placed at rest at two points (P1 and P2) ina stationary gravitational field emit light (at the Bohr frequency) from the same atomictransition. The gravitational potentials at the two points are Φ1 and Φ2. Let the coordinatetime separation between successive wave crests measured by an observer at P1 be ∆t1.Since the time of travel between P1 and P2 will be the same for successive wave crests in astationary field, the coordinate time separation also will be ∆t1 at P2. The coordinate timeinterval ∆t2 between successive crests emitted from the atom located at P2, and measuredby an observer at P2, will equal ∆t1 because the atoms are identical. The gravitationalredshift (ν2 − ν1)/ν1 = Φ2 − Φ1 follows from the equality ∆t1 = ∆t2.

absorbed. The total energy is the sum of the rest energy, the gravitational potential energy,and the kinetic energy. Since the kinetic-energy part is negligible for the heavy masses,energy conservation gives

∆M1

(

c2 +Φ1

)

+∆M2

(

c2 +Φ2

)

= 0, (6.63)

and when Eqs. (6.61) and (6.62) are used to eliminate the rest mass changes, we finallyobtain

hν1

(

1 +Φ1

c2

)

= hν2

(

1 +Φ2

c2

)

. (6.64)

The result in Eq. (6.64) is in agreement with Eq. (6.60).If one considers the quantity

Etot = hν

(

1 +Φ

c2

)

(6.65)

as the total energy of the photon in a weak static gravitational field, one may say that the


photon in the gravitational field has a “kinetic” energy

Ekin = hν, (6.66)

and a “potential” energy

Epot = hνΦ

c2, (6.67)

and that the sum of the two is a conserved quantity. Remembering Bohr’s lesson concerningthe definition of a phenomenon, the potential energy concept for the photon is withoutfoundation unless the whole experimental setup is included in our considerations. The in-troduction of the (kinetic) energy concept for photons in Special Relativity also relates toa closed event (phenomenon). For a general static field, where the frequency ratio is givenby Eq. (6.58), one may ascribe a potential energy

Epot = hν[

(−g00(x))1

2 − 1]

(6.68)

to the photon.

7

The space-time of general relativity

In preparation for a generalization of the Maxwell–Lorentz theory to curved space-time, webriefly discuss the space-time structure of general relativity. The presentation will enablethe reader without previous knowledge of general relativity to understand basic aspects ofclassical electrodynamics (and optics in the absence of charged particles) in curved space-time. For a comprehensive treatment of the fundamental aspects of general relativity thereader may consult, e.g., [241, 238, 93]. A valuable introduction to general relativity is givenin [47].

7.1 Tensor fields

To generalize the tensor-field description one must work with curvilinear rather than rec-tilinear coordinates. Let x ≡ xµ (µ = 0 − 3) be such coordinates of a given space-timepoint. A transformation to another set (x′ ≡ x′µ) of curvilinear coordinates of the samepoint, i.e.,

xµ → x′µ = fµ(x), (7.1)

transforms the differential element dxµ according to

dxµ → dx′µ =∂x′µ

∂xνdxν , (7.2)

assuming differentiability of the transformation in Eq. (7.1). A contravariant vector V µ(x)(tensor of rank one) is a set of four quantities which transform in the same way as dxµ:

V µ(x) → V ′µ(x′) =∂x′µ

∂xνV ν(x), (7.3)

where, as indicated, the quantities may be functions of x. We call V µ(x) a vector field. Ascalar (function) [tensor (field) of rank zero] S(x), is invariant under a coordinate transfor-mation, that is

S(x) → S′(x′) = S(x). (7.4)

We always assume that any coordinate transformation has an inverse

x′µ → xµ = gµ(x′), (7.5)

with

dx′µ → dxµ =∂xµ

∂x′νdx′ν . (7.6)

67


From matrix theory we know that the existence of the inverse transformation requires thatthe Jacobian (J), i.e., the determinant of the matrix ∂x′µ/∂xν does not vanish:

J ≡ Det

∂x′µ

∂xν

≡∣

∣

∣

∣

∂x′µ

∂xν

∣

∣

∣

∣

6= 0. (7.7)

A covariant vector Wµ(x) (field) is a rank one tensor (field) which components trans-form according to

Wµ(x) →W ′µ(x

′) =∂xν

∂x′µWν(x). (7.8)

The transformation coefficients in Eq. (7.8) are the inverse of those in Eq. (7.3). Since, fora scalar field

∂′µS′(x′) ≡ ∂S′(x′)

∂x′µ=∂S(x)

∂x′µ=

∂xν

∂x′µ∂S(x)

∂xν=

∂xν

∂x′µ∂νS(x), (7.9)

the derivative of a scalar [∂µS(x)] transforms like a covariant vector, and not like a con-travariant vector. Eqs. (7.3) and (7.8) are generalizations of those in Minkowski space, i.e.,Eqs. (3.22) and (3.23) with derivatives given by the Lorentz transformation [Eqs. (3.31) and(3.32)]. Like in Special Relativity [Eq. (3.25)] the product of a covariant and a contravariantvector is an invariant in general relativity:

V ′µ(x′)W ′µ(x

′) = V ν(x)Wν (x). (7.10)

The metric tensor can be used to define a covariant vector when a contravariant is given.Thus, the infinitesimal squared distance takes the form of a scalar product,

ds2 = dxµdxµ, (7.11)

if one makes the definition

dxµ ≡ gµνdxν . (7.12)

Generalized to arbitrary vectors, one is led to the definition

Vµ(x) ≡ gµν(x)Vν(x). (7.13)

The inverse metric gµν(x), introduced in Eq. (6.5), permits one to obtain a contravariantvector from a covariant one. Hence,

V µ(x) ≡ gµν(x)Vν(x), (7.14)

as the reader may show by multiplying Eq. (7.13) by g−1αµ from the left (again with summa-

tion over the repeated index µ).The generalization of tensors in Minkowski space to general relativity is obvious, cf. the

examples given for rank two tensors in Eqs. (3.28)-(3.30). The invariance of ds2 [Eq. (6.3)]readily shows that the metric gµν(x) transforms as

g′µν(x′) =

∂xα

∂x′µ∂xβ

∂x′νgαβ(x), (7.15)

and therefore is a covariant second rank tensor. By using the transformation in Eq. (7.14)on the infinitesimal vector components dxµ and dxν , and thereafter Eq. (6.5), it appearsthat ds2 [Eq. (6.3)] can be written in the alternative form

ds2 = gµνdxµdxν . (7.16)

The invariance of ds2 then shows that gµν(x) transform as a contravariant tensor, indeed.

The space-time of general relativity 69

7.2 Covariant derivative

The derivative of a vector V ν(x), ∂µV ν(x), is not a tensor in general. To prove thisone makes use of Eq. (7.3) to transform V ′ν(x′), and Eq. (7.8) for the transformation of thecovariant vector ∂′µ [see Eq. (3.27)]. Hence, one obtains

∂′µV′ν(x′) = (∂′µx

α)∂α[

(∂βx′ν)V β(x)

]

= (∂′µxα)(∂βx

′ν)∂αVβ(x) + (∂′µx

α)(∂α∂βx′ν)V β(x).

(7.17)

Only when the last term in this equation vanishes will the covariant derivative of a con-travariant vector transform as a tensor. But this term is not zero unless

∂α∂βfν(x) = 0. (7.18)

To satisfy Eq. (7.18) the coordinate transformation in Eq. (7.1) must be linear. This isfulfilled for the Lorentz transformation in Minkowski space [Eq. (3.1)], not unexpectedly.

An alternative version of the Principle of Equivalence, known as the Principle of GeneralCovariance states that the forms of physical laws in Special Relativity (gαβ(x) = ηαβ)preserve their forms under a general coordinate transformation (x→ x′ ⇒ ηαβ → gαβ(x)).In relation to electrodynamics, as described by the Maxwell–Lorentz equations in (3.59) and(3.66), a generalization from flat to curved space-time first of all requires the replacementof the usual derivative, ∂µ, by a covariant derivative, µ. We define µ in terms of itsaction on a contravariant vector V ν(x) as follows:

µVν(x) ≡ ∂µV

ν(x) + Lνµα(x)V

α(x), (7.19)

where Lνµα(x) is a linear connection. The purpose of the last term in Eq. (7.19) is to “correct”

the nontensorial form ∂µV ν(x) in such a manner that µVν(x) becomes a tensor. The

connection is called linear because the last term in Eq. (7.19) is linear in V α(x). SinceµV

ν(x) is required to be a tensor (field), and ∂µV ν(x) is not, the connection, Lνµα

cannot be a tensor.The linear connection Lν

µα can be expressed in terms of the metric tensor and its firstderivatives if Eq. (7.19) shall be compatible with the geodesic equation

d2xµ

dσ2+ Γµ

αβ

dxα

dσ

dxβ

dσ= 0, (7.20)

here written in terms of an affine parameter σ. The form in Eq. (7.20) covers both time-likeand null geodesics. The time-like geodesic equation, given in Eq. (6.30), is recovered withthe choice σ = aτ + b. The quantity

T µ(σ) =dxµ(σ)

dσ(7.21)

is the µth component of a tangent vector, T µ(σ, to the geodesics. Since

dT µ

dσ=

dT µ

dxαdxα

dσ= Tα∂αT

µ, (7.22)

the geodesic equation may be written in the form

Tα(

∂αTµ + Γµ

αβTβ)

= 0. (7.23)


If one applies Eq. (7.19) to the four-vector T µ, and inserts the resulting expression for∂αT

µ into Eq. (7.23) one obtains

Tα[

αTµ +

(

Γµαβ − Lµ

αβ

)

T β]

= 0. (7.24)

Because dxµ is a tensor [see Eq. (7.6)], T µ will be a tensor. In turn this means that thequantity in the square bracket of Eq. (7.24) must be a tensorial component. In consequence,Γµ

αβ−Lµαβ must be a tensor. The Christoffel symbol Γµ

αβ is symmetric in the lower indices,

and therefore the antisymmetric part of the linear connection (Lµαβ − Lµ

βα)/2 is a tensor.

The quantity Lµαβ − Lµ

βα is called the torsion tensor. Restricting ourselves to symmetricconnections, the torsion tensor vanishes, and Eq. (7.24) takes the tensorial form

Tα α Tµ = 0, (7.25)

provided

Lµαβ = Γµ

αβ . (7.26)

When the affine connection equals the Christoffel symbol, and the metric is given by Eq.(6.3), the space is called a Riemann space, or a pseudo-Riemann space (because the metric isindefinite). In order to be compatible with the geodesics (of extremum length), and torsion-free connection, the elements of the covariant derivative of a contravariant vector must begiven by

µVν(x) = ∂µV

ν(x) + Γνµα(x)V

α(x). (7.27)

For the covariant derivative of a covariant vector one has [241]

µVν(x) = ∂µVν(x) − Γαµν(x)Vα(x). (7.28)

The concept covariant derivative can be extended from rank one tensors to tensors of ar-bitrary rank [241, 238, 93, 47]. For scalars the covariant derivative is just the ordinarygradient; cf. Eq. (7.9).

7.3 Parallel transport

Starting from the covariant derivative ≡ µ, we now define the directional covariantderivative along a curve xµ(σ) [independent variable given as the affine parameter σ], asfollows:

D

Dσ≡ dxα

dσα . (7.29)

We next define parallel transport of a tensor along the path xµ(σ) to be the require-ment that the covariant derivative of this tensor along the path vanishes. Note that thecomponents of the tangential vector, Tα(σ) = dxα

dσ [Eq. (7.21)] enter Eq. (7.29) in a directmanner.

To exemplify the concept of parallel transport, let us consider the parallel transportrequirement for the tangential vector, viz.,

DT µ

Dσ= 0, (7.30)


in the case where the linear connection is the Christoffel symbol. We immediately realizethat

DT µ

Dσ= Tα α T

µ = 0, (7.31)

but this is just the geodesic equation. We thus have an elegant formulation of the variationalprinciple for a metric geodesic: A geodesic is a curve along which the tangent vector isparallel transported. For time-like paths it is often convenient to write the geodesic equationin terms of the particle four-velocity

vµ =d

dτxµ, (7.32)

that is,

vα α vµ = 0. (7.33)

Alternatively, one may use the four-momentum

pµ = mvµ, (7.34)

where m is the particle’s rest mass. The geodesic equation

Dpµ

Dτ= pα α p

µ = 0 (7.35)

now, in a certain sense, expresses the idea that a freely falling particle in general relativitykeeps propagating in the direction in which its momentum is pointing. In flat space-time (orin a local inertial frame) we recover the fact that the particle in Euclidean space is movingalong a straight line with constant momentum.

The form given in Eq. (7.35) is correct also for null geodesics, assuming pµ to be thefour-momentum of the “light particle.” If we follow the conventional reasoning for particletrajectories, the affine parameter (σ) is chosen such that

?pµ = dxµ/dσ (7.36)

equals the four-momentum of the “light particle.” One should emphasize here that in thelanguage of photon physics, pµ is the well-defined four-momentum of a monochromaticphoton. The question mark on top of the equality sign, is meant to indicate that a monochro-matic photon (in Special Relativity) is completely unlocalized in space, so that the xµ inEq. (7.36) is a meaningless quantity. A photon can only be weakly localized in space-time,and the best localized photons are far from being monochromatic. The best localized freephoton is not in an eigenstate for the momentum operator. This has the implication thatthe left side of Eq. (7.36) tends to be ill-defined, once the right-hand side becomes betterdefined. We shall return to the photon localization problem in Part V.

7.4 Riemann curvature tensor

In the covariant theory of photon wave mechanics and quantum electrodynamics one startsfrom the Maxwell–Lorentz equations in the four-potential description. In flat space-time


the important Lorenz gauge formulation is reached making use of the commutativity ofderivatives; i.e.,

[∂µ, ∂ν ] ≡ ∂µ∂ν − ∂ν∂µ = 0. (7.37)

The most natural generalization of the Maxwell–Lorentz equations to curved space-time isgiven by the so-called minimal substitution rule: ηµν → gµν , ∂µ → µ. In the potentialformulation of electrodynamics one encounters an important deficiency in this rule becausethe covariant derivatives µ and ν do not commute as we shall prove below.

The forms given in Eqs. (7.27) and (7.28) for the covariant derivatives of contra andcovariant vectors also hold if the Christoffel symbol is replaced by the more general linearconnection, Lν

µα (Lαµν), cf. Eqs. (7.19) and (7.27). This implies that the covariant derivative

of a mixed second rank tensor (field), T νµ (x), generally is given by

σTν

µ (x) = ∂σTν

µ (x)− Lλσµ(x)T

νλ (x) + Lν

σλ(x)Tλ

µ (x). (7.38)

Let T µβ = βA

µ be our tensor, and then apply Eq. (7.38) to this tensor. Thus,

α (βAµ) = ∂α (βA

µ)− Lλαβ λ A

µ + Lµαλ β A

λ

= ∂α

(

∂βAµ + Lµ

βλAλ)

− Lλαβ (∂λA

µ + LµλσA

σ) + Lµαλ

(

∂βAλ + Lλ

βσAσ)

= ∂α∂βAµ +

(

∂αLµβλ

)

Aλ + Lµβλ∂αA

λ − Lλαβ∂λA

µ

+ Lµαλ∂βA

λ − LλαβL

µλσA

σ + LµαλL

λβσA

σ (7.39)

An analogous expression for β (αAµ) is obtained by making the index exchange α↔ β

in Eq. (7.39). If we assume that the connection is torsion-free, i.e.,

Lµαβ = Lµ

βα, (7.40)

the reader may easily prove to herself that

[α,β]Aµ ≡ (α β −β α)A

µ

=(

∂αLµβλ

)

Aλ − (∂βLµαλ)A

λ + LµαλL

λβσA

σ − LµβλL

λασA

σ

=(

∂αLµβν

)

Aν − (∂βLµαν)A

ν + LµαλL

λβνA

ν − LµβλL

λανA

ν , (7.41)

where the last member of the equation follows after relabelling of certain dummy indices.One usually writes Eq. (7.41) in the form

[α,β]Aµ(x) = Rµ

ναβ(x)Aν (x), (7.42)

where

Rµναβ = ∂αL

µβν − ∂βL

µαν + Lµ

αλLλβν − Lµ

βλLλαν (7.43)

is the so-called Riemann curvature tensor. The quantity Rµναβ is indeed a tensor, despite

the fact that the linear connection is not a tensor, because the left side of Eq. (7.42) and Aν

are tensor components. The result above is easily extended to the case where the torsiontensor

Lναβ − Lν

βα ≡ 2Lν[αβ] (7.44)


is nonvanishing. Hence,

[α,β]Aµ(x) = Rµ

ναβ(x)Aν (x) − 2Lν

[αβ](x)ν Aµ(x). (7.45)

The inclusion of the torsion term does not change the circumstance that Rµναβ is a ten-

sor because the antisymmetric part of the linear connection is a tensor. In physics, theChristoffel symbol is the important connection, and the related Riemann curvature tensor,

Rµναβ = ∂αΓ

µβν − ∂βΓ

µαν + Γµ

αλΓλβν − Γµ

βλΓλαν , (7.46)

is the one of interest in photon wave mechanics. Since the Christoffel symbol depends on firstderivatives of the metric tensor, the Riemann curvature tensor clearly depends on secondderivatives of gµν(x).

The notion of curvature can be defined in terms of the failure of the commutativity ofsuccessive covariant differentiations on tensor fields, cf. Eq. (7.42). Lack of commutativityof derivatives implies that a vector parallel transported around an infinitesimal closed curvedoes not return to its original value [206, 238, 47].

7.5 Algebraic properties of the Riemann curvature tensor

In order to investigate the algebraic properties of the curvature tensor it is convenient toconsider, instead of Rµ

ναβ given in Eq. (7.46), its fully covariant form

Rµναβ = gµσRσναβ . (7.47)

I leave the main part of the calculation of Rµναβ as an exercise for the reader. Just a fewwords on the strategy plus an important relation are given here. The expression for theChristoffel symbol [Eq. (6.31)] is inserted into the first two terms on the right side of Eq.(7.46) [with µ replaced by σ]. For the term ∂αΓ

σβν we then obtain

gµσ∂αΓσβν =

1

2gµσ

(

∂αgσλ)

(∂βgλν + ∂νgλβ − ∂λgβν)

+1

2gµσg

σλ (∂α∂βgλν + ∂α∂νgλβ − ∂α∂λgβν) . (7.48)

The terms with second derivatives of the metric tensor we leave as they stand. The quantitygµσ∂αg

σλ is rewritten by means of the relation

∂α(

gµσgσλ)

= ∂αδλ

µ = 0. (7.49)

Hence,

gµσ∂αgσλ = −gσλ∂αgµσ = −gσλ

(

Γηαµgησ + Γη

ασgηµ)

. (7.50)

The last member of Eq. (7.50) follows from the fact that the covariant derivative of themetric tensor is zero. We shall prove this directly in Sec. 7.7, where the metric compatibilityconcept is introduced and discussed. The same kind of calculation as indicated above is usedto rewrite the term −gµσ∂βΓσ

αν . At this stage we have terms in Rµναβ containing secondderivatives of the metric tensor, and terms containing products of two Christoffel symbols


and the metric. The final result for the fully covariant form of the curvature tensor readsas follows:

Rµναβ =1

2(∂β∂νgµα − ∂β∂µgνα − ∂α∂νgµβ + ∂α∂µgνβ) + gησ

(

ΓηαµΓ

σνβ − Γη

βµΓσνα

)

.

(7.51)

Remembering that the Christoffel symbol is symmetric in the lower indices, the result inEq. (7.51) for the curvature tensor has a symmetry

Rµναβ = Rαβµν , (7.52)

an antisymmetry

Rµναβ = −Rνµαβ = −Rµνβα = Rνµβα, (7.53)

and a cyclicity

Rµναβ +Rµβνα +Rµαβν = 0 (7.54)

in the last three indices.By contracting Rµναβ with gµα one gets the symmetric Ricci tensor

Rνβ = gµαRµναβ . (7.55)

Contraction of Rµναβ with both gµν and gαβ gives zero, as the reader may prove from theantisymmetry property [Eq.(7.53)]. From the antisymmetry property it also follows that(essentially) the only way to construct a nonzero (Ricci) scalar (R) from the curvaturetensor by contracting with the metric tensor twice is as follows:

R = gµαgνβRµναβ . (7.56)

It can be shown that the constraints in Eqs. (7.52)-(7.54) for a N × N matrix results inN2(

N2 − 1)

/12 independent components. For N = 4, the curvature tensor thus has 20independent components.

7.6 Einstein field equations in general relativity

From the Ricci tensor and the scalar curvature one defines the Einstein tensor, Gµν, via

Gµν ≡ Rµν − 1

2Rgµν . (7.57)

It turns out [206, 241] that the variational principle

δ

∫

R√−gd4x = 0 (7.58)

leads to the Euler–Lagrange equations

Gµν = 0. (7.59)


In general relativity, as well as in Special Relativity, the so-called energy-momentum tensor,Tµν, defined in terms of the density and current of the energy-momentum four-vector,pµ of a particle, plays an important role [206, 241, 238, 93, 47]. Hence, the Einstein fieldequations in general relativity, i.e. [206, 241, 238, 93, 47],

Gµν = 8πGTµν (7.60)

relate the Einstein and energy-momentum tensors. The quantity G is Newton’s gravitationalconstant, and units with c = 1 are used. In c 6= 1 units the factor 8πG is replaced by 8πG/c4.

It is often useful to rewrite the Einstein field equations in a slightly different form. Forthis purpose, let us double contract the Einstein tensor Gµν with gµν :

gµνGµν = gµνRµν − 1

2gµνgµνR. (7.61)

The first part of the right side of this equation is just the Ricci scalar [cf. Eqs. (7.55) and(7.56)]. In the second part

gµνgµν = 4. (7.62)

Hence,

gµνGµν = −R. (7.63)

By introduction of the scalar

T ≡ gµνTµν , (7.64)

the contracted Einstein equation gives

R = −8πGT. (7.65)

By eliminating R in favor of T in Eq. (7.57) one realizes that Einstein field equations canbe written as follows:

Rµν = 8πG

(

Tµν − 1

2gµνT

)

. (7.66)

In vacuum the Einstein equations therefore are simply

Rµν = 0. (7.67)

It follows readily from the symmetry in Eq. (7.52) that the Ricci tensor is symmetric (asalready mentioned), and at first sight this seems to imply that there are ten independentEinstein field equations. The Bianchi identity [241, 47]

µGµν = 0, (7.68)

which we do not prove here, represents four constraints on the functions Rµν(x), so thereare in fact only six independent Einstein equations. In turn this means that the symmetricmetric tensor, gµν, with its ten independent elements has four nonphysical degrees offreedom.


7.7 Metric compatibility

A linear connection, Lνµα, is called metric compatible if the covariant derivative of the metric,

gµν , is zero, everywhere, that is

αgµν(x) = 0. (7.69)

Metric compatibility makes it easy to raise or lower an index on a tensor behind a covariantderivative. Thus, for a contravariant vector field V α one has

gνα µ Vα = µ (gναV

α) = µVν . (7.70)

If we assume that the connection is torsion free the metric compatibility condition in Eq.(7.69) allows us to derive a unique expression for the connection coefficients. To prove this,we start from the general expression for the covariant derivative of the metric tensor, viz.,

αgµν = ∂αgµν − Lλαµgλν − Lλ

ανgµλ = 0. (7.71)

Cyclic permutation of the indices α, µ, and ν in Eq. (7.71) leads to the two relations

µgνα = ∂µgνα − Lλµνgλα − Lλ

µαgνλ = 0, (7.72)

νgαµ = ∂νgαµ − Lλναgλµ − Lλ

νµgαλ = 0. (7.73)

By subtracting Eqs. (7.72) and (7.73) from Eq. (7.71), one obtains

∂αgµν − ∂µgνα − ∂νgαµ + 2Lλµνgλα = 0, (7.74)

under the assumption that the connection is symmetric. Now, the connection is easilydetermined by multiplying Eq. (7.74) by the inverse metric gσα [satisfying Eq. (6.5)]. Hence,

Lσµν =

1

2gσα (∂µgνα + ∂νgαµ − ∂αgµν) ≡ Γσ

µν . (7.75)

Metric compatibility therefore requires that the connection must be the Christoffel symbol,provided we limit ourselves to torsion-free connections.

7.8 Geodesic deviation of light rays

Let us consider a one-parameter (s) family of geodesic light rays. For a given s, the curvexµs = xµs (σ) satisfies the geodesic equation given in Eq. (7.20), σ being a suitable affineparameter for the null geodesics. The collection of the geodesic curves, which we assume donot cross, defines a smooth two-dimensional surface and we therefore write xµs (σ) ≡ xµ(σ, s).The quantity

T µ(σ, s) =∂xµ(σ, s)

∂σ(7.76)

is the µ-component of a tangent vector, T µ, to the geodesics (fixed s). The first-orderchange in the geodesic curves of neighboring rays, i.e.,

xµ(σ, s+ δs)− xµ(σ, s) =∂xµ(σ, s)

∂sδs, (7.77)


thus is characterized by the so-called deviation vector, Sµ, having components

Sµ(σ, s) =∂xµ(σ, s)

∂s, (7.78)

with fixed σ. A schematical illustration of the geodesic deviation concept is shown in Fig.7.1.

>>

d

c Tmm

+ ss

s

s

FIGURE 7.1Schematic illustration of the geodesic deviation concept. Two neighboring geodetic lines(light rays) parameterized by s and s+δs, and with affine parameter σ, are shown. The whitearrow indicates the local tangent vector T µ(σ, s) at (σ, s). The first-order change (devia-tion) in the geodesic curves of neighboring rays (for fixed σ), χµ(σ, s), is indicated by theblack arrow. In terms of the deviation vector Sµ(σ, s) one has χµ(σ, s) = Sµ(σ, s)δs.Initially parallel geodesics will fail to remain parallel if the Riemann curvature tensor Rµ

αβν

is nonvanishing.

The rate of change of the deviation Sµ along infinitesimally nearby geodesics is givenby the directional derivative

V µ ≡ DSµ

Dσ= Tα α S

µ. (7.79)

Loosely speaking, V µ may be interpreted as the “relative velocity” with which the nearbygeodesics move relative to each other when we go along the rays. Similarly, the directionalderivatives of V µ, i.e.,

Aµ =DV µ

Dσ= T β β V

µ = T β β (Tα α Sµ) (7.80)

characterize the “relative acceleration” Aµ. It turns out, as we shall soon realize, that therelative acceleration is zero if and only if the Riemann curvature tensor vanishes. To obtainthe connection between the relative acceleration and the curvature tensor, it is convenientto make use of the relation

Tα α Sµ = Sα α T

µ. (7.81)

To prove Eq. (7.81) we utilize the expression given in Eq. (7.27) for the covariant derivativeof a contravariant vector. Hence, we get

Tα α Sµ − Sα α T

µ = Tα(

∂αSµ + Γµ

αβSβ)

− Sα(

∂αTµ + Γµ

αβTβ)

= Tα∂αSµ − Sα∂αT

µ = 0 (7.82)


where the third member of this equation follows from the symmetry of the Christoffelsymbol, Γµ

αβ = Γµβα. That the third member is zero is seen by inserting the expressions for

T µ [Eq. (7.76)] and Sµ [Eq. (7.78)], and making use of the commutativity of the ordinaryderivatives ∂/∂s and ∂/∂σ. We have thus proved the assertion in Eq. (7.81). By means ofEq. (7.81), the acceleration can be written as follows:

Aµ = T β β (Sα α Tµ) = T β (βS

α) (αTµ) + T βSα β αT

µ, (7.83)

using in the last step Leibnitz rule for differentiation. By means of Eq. (7.81) and Leibnitz’rule (once more), we obtain

T β (βSα) (αT

µ) = Sβ (βTα) (αT

µ)

= Sβ [β (Tα α T

µ)− Tα β αTµ]

= −SαT β α βTµ. (7.84)

The last member of Eq. (7.84) follows from the geodesic equation [written in the form in Eq.(7.31)], and interchange (α ↔ β) of dummy indices in the nonvanishing term. By insertingEq. (7.84) into Eq. (7.83), the expression for the acceleration takes the form

Aµ = SαT β [β,α]Tµ = SαT βRµ

νβαTν, (7.85)

using the result given in Eq. (7.42) for the commutator of two covariant derivatives. Thefinal result

Aµ ≡ D2Sµ

Dσ2= Rµ

αβνTαT βSν (7.86)

is the geodesic deviation equation for light rays. The same result of course is valid for time-like geodesics, where the proper time (τ) may be used as affine parameter (σ). Initiallyparallel geodesics (geodesics for which V µ initially is zero) will fail to remain parallel onlyon curved surfaces, that is if and only if Rµ

αβν = 0.

8

Electromagnetic theory in curved space-time

8.1 Vacuum Maxwell equations in general relativity

We recall from Sec. 3.4 that the set of free Maxwell equations in the framework of SpecialRelativity can be written in manifest covariant form as follows:

∂αFαβ = 0, (8.1)

∂αFβγ + ∂βFγα + ∂γFαβ = 0, (8.2)

where Fαβ and Fαβ are the field tensor written in contravariant [Eq. (3.53)] and covariant[Eq. (3.61)] form, respectively. For brevity, we have omitted the subscript T on the fieldtensor; compare Eqs. (3.58) and (8.1).

We extend the Maxwell theory to general relativity by supposing that the extended fieldtensor behaves as a tensor under general coordinate transformations and that it reducesto the already introduced field tensor in locally inertial Minkowskian coordinates. By theminimal substitution

∂α → α, (8.3)

i.e., with all derivatives replaced by covariant derivatives, we finally reach the vacuumMaxwell equations in general relativity:

αFαβ(x) = 0, (8.4)

αFβγ(x) +βFγα(x) +γFαβ(x) = 0. (8.5)

It is now being understood that indices are to be raised and lowered with the metric tensorof general relativity, and not the Minkowski metric. For the field tensor we thus have

Fαβ(x) = gαµ(x)gβν(x)Fµν (x). (8.6)

While the connection between the covariant and contravariant field tensors is trivial inMinkowski space, it is far from so in curved space-time, as we shall realize in Sec. 8.3.In a certain reformulation, we shall see that Eq. (8.6) in a sense appears as a kind ofconstitutive relation for curved space-time in vacuum. The tensor equations (8.4)–(8.6) aretrue (outside matter) in the absence of gravitation and generally covariant and thereforealso true in arbitrary gravitational fields. In the presence of charged matter, Eq. (8.4) mustbe replaced by

αFαβ(x) = −µ0J

β(x), (8.7)

where Jβ(x) is a suitable extension of the Minkowski current density four-vector to curvedspace-time.

79


8.2 Covariant curl and divergence in Riemann space

For the purpose of photon wave mechanics, e.g., it is useful to seek to rewrite the vacuumMaxwell equations [Eqs. (8.4) and (8.5)] in such a manner that only normal derivatives(∂α) occur. With this aim, let us first study the concepts covariant curl and divergence,and let us assume that the connection is given by the Christoffel symbol.

Recalling that the covariant derivative of a covariant vector is given by

µVν = ∂µVν − ΓσµνVσ, (8.8)

cf. Eq. (7.28), and that the Christoffel symbol is symmetric in the lower indices, it immedi-ately follows that the covariant curl of of the vector V µ is identical to the ordinary curlin four dimensions, i.e.,

µVν −νVµ = ∂µVν − ∂νVµ. (8.9)

In order to determine the covariant divergence of a contravariant vector, viz.,

µVµ = ∂µV

µ + ΓµµαV

α, (8.10)

It first reminds the reader about a few results from matrix calculus. Let the determinant ofthe nonsingular matrix aij be a. Expanded along the ith row, the determinant is

a =∑

j

aijAij , (8.11)

where Aij is the cofactor [minor with sign (−1)i+j ] to aij . This cofactor relates to the jithelement of the reciprocal matrix (bij) to aij as follows:

Aij = abji. (8.12)

Assume now that the elements of aij are functions of some coordinates, xk. Differenti-ation of the determinant, a = a(aij(xk)), with respect to xk then gives

∂a

∂xk=

∂a

∂aij

∂aij∂xk

= Aij∂aij∂xk

, (8.13)

where the summation over i and j is implicit. The last member of Eq. (8.13) results fromEq. (8.11). By means of Eq. (8.12), one finally obtains

∂a

∂xk= abji

∂aij∂xk

. (8.14)

Applied to the symmetric metric gλµ, with inverse gλµ, we have the result which isneeded below, viz.,

∂αg = ggµλ∂αgλµ. (8.15)

Let us return now to Eq. (8.10). From Eq. (6.31), one obtains for the contracted Christoffelsymbol

Γµµα =

1

2gµλ (∂µgλα + ∂αgλµ − ∂λgµα) =

1

2gµλ∂αgλµ, (8.16)

Electromagnetic theory in curved space-time 81

since the first and last term in the parenthesis cancel, as the reader may realize by inter-change of dummy indices in the last term [λ → µ, µ → λ]. By combining Eqs. (8.15) and(8.16), it appears that

Γµµα =

1

2(−g)∂a(−g) =1√−g ∂α

√−g. (8.17)

When this result is inserted into Eq. (8.10), and the dummy index is changed from α to µin the last term, one obtains

µVµ = ∂µV

µ +1√−g

(

∂µ√−g

)

V µ, (8.18)

and then the following compact expression for the covariant divergence of a contravariantvector:

µVµ =

1√−g ∂µ(√−gV µ

)

. (8.19)

If V µ vanishes at infinity, one gets as consequence of Eq. (8.19)∫

(µVµ)

√−gd4x = 0, (8.20)

where the integration extends over the entire space-time domain. It appears from Eq.(8.20)that the infinitesimal volume element

√−gd4x is an invariant under coordinate transforma-tion (x→ x′), since µV

µ is a scalar. One may arrive at this conclusion without referenceto Eq. (8.20): From integral calculus it is known that under a general coordinate transfor-mation x→ x′, the infinitesimal volume element d4x changes to [241]

d4x′ = |J |d4x, (8.21)

where |J | is the numerical value of the Jacobian [given in Eq. (7.7)]. One can relate J tothe determinant g and g′ of the metric tensor in the two coordinate systems. By taking thedeterminant of Eq. (7.15), one finds

g′ = J−2g, (8.22)

and

|J | =√ −g

−g′ . (8.23)

Inserting this result into Eq. (8.21), one obtains√

−g′d4x′ = √−gd4x, (8.24)

as expected.

8.3 A uniform formulation of electrodynamics in curved and flatspace-time

8.3.1 Maxwell equations with normal derivatives

It is possible, as we now shall realize, to rewrite the Maxwell equations in curved space-time in such a manner that only normal derivatives appear. One interesting outcome of


such a reformulation relates to the circumstance that the set of vacuum equations thenmay be given in a form mathematically resembling those of standard Maxwell–Lorentzelectrodynamics in the presence of charged matter. In the new picture four fields E, B, D,and H appear, and the connection between the covariant and contravariant field tensors ina certain sense becomes equivalent to a set of constitutive equations, D = D (E,B) andH = H (E,B), among the fields.

To achieve our goal of rewriting the free Maxwell equations in standard form use is madeof the formula for the covariant derivative of a covariant second rank tensor, Tµν (metricconnection):

σTµν = ∂σTµν − ΓλµσTλν − Γλ

νσTµλ. (8.25)

The relation in Eq. (8.25) is the obvious extension of Eq. (7.28) to second rank tensors.Applied to the covariant derivatives in Eq. (8.5), one has

αFβγ = ∂αFβγ − ΓλβαFλγ − Γλ

γαFβλ, (8.26)

and by cyclic permutation (α → β → γ → α)

βFγα = ∂βFγα − ΓλγβFλα − Γλ

αβFγλ, (8.27)

γFαβ = ∂γFαβ − ΓλαγFλβ − Γλ

βγFαλ. (8.28)

Due to the symmetry of the Christoffel symbol (Γµαβ = Γµ

βα), and the antisymmetry of thefield tensor,

Fβα(x) = −Fαβ(x), (8.29)

addition of Eqs. (8.26), (8.27), and (8.28) gives

αFβγ +βFγα +γFαβ = ∂αFβγ + ∂βFγα + ∂γFαβ . (8.30)

The covariant derivatives in the field equation in (8.5) thus have been eliminated in favor ofnormal derivatives. The form of Eq. (8.5) is transformed into exactly the same form as thatof the corresponding field equation in flat space [Eq. (8.2)]. It is clear from the derivationabove that a relation of the form in Eq. (8.30) holds for all torsion-free connections, Γµ

αβ =

Γµβα, and an arbitrary antisymmetric covariant tensor, Tµν = −Tνµ.In order to rewrite the Maxwell equation given in Eq. (8.4), we form from the covariant

derivative of the contravariant tensor T µν (metric connection), viz.,

σTµν = ∂σT

µν + ΓµσλT

λν + ΓνσλT

µλ, (8.31)

the covariant divergence

µTµν = ∂µT

µν + ΓµµλT

λν + ΓνµλT

µλ. (8.32)

If T µλ is antisymmetric (T µλ = −T λµ), we have for the last term in Eq. (8.32)

ΓνµλT

µλ = −ΓνµλT

λµ = −ΓνλµT

λµ = −ΓνµλT

µλ, (8.33)

a relation which shows that ΓνµλT

µλ is zero. By means of Eq. (8.17), we hence end up withthe result

µTµν = ∂µT

µν +1√−g

(

∂λ√−g

)

T λν =1√−g∂µ

(√−gT µν)

, (8.34)


for an antisymmetric tensor. Since the field tensor Fαβ is antisymmetric, the covariantderivative, α, in Eq. (8.4) thus can be eliminated in favor of the normal derivative, ∂α,using Eq. (8.34).

By now, we have achieved the goal of rewriting the free Maxwell equations in curvedspace-time in terms of normal derivatives only:

∂α

(

√

−g(x)Fαβ(x))

= 0, (8.35)

∂αFβγ + ∂βFγα + ∂γFαβ = 0, (8.36)

with the relation

Fαβ(x) = gαγ(x)gβδ(x)Fγδ(x). (8.37)

In the presence of charged matter, Eq. (8.35) must be replaced by

∂α

(

√

−g(x)Fαβ(x))

= −µ0

√

−g(x)Jβ(x), (8.38)

whereas Eq. (8.36) of course stays unaltered. The current density Jβ(x) is a four-currentdensity in arbitrary coordinates. It is required that Jβ(x) behaves as a tensor (of rankone) under general coordinate transformations, and reduces to the flat-space form in locallyinertial Minkowski coordinates. The presence of the factor

√

−g(x) on the right side of Eq.(8.38) is a consequence of the insertion of Eq. (8.34) [for Tαβ = Fαβ ] into Eq. (8.7).

8.3.2 Maxwell equations with E, B, D, and H fields

The covariant field tensor Fαβ(x) has at the outset six independent components, andwritten in the form

Fαβ(x) ≡

0 −E1(x)/c −E2(x)/c −E3(x)/cE1(x)/c 0 B3(x) −B2(x)E2(x)/c −B3(x) 0 B1(x)E3(x)/c B2(x) −B1(x) 0

, (8.39)

the component “names” correspond to those of the electric and magnetic fields of theMaxwell theory in flat space-time. With the component notation in Eq. (8.39), it is clearfrom the discussion in Sec. 3.4 that Eq. (8.36) is equivalent to the equations

∇×E(x) = − ∂

∂tB(x), (8.40)

∇ ·B(x) = 0. (8.41)

With the metric tensor of curved space-time, the relations between the components ofthe covariant and contravariant field tensors can be determined. In Minkowskian space therelations are given by Eqs. (3.53) and (3.61), or in compact form by Eq. (3.62). In hindsight,let us write the antisymmetric contravariant field tensor in the form

Fαβ(x) ≡ µ0√

−g(x)

0 cD1(x) cD2(x) cD3(x)−cD1(x) 0 H3(x) −H2(x)−cD2(x) −H3(x) 0 H1(x)−cD3(x) H2(x) −H1(x) 0

. (8.42)


The so defined new field vectors D(x) and H(x) satisfy in empty space the equations

∇ ·D(x) = 0, (8.43)

∇×H(x) =∂

∂tD(x), (8.44)

as the reader may verify by inserting Eq. (8.42) into Eq. (8.35); [setting β = 0 in Eq. (8.35),Eq. (8.43) emerges, and with β = 1, 2, and 3, the three Cartesian components of Eq. (8.44)are obtained]. The free-space Maxwell equations in curved space-time now are given by theset in Eqs. (8.40), (8.41), (8.43), and (8.44).

The contravariant current density for a system of charged point particles (charge qn forparticle number n) has components of the form [241]

Jµ(x) =1

√

−g(x)∑

n

qn

∫

δ(4) (x− xn) dxµn (8.45)

in general relativity. The quantity δ(4) (x− xn) is the four-dimensional delta function in ageneral coordinate system, and the integral is taken along the trajectory of the nth particle.The current density components in Special Relativity are obtained setting g(x) = −1 in Eq.(8.45). In terms of charge (ρ(x)) and current (J(x)) densities, i.e., with four-vector

Jµ(x) = (cρ(x),J(x)) , (8.46)

Eq. (8.38) shows that the generalizations of Eqs. (8.43) and (8.44) are

∇ ·D(x) = ρ(x), (8.47)

∇×H(x) = J(x) +∂

∂tD(x). (8.48)

In a formulation where only normal derivatives occur, the complete set of dynamical equa-tions for the electromagnetic field in general relativity thus consists of Eqs. (8.40), (8.41),(8.47), and (8.48). These equations of general relativity are form-identical to the macroscopicMaxwell equations with “free” charge and current density ρ(x) and J(x) [101].

8.3.3 Microscopic Maxwell–Lorentz equations in curved space-time

At this point, it is instructive to compare the dynamical field equations set up for generalrelativity in the previous subsection with the well-known microscopic and macroscopic formsvalid in Special Relativity.

The microscopic Maxwell–Lorentz equations in flat space-time usually are written in theform [56, 57, 133, 101, 113]

∇×E(r, t) = − ∂

∂tB(r, t), (8.49)

∇ ·B(r, t) = 0, (8.50)

∇ · E(r, t) = ǫ−10 ρ(r, t), (8.51)

∇×B(r, t) = µ0J(r, t) + c−2 ∂

∂tE(r, t), (8.52)

where ρ(r, t) and J(r, t) are the microscopic charge and current densities. In flat space, acomparison of Eqs. (3.53) and (8.42) shows that D = ǫ0E and B = µ0H. By inserting theserelations into Eqs. (8.47) and (8.48), the dynamical field equations in curved space reduce to


the microscopic Maxwell–Lorentz equations. An interesting formulation appears in generalrelativity if one replaces the microscopic charge and current densities in Eqs. (8.47) and(8.48) by the so-called generalized polarization (Pgen) and magnetization (Mgen) fields via[115]

ρ(r, t) = −∇ ·Pgen(r, t), (8.53)

J(r, t) =∂

∂tPgen(r, t) +∇×Mgen(r, t). (8.54)

Such a replacement in flat space-time is discussed in detail in my book on the quantum the-ory of near-field electrodynamics [127]. Here, I only emphasize that Pgen and Mgen containall orders in a multipole expansion, not just the electric- and magnetic-dipole contributions.The quantities Pgen andMgen are associated to matter in general relativity so let us renamethem:

Pgen(r, t) ≡ Pmatt(r, t), (8.55)

and

Mgen(r, t) ≡ Mmatt(r, t). (8.56)

When the relations in Eqs. (8.53) and (8.54) [with the renaming in Eqs. (8.55) and (8.56)]are inserted in Eqs. (8.47) and (8.48), it is realized that the general set of dynamical fieldequations in curved space-time can be written in the compact form

∇×H(x) =∂

∂tD(x), (8.57)

∇ ·D(x) = 0, (8.58)

∇× E(x) = − ∂

∂tB(x), (8.59)

∇ · B(x) = 0, (8.60)

where

D(x) = D(x) +Pmatt(x) = ǫ0E(x) +Pcurv(x) +Pmatt(x), (8.61)

H(x) = H(x) −Mmatt(x) = µ−10 B(x) −Mcurv(x) −Mmatt(x), (8.62)

and E(x) ≡ E(x), B(x) ≡ B(x). [The change to calligraphic letters for the electric andmagnetic fields is just made to make the notation in Eqs. (8.57)-(8.60) uniform.] In the lastmember of Eq. (8.61) the difference between D(x) and E(x) in general relativity has beenexpressed in terms of a curvature “polarization” Pcurv(x) ≡ D(x)−ǫ0E(x). In Eq. (8.62) wehave introduced a curvature “magnetization” by Mcurv ≡ µ−1

0 B(x)−H(x). As the readermay have noticed, the Maxwell–Lorentz equations in general relativity, when written asin Eqs. (8.57)–(8.60), take the same form as in macroscopic electrodynamics in flat spacein the absence of “free” charges [101]. The extensions: Macroscopic Maxwell equations ⇒Microscopic Maxwell–Lorentz equations ⇒ Electrodynamic equations in curved space-time,finally have been expressed in a uniform language.

8.3.4 Constitutive relations in curved space-time

Our reformulation of classical electrodynamics in general relativity makes it natural to seekto describe the influence of space-time curvature via some kind of constitutive relations. In


electrodynamics in flat space, constitutive relations often are useful when one wants toquantize a coupled field-matter system, and thus replace photons by polaritons (quanta ofthe coupled system). Perhaps, some of the quantization methods known from condensed-matter electrodynamics can be transferred to general relativity when the electrodynamicshere is formulated in terms of constitutive relations.

The forms given for Fαβ and Fαβ in Eqs. (8.39) and (8.42) makes it clear that theconstitutive relations for curved space-time are hidden in the connection between the fieldtensor and its dual partner [Eq. (8.37)].

A constitutive relation D = D(E,B) is obtained via a calculation of the componentsF 0i(= −F i0) = cµ0Di/

√−g, i = 1, 2, 3, as a function of the elements in Fαβ. Elementsof the type Fi0(= −F0i) give contributions to the D-field from the electric field (E), andelements Fij(= −Fji) link D to the magnetic field (B). The constitutive relation can bewritten in a variety of equivalent forms, e.g., as follows:

Di = ǫ0εijEj + ǫ0c√−gG0 · Gi ×B, (8.63)

with the abbreviations

εij =√−g

(

gi0gj0 − g00gij)

, (8.64)

and

Gµ =(

gµ1, gµ2, gµ3)

, µ = 0− 3. (8.65)

The first part of Eq. (8.63) indicates that curved space-time in the present formulationbehaves like a kind of dielectric medium, the ijth element of the (relative) dielectric tensorbeing given by Eq. (8.64). The second part of Eq. (8.63) endows curved space-time also withgyromagnetic properties [3], and among the four vectors Gµ (µ = 0 − 3) given in terms ofelements of the metric tensor the Gi’s (i = 1−3) appear as a set of gyration vectors [3]. If theelectromagnetic field is so weak that it does not influence the (local) curvature of space-time,the gyrotropic constitutive relation will be linear. The presence of mass (in this concept’sgeneral sense) curves space-time. Hence, because an electromagnetic field contains energyand therefore possesses an equivalent mass given by Einstein’s famous formula E = mc2,light affects the curvature. For a photon of numerical momentum p, the equivalent massis m = p/c. Since the metric tensor is symmetric, Eq. (8.64) shows that also the dielectrictensor of curved space-time must be symmetric, εij = εji.

The other part of the constitutive relation, viz., H = H(E,B), is obtained relating theelements F ij(= −F ji), i, j = 1 − 3, to the elements of Fαβ. Elements of the Fij givecontributions to the H-field from B, and elements F0i(= −Fi0) contribute to the cross-coupling between H and E. Starting from

F ij = gikgjlFkl +(

gi0gjk − gj0gik)

F0k, (8.66)

the relations F ij = µ0εijmHm/

√−g, Fkl = εklnBn and F0k = Ek/c give

µ0√−g εijmHm = gikgjlεklnBn +

1

c

(

gi0gjk − gikgj0)

Ek. (8.67)

To determine the components of the H-field from Eq. (8.67), one multiplies the equationby εija and makes a summation over i and j. Since (with the summation over these indiceskept implicit),

εijaεijb = 2δab, (8.68)


where δab is the Kronecker delta (a, b = 1− 3) one finds

2µ0√−gHa = εijagikgjlεklnBn +2

cgi0gjkεijaEk. (8.69)

The form of the factor to Ek was obtained from the rewriting

εija(

gi0gjk − gikgj0)

= gi0gjk(

εija − εjia)

= 2gi0gjkεija. (8.70)

By some obvious changes in the index labelling, one finally reaches the form

µ0Hi =

√−gc

[

1

2gkmglnεkliεmnj (cBj) + εkligk0gljEj

]

(8.71)

for the constitutive relation. The connection between Hi and Bj implies that curved space-time has an effective permeability in the present formulation. If one denotes the ijth com-ponents of the inverse (relative) permeability tensor µ by µ−1

ij it appears from Eq. (8.71)that

µ−1ij =

√−g2

gkmglnεkliεmnj . (8.72)

Using the abbreviation in Eq. (8.72), the constitutive relation in Eq. (8.71) reads as follows:

µ0Hi = µ−1ij Bj +

√−gc

εkligk0gljEj . (8.73)

From the connection between the contravariant and covariant field tensors [Eq. (8.37)], wehave reached by now the constitutive relations in Eqs. (8.63) and (8.73).

8.3.5 Remarks on the constitutive relations in Minkowskian space

Let us make a pause in our considerations concerning the constitutive relations of curvedspace-time, and return to flat space. For later comparison, it is sufficient here to limitourselves to a brief summary of certain aspects of the well-known dielectric constitutiverelation

D(r, t) = ǫ0

∫ ∞

−∞ε(r, r′, t, t′) ·E(r′, t′)d3r′dt′, (8.74)

where ε(r, r′, t, t′) is the generalized dielectric function (tensor) of the medium (particlesystem) under study [113, 115, 3, 105]. The relation between E and D is called linear if ε isfield independent. As indicated in Eq. (8.74), the dielectric tensor is a function of two spacecoordinates, r and r′, in general. This means that the D(r, t)-field in a certain point r inspace depends on the electric field E(r′, t′) in surrounding points, r′. When written in thisspatially nonlocal form all magnetic effects can be included [127, 115]. It is possible also incurved space-time to eliminate the magnetic field in the constitutive relation in Eq. (8.63)with the help of the Maxwell equation∇×E = −∂B/∂t. The resulting constitutive equationhas a spatially nonlocal dielectric tensor, but it is simple in the sense that the D-field at ronly depends on the electric field and this field’s first (spatial) derivatives at r. The readerneed not be concerned with the subtleties related to the incorporation of magnetic effectsin ε(r, r′, t, t′). Physically, the study of local-field phenomena in electrodynamics requiresthe use of a spatially nonlocal dielectric function. If local-field effects [113] can be neglectedthe constitutive relation in Eq. (8.74) reduces to a spatially local form

D(r, t) = ǫ0

∫ ∞

−∞ε(r, t, t′) ·E(r, t′)dt′, (8.75)


with a new dielectric tensor, ε(r, t, t′), which only depends on r. Local-field effects oftenplay an important role in electrodynamics, and the requirements for reducing Eq. (8.74)to Eq. (8.75) are not easy to establish in general [113, 115, 3, 105]. The medium now ischaracterized as inhomogeneous. Although the connection in Eq. (8.75) is local in space, it isstill nonlocal in time. If the electrodynamic properties of the medium are time independent,the dielectric function can depend only on the time difference τ = t− t′, i.e.,

ε(r, t, t′) = ε(r, t− t′) (8.76)

in the case of an inhomogeneous medium. When the dielectric tensor possesses translationalinvariance in time [Eq. (8.76)], the state of the medium is characterized as stationary. Ifone assumes that the dielectric function has delta-function support in time everywhere inspace, that is

ε(r, t− t′) = ε(r)δ(t− t′), (8.77)

the connection between E and D becomes instantaneous in time:

D(r, t) = ǫ0ε(r) ·E(r, t). (8.78)

The new inhomogeneous dielectric tensor, ε(r), is time independent. For a detailed accountof linear microscopic constitutive relations in electrodynamics, the theory’s relation to so-called response theory, and the roles of Einstein causality and microcausality, the reader isreferred to my book Quantum Theory of Near-Field Electrodynamics [127].

8.3.6 Permittivity and permeability for static metrics

We now return to the vacuum constitutive relations of curved space-time. Since gαβ =gαβ(xµ) it appears from Eqs. (8.63) and (8.71) that the constitutive relations always arelocal in both space and time. The possibility to bring in nonlocality in the formalism byeliminating B in Eq. (8.63) in favor of E by means of the dynamical field equation in (8.59)is of no interest for what follows. The same applies to the elimination of the E-field in Eq.(8.71). The locality of the constitutive relations in space and time endows curved space-timewith unusual “material” properties seen in the perspective of flat-space electrodynamics.

Space-time is said to be stationary if there exist a special coordinate system in whichthe metric is visibly time independent. Let gµν be the metric tensor in this particularcoordinate system. Then,

∂0gµν.= 0 (8.79)

where notation.= means that the equation holds only in the special coordinate system.

From Eq. (8.79) it follows that

gµν(xµ) .= gµν(x). (8.80)

Even though the metric is stationary, the line element will still in general contain crossterms between time and space that is

ds2.= g00(x)

(

dx0)2

+ 2g0i(x)dx0dxi + gij(x)dx

idxj . (8.81)

If ds2 furthermore is invariant under time reversal

x0 → x′0 = −x0, (8.82)


we have

ds2.= g00(x)

(

dx0)2 − 2g0i(x)dx

0dxi + gij(x)dxidxj . (8.83)

A comparison of Eqs. (8.81) and (8.83) then shows that

g0i(x).= 0, i = 1− 3. (8.84)

If space-time is stationary (translationally invariant in time) and invariant under timereversal it is called static. For a static space-time

ds2.= g00(x)

(

dx0)2

+ gij(x)dxidxj . (8.85)

In static space-time, the last term in Eq. (8.63) vanishes because G0.= 0. The constitu-

tive equation then takes the simple form

Di.= ǫ0εijEj , (8.86)

with the ijth element of the dielectric tensor given by

εij.= −√−gg00gij . (8.87)

For static space-time the cross terms between the H- and E-fields in Eq. (8.71) vanish sothat the constitutive relation in Eq. (8.71) only involves magnetic quantities, i.e.,

µ0Hi.= µ−1

ij Bj . (8.88)

The elements of the inverse magnetic permeability, given so far by the compact expressionin Eq. (8.72) can also be written as follows:

µ−1ii =

√−g[

gjjgkk − (gjk)2]

, i 6= j 6= k, (8.89)

µ−1ij =

√−g[

gikgjk − gkkgij]

, i 6= j 6= k, (8.90)

as I urge the reader to prove for herself. When using the expressions given for the on [Eq.(8.89)] - and off [Eq. (8.90)] - diagonal elements in Eq. (8.88) one must remember that thegij ’s refer to the special coordinate system. In the flat space-time limit, where gµν = ηµν ,one of course regains the usual relations D = ǫ0E and B = µ0H.

8.4 Permittivity and permeability in expanding universe

Observations shows that our universe is approximately homogeneous and isotropic. Thehomogeneity and isotropy here refer to symmetries of space and not of space-time. Thesimplest example of such a cosmological geometry is described by the line element

ds2 = −c2dt2 + a2(t)(

dx2 + dy2 + dz2)

, (8.91)

where a(t) is the so-called scale factor. The metric described by Eq. (8.91) is called theflat Robertson–Walker metric [241, 238, 93, 47, 205, 239], where flat here refers to space(t = const spatial slices) not space-time. If a(t) increases in time, the line element given byEq. (8.91) describes an expanding universe. Thus, the physical distance, d = d(t), between

events with coordinate distance ∆d =(

∆x2 +∆y2 +∆z2)1/2

is given by d(t) = a(t)∆d.


For increasing a(t), this is the sense in which Eq. (8.91) describes an expanding universe.The scale factor a(t) must obey the Einstein field equations given in Eq. (7.66), and to ex-amine the time evolution one must consider simplified models describing the role of matter,radiation, and vacuum [241, 47].

Since the metric described by Eq. (8.91) has no off-diagonal elements, the constitutiverelations in Eqs. (8.63) and (8.71) have no cross coupling terms (D ↔ B, H ↔ E), andthe permittivity and permeability tensor both become diagonal. With g00 = g00 = −1, andgii = g−1

ii = a−2(t), one obtains in the Robertson–Walker coordinates from Eq. (8.64)

εii(t).= −√−gg00gii .= a(t), i = 1− 3, (8.92)

since g = −a6(t). The permittivity tensor of the expanding (or contracting) universe henceequals the scale factor, multiplied by the unit tensor, in the flat Robertson–Walker metric.It appears from Eq. (8.89) that the elements of the permeability tensor, necessarily equal,are given by

µii(t).= a(t), i = 1− 3. (8.93)

Note that εii(t) = µii(t) in a flat expanding universe (eu). The speed of light is everywherethe same, and given at time t by

ceu(t) =c

√

εii(t)µii(t)=

c

a(t), (8.94)

provided a(t) changes slowly in time. Qualitatively, the time derivative of a(t) at the presentage t0 of the universe is given by a(t0) ≃ a(t0)/tH , where tH is the Hubble time at t0 [238,47]. Since a(t) thus changes appreciably only on a cosmological time scale, the expressiongiven for ceu(t) in Eq. (8.94) is an extremely good approximation. The result for the speedof light is in agreement with the fact that a pulse of light travels on null geodesics:

0 = −c2dt2 + a2(t)dr2, (8.95)

where dr2 = dx2 + dy2 + dz2. It of course also appears from Eq. (6.53) [with c reinserted]that our massless particle travels a distance dr in a time dt = a(t)dr/c. By integration ofEq. (8.95) one obtains

R =

∫ to

te

c

a(t)dt, (8.96)

where te and to are times of emission (e) and observation (o) for a light pulse traveling thecoordinate distance R.

At this stage it may be interesting to compare the eikonal approach in flat space (Sec.5.2) to the result in Eq. (8.96). Thus, if one integrates the differential relation given in Eq.(5.36) using a space independent but time dependent refractive index, i.e., n = n(t), oneobtains, for a distance R =

∫

ds and times te and to,

R =

∫ to

te

c

n(t)dt. (8.97)

The close relationship between the results in Eqs. (8.96) and (8.97) was to be expectedbecause space is flat in the Robertson–Walker model.

On the basis of Eq. (8.96) it is easy to derive the expression for the cosmological redshift[238, 47].


8.5 Electrodynamics in potential description. Eikonal theory andnull geodesics

In the present section we shall see that it is possible to reformulate the Maxwell theoryin curved space-time in terms of the so-called vector potential, Aµ(x). In flat space, thepotential description plays an important role in both classical and quantum electrodynamicsas we shall realize later on in this book. In the context of photon physics, a beautiful wavemechanical description of photons emerges from Aµ(x). In Part III, we study photonwave mechanics in the field-unquantized limit paying particular attention to the potentialformulation. If the reader wants to familiarize herself with the potential description ofelectrodynamics Sec. 8.5 may be studied after having read Part III.

The content in the Maxwell equation in Eq. (8.7) may also be expressed in terms of themixed (Fα

β(x)) or the covariant (Fαβ(x)) field tensor. Thus,

αFαβ(x) = −µ0Jβ(x), (8.98)

and

αFαβ(x) = −µ0Jβ(x). (8.99)

Using Eq. (6.5) the reader may convince herself by an explicit calculation that the left-handsides of Eqs. (8.98) and (8.99) are equal. Hence,

αFαβ = δαν α F

νβ = gαµgµν α F

νβ = µFµβ = αFαβ . (8.100)

For what follows the form given in Eq. (8.99) is a convenient starting point. A covariantvector potential field, Aµ(x), now is introduced via the prescription

Fαβ(x) = ∂αAβ(x)− ∂βAα(x). (8.101)

The antisymmetry of the field tensor is manifest from Eq. (8.101). With the notation

Aµ(x) = (A0,A) = (−φ/c,A), (8.102)

one obtains from Eqs. (8.39) and (8.101) the relations

E = −∂A∂t

−∇φ, (8.103)

B = ∇×A. (8.104)

Thus, in Minkowski space one recovers the usual connections between the vector (A) andscalar (φ) potentials and the electric and magnetic fields; see Sec. 10.1. By means of Eq.(8.9), expressing the identity of the ordinary and covariant curls, the elements of the fieldtensor can be written in the manifest covariant form

Fαβ(x) = αAβ(x)−βAα(x). (8.105)

By inserting Eq. (8.105) into Eq. (8.99), one gets

α α Aβ(x)−α β Aα(x) = −µ0Jβ(x). (8.106)

The Maxwell equation in Eq. (8.99) by now has been rewritten in terms of the covariantvector potential field.


It is an important advantage of the potential formulation of electrodynamics that theother field equation, viz. Eq. (8.5), is automatically satisfied. Before this is shown by anexplicit calculation, I remind the reader of the fact that the two homogeneous Maxwellequations in flat space-time [Eqs. (2.1) and (2.4)], also are satisfied identically using theconnections in Eqs. (8.103) and (8.104). In order to demonstrate that Eq. (8.5) is satisfiedidentically when the various components of the field tensor are given by Eq. (8.105), onemakes use of the local commutator relation

α β Aγ(x) −β α Aγ(x) = Rγδαβ(x)Aδ(x) = −Rδγαβ(x)A

δ(x), (8.107)

and two similar ones obtained by cyclic permutation of the indices (α, β, γ). The first mem-ber of Eq. (8.107) follows from Eq. (7.42), by lowering the upper µ-index with the metrictensor. The second member originates from the antisymmetry of the Riemann curvaturetensor. A little work now tells us that Eq. (8.5) takes the form

(Rδαβγ(x) +Rδβγα(x) +Rδγαβ(x))Aδ(x) = 0, (8.108)

but the cyclicity of the curvature tensor [Eq. (7.54)] then shows that eq. (8.108) is identicallysatisfied. In the potential formalism of electrodynamics in curved space-time, one is left withthe generally covariant field equation in (8.106), in agreement with the conclusion reachedin Minkowski space (and for a local inertial system in general relativity, of course).

In Minkowski space the relations in Eqs. (8.103) and (8.104) do not determine thevector and scalar potentials completely, as might be well-known to the reader. Thus, acertain kind of transformation of the potentials, called gauge transformations, leaves theelectric and magnetic fields unaltered. Gauge transformations are of great importance inphoton physics, and we shall discuss these in various contexts later on, starting in PartIII, where we examine the role of gauge transformations in photon wave mechanics basedon the four-potential. The freedom in the choice of potentials enables one to put certainrestrictions on the gauge transformation. In a group of gauges, named the Lorenz gauges,one requires that the potentials satisfy the covariant constraint

∂αAα(x) = 0. (8.109)

This so-called Lorenz condition may be generalized to curved space-time replacing theordinary derivative by the covariant derivative:

αAα(x) = 0. (8.110)

The Lorenz condition in Eq. (8.110) simplifies the field equation given in Eq. (8.106). Inorder to utilize Eq. (8.110) to reduce Eq. (8.106), first we write the commutator relation inEq. (7.42) in the form

[α,β]Aµ(x) = R ναµ β(x)Aν (x), (8.111)

and then we make a contraction (µ→ α). Hence,

[α,β ]Aα(x) = R ναα β(x)Aν (x). (8.112)

The quantity Rνβ(x) = R να

α β(x) is the Ricci tensor [cf. Eq. (7.55)]. The Lorenz conditionin Eq. (8.110) finally reduces Eq. (8.112) to

α β Aα(x) = Rνβ(x)Aν(x). (8.113)


When this result is used in Eq. (8.106), we finally reach the following field equation in theLorenz gauge

β β Aα(x)−Rβα(x)Aβ(x) = −µ0Jα(x). (8.114)

Before proceeding with the general analysis, I want to emphasize a deficiency in theminimal substitution rule in (8.3). In Minkowski space the potential satisfies the field equa-tion

∂β∂βAα(x)− ∂β∂αAβ(x) = −µ0Jα(x), (8.115)

which also may be written in the form

∂β∂βAα(x)− ∂α∂

βAβ(x) = −µ0Jα(x), (8.116)

since the normal derivatives commute [Eq. (7.37)]. The proof that Eq. (8.116) is correctis given in Sec. 12.1. The result above may also be obtained inserting the contravariantflat space-time form of Eq. (8.101) into Eq. (3.59), and then lowering the index on thefour-current density. The Lorenz condition in Eq. (8.109) simplifies Eq. (8.116) to

∂β∂βAα(x) = −µ0Jα(x). (8.117)

Now, if the minimal substitution rule is made in Eq. (8.117) one does not obtain the fieldequation in (8.114), but one without the Ricci tensor term. In the present instance, we mustdecide in favor of Eq. (8.114), because this implies charge conservation, while the alternativefield equation without the Ricci term conflicts with this. Perhaps, this conclusion is not sosurprising after all, since we already knew that once two consecutive derivatives are involvedthe commutativity known from flat space-time is lost [Eq. (7.37) versus Eq. (7.42)].

To demonstrate that the Maxwell equations in general relativity imply charge conser-vation, it is convenient to start from the inhomogeneous field equation written in terms ofordinary derivatives, i.e., from Eq. (8.38). By differentiation we have

∂β∂α(√−gFαβ

)

= −µ0∂β(√−gJβ

)

. (8.118)

Since ordinary derivatives commute, and the field tensor is antisymmetric, we have

∂β∂α(√−gFαβ

)

= ∂α∂β(√−gFαβ

)

= ∂β∂α(√−gF βα

)

= −∂β∂α(√−gFαβ

)

. (8.119)

The equality of the first and last member of this equation shows that the left side of Eq.(8.118) is zero. Therefore, ∂β

(√−gJβ)

= 0, or equivalently

βJβ(x) = 0 (8.120)

in view of Eq. (8.19). The result in Eq. (8.120) expresses the charge conservation in curvedspace-time. In Minkowski space, Eq. (8.120) reads

∂βJβ(x) = 0, (8.121)

and this states the well-known charge conservation (equation of continuity) in flat space-time. The charge conservation condition in Eq. (8.121) of course is inherent in the ordinaryset of Maxwell–Lorentz equations [101, 127].

Starting from Eq. (8.114) contact to the geometrical optics approximation is obtainedassuming that the space-time scale of variation of the electromagnetic potential is much


smaller than that of the curvature. In free space, and with neglect of the Ricci tensor term,the field equation is reduced to

β β Aα(x) = 0. (8.122)

In line with the discussion in Sec. 5.2, we seek (approximate) solutions to Eq. (8.122) of theform

Aα(x) = A0αe

iS(x), (8.123)

where the amplitude A0α is nearly constant. The eikonal, S(x), here is defined as a func-

tion of space-time, and without a prefactor in the exponent [compare to Eqs. (5.19) and(5.20)]. Substituting Eq. (8.123) into Eq. (8.122) and neglecting the small terms containingderivatives of A0

α, it appears that the phase, S(x), must satisfy

β β S(x) + i(

βS(x))

(βS(x)) = 0. (8.124)

Since S(x) is real the two parts of Eq. (8.124) must separately vanish. In Sec. 5.2, the eikonalequation was derived using the first-order Maxwell equations (for E and B), but it mightas well have been derived from the vacuum equation for the electric or magnetic field, orfrom the wave equation for the vector or scalar potential, quantities which are uncoupledin the Lorenz gauge. The eikonal equation in general relativity thus takes the form

βS(x)β S(x) = 0. (8.125)

Since the covariant derivative of a scalar is just the ordinary derivative, Eq. (8.125) isidentical to its flat space-time form, viz.,

∂βS(x)∂βS(x) = 0. (8.126)

Differentiation (∂α) of Eq. (8.126) gives

0 = ∂α(

∂βS∂βS)

= 2(

∂βS)

(∂α∂βS) = 2qβ∂βqα, (8.127)

where

qβ = ∂βS = βS. (8.128)

The quantity qβ is a contravariant normal vector to the surfaces of constant S(x), andqβ (qα) its covariant partner. When Eq. (8.128) is inserted into Eq. (8.125) one seesthat qβ is a null vector:

qβqβ = 0. (8.129)

The surfaces of constant S(x) are called null hypersurfaces. It follows from Eq. (8.129) thatthe normal vector to a null hypersurface is tangent to the hypersurface. A comparison ofEqs. (7.22) and (8.127) indicates that qα = dxα(σ)/dσ is a tangent vector because Eq.(6.52) here becomes

0 = gαβqαqβ = qβq

β , (8.130)

in agreement with Eq. (8.129). In the geometrical optics approximation the argumentsabove suggest that light travels on null geodesics. Light rays thus follow null geodesics infree space in general relativity. This conclusion is in line with the fact that light rays in anInhomogeneous Vacuum in Minkowskian space (Sec. 5.1) satisfy the eikonal equation (seeSec. 5.2). A comparison of Eqs. (5.40) and (8.128) indicates that qβ in a sense plays therole of being the local wave four-vector of a “light particle.”


8.6 Gauge-covariant derivative

In Sec. 7.2 we introduced and discussed the covariant derivative of a vector field. The rela-tions between the normal and covariant derivatives of contravariant and covariant vectorswere given in Eqs. (7.27) and (7.28) in the case where the affine connection equals theChristoffel symbol. The covariant derivative of a vector field transforms as a tensor, andthe concept relates to the Principle of General Covariance which states that the forms ofphysical laws preserve their forms under a general coordinate transformation (x→ x′). ThePrinciple of General Covariance governs the effect of gravitational fields, and the principleis often characterized as a dynamic symmetry. In electrodynamics the effect of the elec-tromagnetic field is governed by another dynamic symmetry called local gauge invariance,and here a gauge-covariant derivative emerges, as we shall see now. Since the considerationsbelow are somewhat outside the book’s main line of development the reader may jumpover the material in this section in a first reading. After having studied Sec. 12.4, whichdeals with massive photon fields, the reader may find it fruitful to return to this section.The Principle of Local Gauge Invariance states that the differential equations governingthe dynamical development of a charged particle field, ψ(x), coupled to an electromag-netic potential, Aµ(x), retain their form when subjected to the combined transformation[209, 211, 127, 53]

ψ(x) ⇒ ψ(x) = exp(

iq

~χ(x)

)

ψ(x), (8.131)

Aµ(x) ⇒ Aµ(x) = Aµ(x) + ∂µχ(x), (8.132)

where χ(x) is an arbitrary so-called gauge function, and q is the charge of the particle field.The dynamical equation for the particle field involves derivatives ∂µψ(x) of ψ(x), and undera gauge transformation we have

∂µψ(x) ⇒ ∂µψ(x) = exp(

iq

~χ(x)

)

[

∂µ +iq

~∂µχ(x)

]

ψ(x). (8.133)

The derivatives ∂µψ(x) thus do not transform like ψ(x) under a gauge transformation,just as the derivatives of a vector field, ∂µV

ν(x), do not transform as a tensor field ingeneral relativity. It is the presence of the term proportional to ∂µχ(x) in Eq. (8.133) thatprevents ∂µψ(x) from transforming like ψ(x). Since the quantity ∂µχ(x) also appears inthe transformation of the four-potential [Eq. (8.132)], it is obvious that by means of newso-called gauge-covariant derivatives (operators) defined by

Gµ ≡ ∂µ − iq

~Aµ(x), µ = 0− 3, (8.134)

Gµψ(x) will transform as ψ(x) under an arbitrary gauge transformation. Hence,

Gµψ(x) ⇒ G

µ ψ(x) =

(

∂µ − iq

~Aµ(x)

)

ψ(x) = exp(

iq

~χ(x)

)

Gµ ψ(x), (8.135)

as the reader readily may verify using Eqs. (8.132) and (8.133). The quantity

πµ ≡ ~

iG

µ = ~i∂µ − qAµ(x) (8.136)

may be recognized as the kinematic four-momentum operator of the particle, (~/i∂µ)being the canonical (or conjugate) four-momentum operator [211, 127, 53].


To exemplify the presence of the gauge-covariant derivative in the dynamical equationfor a particle field, let us consider the Dirac equation for a spin-1/2 particle in an electro-magnetic field. In compact notation, the Dirac equation for the four-component spinor, ψ,may be written as follows [88, 53, 46]:

(

iγµ Gµ −QC

)

ψ(x) = 0, (8.137)

where

QC =mc

~(8.138)

is the Compton wave number of the particle (rest mass m), and

γµ = β (1,α) , (8.139)

β and α = (α1, α2, α3) being the four-dimensional realization of the four Dirac matrices.Related to the Dirac equation one has a conserved gauge-invariant four-current density

Jµ(x) = qcψ∗(x)γ0γµψ(x), (8.140)

where ψ∗ is the adjoint of ψ. The conservation of the four-current (equation of continuityfor charge) is expressed as

∂µJµ(x) = 0. (8.141)

The covariant form of the Dirac equation, and a charged spin-1/2 particle’s coupling to theelectromagnetic field, are discussed in Sec. 25.2, and the Gordon decomposition of the Diracfour-current density into its spatial and generalized spin parts is established in Sec. 25.3.

Part III

Photon wave mechanics

9

The elusive light particle

Classical electromagnetics in free space leaves no room for accommodation of a light particle.To underline this, it is sufficient to remember that Einstein in order to reach his light-quantum (energy-quantum) hypothesis had to mix classical theoretical physics with a pieceof experimental information that defies description in classical terms. In his 1905-paperEinstein concluded [60]:Monochromatic radiation of low density (within the range of validityof Wien’s radiation formula) behaves thermodynamically as though it consisted of a numberof mutually independent energy quanta of magnitude hν. The introduction of light quantain the Wien regime (the high-frequency part of the Planck blackbody radiation law) isthe first step toward the concept of electromagnetic radiation as a Bose gas of photons[180, 39, 68, 69]. Although a photon gas obeys Bose–Einstein statistics for all frequenciesthe statistical independence assumed by Einstein in 1905 is not true in general. Einsteinused Boltzmann statistics when he conjectured the existence of light quanta. In the Wienregime Boltzmann and Bose–Einstein statistics lead to the same conclusion since the meannumber of particles (photons) with a given energy is much less than one. The circumstancethat the light quanta are not statistically independent in Bose–Einstein counting finds itsexplanation in quantum mechanics. Hence, the statistical correlation is induced by therequirement of totally symmetric wave functions.

The inadequacy of classical electrodynamics also showed up when Bohr in 1913 on thebasis of the Rutherford picture of the atom, presented his revolutionary thoughts “On theConstitution of Atoms and Molecules” [24, 25, 26, 181]. Thus, his principal assumptions: (1)That the dynamical equilibrium of the systems in the stationary states can be discussed byhelp of the ordinary mechanics, while the passing of the systems between different stationarystates cannot be treated on that basis, and (2) That the later process is followed by theemission of a homogeneous radiation, for which the relation between the frequency and theamount of energy emitted is the one given by Planck’s theory [26].

We realized in Part II that at high frequencies, where the Maxwell–Lorentz theory makescontact to geometrical optics, the classical description of light rays and null geodesics con-tains certain particle-like features. In Minkowskian (flat-space) vacuum the light rays arestraight lines, and as such not very interesting. The real bonus of geometrical optics relatesto what I have called inhomogeneous vacuum, i.e., a matter-filled space where the (lin-ear) interaction between light and matter is described in terms of a frequency independentspatially inhomogeneous (linear) susceptibility; cf. the discussion in Sec. 5.1. This essentialnecessity of the presence of a massive-particle background (matter-filled space) for the prop-agation of particle-like light in geometrical optics opens the door a little to a basic principleof photon physics: A free photon is an abstraction. Only during its interaction with matter(charged massive particle(s)) does the photon come to “existence” (can be registered). Inthe words of Bohr [29, 30, 31]: “No elementary phenomenon is a phenomenon until it is aregistered (observed) phenomenon.” The epistemological lesson of Bohr is exemplified bythe photon beam splitter experiment shown in Fig. 9.1.

In Chapt. 32 we shall briefly discuss the quantum theory of measurements, essentiallythe model of von Neumann [168, 157]. However, according to Bohr “measuring instrumentsmust be described classically” [33, 34]. The difficulties arising when seeking to describe a

99


S

M1

M2

DA

DB

R

R

1 2

FIGURE 9.1A single-photon wave packet (WP) emitted by a source (S) interacts with a half-silveredmirror (M1). Conceived wrongly(!) as a particle, the photon is (with the same probability)either reflected by or transmitted through M1. Conceived also wrongly(!) as a wave, theone-photon field (beam) is split into reflected and transmitted parts (of equal intensity).Since “no elementary phenomenon is a phenomenon until it is a registered phenomenon” itis wrong to attribute any physical tangibility to the process until a (macroscopic) detectionsystem is added to the S+M setup. Via reflection in the (ideal) mirrors R, the field/particleghost is registered in the macroscopic photon counters DA and DB, placed as shown. Theclosed arrangement S+WP +M1+R+R+DA+DB is a phenomenon, (1). Since WP is asingle-photon wave packet, registration occurs either in DA or DB, never in both counters.Hence, in this phenomenon the photon is a particle since one can state by which route thephoton came. If the arrangement is changed by placing a half-silvered mirror M2 in thecrossing domain, the closed arrangement S +WP +M1 +R+R+M2 +DA +DB is a newphenomenon, (2). Now the detector DA never registers anything. This is the fingerprint ofdestructive/constructive interference. (2) hence tells us that the photon is a wave. Together,the phenomena (1) and (2) constitute a rational quantum physical generalization of theclassical picture of electromagnetics.

macroscopic system by quantum theory have been discussed extensively over the years [247],but these difficulties are not addressed in this book.

Notwithstanding that a free photon is not a phenomenon, and that (at least a pieceof) quantum physics must be mixed into classical electrodynamics to have a chance ofcatching a glimpse of a photon, here, in Part III, we take on a new search for a light particlestarting from the set of free-space classical Maxwell equations. Thus, we examine whetherit is possible to reinterpret the free-space wave theory of Maxwell in such a manner thatthe theory appears as a wave mechanical theory of a light particle. The “only” quantityforeign to classical electrodynamics which we need to introduce is Planck’s constant, h,(the elementary quantum of action). The formalism we end up with is called photon wavemechanics, or alternatively the first-quantized theory of the photon.

To set up a wave mechanical equation for a free photon one needs in the first place aquantity which may qualify as a photon wave function. The (chosen) wave function nec-essarily must be closely related to the electromagnetic field vectors, e.g., to E, B, or acombination of E and B. Conceptually various choices may qualify as a photon wave func-tion in free space, and over the years different choices have been made. A choice is one thing,

The elusive light particle 101

but to claim that a given choice is the photon wave function is quite another matter. If thedifferent choices satisfy certain necessary criteria, for instance that the light particle wavefunction must propagate with the vacuum speed of light, one cannot justify giving priorityto one particular choice in global vacuum. In order to distinguish between the capabilities ofthe various choices it is necessary to consider the photon-matter interaction, rememberingagain what we have learned from Bohr’s deep insight into what qualifies as a phenomenonin quantum physics. Although a free photon is an abstraction, one must not forget that allelectrodynamic interactions between charged matter systems (photon sources/drains) aredelayed, as dictated by the speed of light in vacuum.

In a fundamental quantum physical context the field-matter interaction appears viathe vector (A) and scalar (φ) potentials, not via E and B. Thus, the Aharonov–Bohmeffect [5, 90] has demonstrated that a charged particle (e.g., an electron) may be influencedelectrodynamically in a region of space where E = B = 0, provided the transverse partof the vector potential, AT , is nonvanishing in this region. This result agrees with thefact that the vector and scalar potentials appearing in the relativistic Dirac equation (ornonrelativistic Schrodinger equation) describing a charged spin-1/2 particle coupled to anelectromagnetic field cannot be eliminated completely in favor of E and B. In order tocapture the Aharonov–Bohm effect, and for other reasons to be discussed later on, I haveargued that perhaps it would be most natural to link the free photon wave function conceptdirectly to the transverse part of the vector potential [125, 128, 129].

In Chapt. 10 the photon wave mechanical formalism based on AT is discussed. In freespace there is no net effect of the longitudinal part of the vector potential (AL) and thescalar potential (φ = cA0) since the longitudinal electric field vanishes identically in ev-ery space-time point. The transverse part of the vector potential is gauge invariant, andbecause elementary particles have positive energies, we take the positive-frequency part of

the transverse vector potential, A(+)T , as the (unnormalized) vectorial wave function of the

light particle (photon). The normalized wave function, a(+)T , satisfies a dynamical equation

of first-order in time, and if this equation is multiplied by ~, we obtain our vectorial quan-tum mechanical wave equation for the photon. In the wave-vector (q) representation, thephoton Hamilton and momentum operators are given by H = c~q(= ~ω) and p = ~q. Theassociated eigenvalues E = ~ω and p = ~q agree with the Einstein–de Broglie relations.In the direct (r)-space representation the dynamical evolution of the photon wave functionis governed by an integro-differential equation in which the Hamiltonian density operatoris singular. Using circularly polarized (helicity) base states one may derive Schrodinger-likewave equations for the two photon helicity species, and show that the transverse photonhas the signature of a spin-1 particle.

It is instructive to compare the wave mechanics for the two photon helicity species withneutrino wave mechanics in an approximation where the small neutrino mass is neglected.The neutrino is a spin-1/2 particle, and in the massless limit, the two-component spinorialneutrino wave function satisfies the Weyl equation [246]. The neutrino has negative (left-handed) helicity, and the structures of the wave equations for the neutrino and the negative-helicity photon are analogous. The antineutrino has positive (right-handed) helicity and itsWeyl equation has the same structure as the wave equation for a positive-helicity photon[88, 46].

In Chapt. 11 we shall associate longitudinal (L) and scalar (S) photons to the positive-frequency parts of the longitudinal vector potential, AL(+), and the scalar potential, A0(+),in the Lorenz gauge. In the wave-vector domain the properly normalized scalar wave func-

tions a(+)L (q; t) = (q/q) · a(+)

L (q; t) and a0(+)(q; t) satisfy wave equations which are form-identical to the one for the two transverse (scalar) photon species. The L- and S-photonsoften are called virtual photons because there is no net effect of these photon types in free


space. In the covariant description of quantum electrodynamics the L- and S-photons playa crucial role in the presence of field-matter interaction. In the context of photon physicsin the rim zone of matter, it is useful to introduce two new dynamical variables by the

unitary transformation a(+)NF = (i/

√2)(a

(+)L − a0(+)) and a

(+)G = (i/

√2)(a

(+)L + a0(+)). The

photon associated to a(+)G is named the gauge (subscript G) photon because this photon

type in vacuum can be eliminated completely by a suitable gauge transformation within

the Lorenz gauge. The photon associated to a(+)NF is called the near-field (subscript NF)

photon [127, 125, 128, 129]. The near-field photon variable is invariant against transfor-

mations within the Lorenz gauge, and in free space a(+)NF (q; t) = 0. In the presence of

field-matter interaction one can relate the positive-frequency part of the component of the

longitudinal electric field along the wave-vector direction to either a(+)NF (q; t) or a

(+)G (q; t).

Since a(+)NF (q; t) = 0 in free space, it is convenient to choose a

(+)NF (q; t). Due to the fact that

E(+)L (r, t) is nonvanishing only in the rim zone of matter (and inside matter) it is reasonable

to call a(+)NF (q; t) a near-field photon variable (wave function).

In the standard model of elementary particles, the spin-one particles, i.e., the photons,the gluons, and the W± and Z0 gauge bosons all are presumed to be massless [242]. Ex-perimentally, the gauge bosons appear to have masses in vacuum. The nonvanishing massof the gauge bosons is associated to the circumstance that the physical vacuum possessescertain properties. In conventional field theory, the state of lowest energy is the vacuum.It turns out however that the vacuum state is non-unique. This degeneracy of the physicalvacuum implies that if one selects one of the lowest energy states as the ground state, theground state is not invariant under the symmetry transformations of the system. This so-called spontaneously broken symmetry leads to the introduction of the Higgs field, and thecoupling of this field to a massless gauge boson gives the gauge boson an apparent mass[209, 96, 97, 98, 243, 156].

The photon remains massless in vacuum since the electromagnetic gauge symmetry isnot spontaneously broken and the photon therefore not coupled to the Higgs field. Never-theless, it is interesting to reflect on whether there is a niche for a photon mass concept inconventional quantum electrodynamics. The question is quite important from a fundamen-tal point of view because a photon only manifests itself via the field-matter interaction. InChapt. 12, a brief analysis of the massive photon field concept is given.

Having in mind that we, in the wave mechanical theory of the photon, give priority toa description where the transverse part of the vector potential (properly normalized) playsthe role as the photon wave function in free space, it is useful to begin the discussion fromthe free-space wave equation for the four-potential, Aµ. Adding to this equation a mass(M) term by the replacement ⇒ − (Mc/~)2, gives us a first guess at a second-orderwave equation for a massive photon field. For M 6= 0, the Lorenz condition can no longerbe considered as a subsidiary condition. The condition must necessarily be satisfied, and byutilizing this the wave equation is reduced to the so-called Proca equation [199], which hasthe form of a quantum mechanical Klein–Gordon wave equation for a spin-one particle. Theelectrodynamical field-matter interaction occurs in the inhomogeneous wave equation for thefour-potential via the four-current density, Jµ, and the question therefore arises whethersituations exist for which Jµ = −KAµ, where K is a positive constant. In a somewhatmore limited context it is interesting to see whether a transverse photon mass can appear.In that case, this would require JT = −KTAT (KT > 0). Linear response theory showsthat this requirement generally is met at high frequencies where the diamagnetic interactiondominates, provided the many-body particle density can be considered homogeneous [127];see Sec. 12.3. The requirement is also satisfied in superconductors, because the paramagneticcontribution to the many-body conductivity vanishes in the superconducting state [212, 45,

The elusive light particle 103

8]. The Meissner effect in BCS superconductivity is related to the diamagnetic effect in bothlinear and nonlinear electrodynamics [107, 108]. Armed with the result for transverse fields,we proceed in Sec. 12.4 to a study of the interaction between the electromagnetic field and arelativistic charged and spinless boson particle. The analysis shows that it is possible to makethe photon field massive also in this case if the norm of the boson particle’s wave functionis constant in space-time. We conclude the chapter on photon mass with a discussion ofthe massive photon propagator, i.e., the propagator associated to the Proca equation. Inthe limit where the mass vanishes one obtains the Feynman photon propagator. In thewave four-vector domain the massive photon propagator can be expressed in a form whichcontains additive contributions from three independent polarization states. In the massivephoton’s rest frame two of the polarizations are transverse and one is longitudinal. The restframe is obtained by a Lorentz velocity boost equal to the massive photon’s group velocityin the original frame.

The birth of quantum field theory in the years 1925-30 did not mean that the search fora wave mechanical description of the assumed point-like light quantum corpuscle was givenup. As mentioned in my brief review entitled “Historical papers on the particle concept oflight” [126], photon wave mechanics dates back to the works of Landau and Peierls [141]and Oppenheimer [178] from 1930-31. From a modern point of view Oppenheimer’s theoryis of particular importance because it relates in a direct manner to the so-called photonenergy wave function formalism, which has received much attention in recent years [16, 123,15, 222, 120]. Starting from Oppenheimer’s approach, a brief account of this formalism isgiven in Chapt. 13.

In Oppenheimer’s note on light quanta, the angular momentum of the photon is ofcentral importance. In 1927 Jordan had (incorrectly) suggested to use a two-componentequation involving the Pauli spin vector (σ) operator to account for the polarization of thelight quanta [103]. Jordan’s proposal essentially is the Weyl equation for a massless neu-trino (spin-1/2 particle). Oppenheimer’s wave equation for the light quantum is obtained byreplacing σ by the spin-1 angular momentum (operator) Σ. In the 3× 3 matrix representa-tion of the components of Σ, the photon wave function becomes a three-component spinordescribing free light quanta composed of positive-helicity species. With the replacementΣ ⇒ −Σ, Oppenheimer’s photon wave equation for negative-helicity species is obtained.The generator of infinitesimal rotations multiplied by ~ (J) which is the sum of the or-bital angular momentum operator (L = −i~r × ∇) and the spin operator (S = ~Σ) is aconstant of motion. The Oppenheimer photon wave equations for the two helicity speciesare form-identical to the dynamical equations for the two Riemann–Silberstein vectors, F±,

and therefore also the equations for these vectors’ positive-frequency parts, F(+)± . This sug-

gests that the wave functions for the photon helicity species in direct space are identified

with properly normalized F(+)± ’s, denoted by f

(+)± =

√

ǫ0/2(e(+)T (r, t)±icb(+)(r, t)). Oppen-

heimer finally takes the six-component bispinor (f(+)+ , f

(+)− ) as the photon wave function.

The Oppenheimer photon wave function is also called the photon energy wave functionbecause the integral of the “density” Φ†Φ over the entire space gives one the total fieldenergy associated to a single Oppenheimer photon.

Since a massless photon certainly is a relativistic object, scalar products involving pho-ton states necessarily must be handled relativistically. In Sec. 13.5, we briefly discuss therelativistic scalar product, and the related Lorentz-invariant integration on the energy shell(which for a massless photon is the light cone). Use of the relativistically invariant scalarproduct in wave-vector (q-)space leads to the satisfactory conclusion that the photon four-momentum and spin in the framework of the Oppenheimer theory appear as quantummechanical mean values of the photon four-momentum [pµ = ~(q,q)] and spin [±~q/q]operators over q-space.

10

Wave mechanics based on transverse vector potential

10.1 Gauge transformation. Covariant and noncovariant gauges

In Sec. 8.4, a covariant vector potential (field), Aµ(x), was introduced [Eq. (8.102)], via aprescription relating to the antisymmetric field tensor [Eq. (8.101)], and it was shown thatone recovers the well-known connections between potentials and fields [Eqs. (8.103) and(8.104)]. Although it was required that the four-component vector potential transforms as avector under coordinate transformation (x→ x′) we did not follow up on this requirement.

Let us now turn to the usual scheme for introducing potentials in classical electrody-namics in Minkowskian space. Only the homogeneous Maxwell–Lorentz equations, viz.,

∇×E(r, t) = − ∂

∂tB(r, t), (10.1)

∇ ·B(r, t) = 0, (10.2)

are needed in this context. Though Eqs. (10.1) and (10.2) have been given before [Eq. (2.1)and (2.4)], it is convenient to repeat these, and a few other ones, in this section. Eq. (10.2)can be satisfied identically by taking

B(r, t) = ∇×A(r, t), (10.3)

where A(r, t) is the so-called vector potential. Given the magnetic field, Eq. (10.3) doesnot determine the vector potential completely. According to Eq. (2.10) [with W = A], weneed to specify also ∇ ·A in order to obtain (up to a space-independent constant) a uniquevector potential. By inserting Eq. (10.3) into Eq. (10.1), it appears that this equation issatisfied provided

E(r, t) = − ∂

∂tA(r, t)−∇φ(r, t), (10.4)

φ(r, t) being the so-called (standard) scalar potential. There always exist functions A andφ such that E and B can be written in the forms in Eqs. (10.3) and (10.4). In fact, thereexists a whole family of potentials for given E and B. Thus, a gauge transformation

A(r, t) ⇒ A′(r, t) = A(r, t) +∇χ(r, t), (10.5)

φ(r, t) ⇒ φ′(r, t) = φ(r, t)− ∂

∂tχ(r, t), (10.6)

from an old, (φ,A), to a new, (φ′,A′), set of potentials with an arbitrary gauge functionχ(r, t) does not change the form of Eqs. (10.3) and (10.4):

B(r, t) = ∇×A(r, t) = ∇×A′(r, t), (10.7)

E(r, t) = − ∂

∂tA(r, t) −∇φ(r, t) = − ∂

∂tA′(r, t)−∇φ′(r, t). (10.8)

105


In order to investigate the question concerning the possible four-vector status of thepotential, and the relation to gauge transformations, we consider the combination

∇ ·A′(r, t) +1

c2∂

∂tφ′(r, t) = ∇ ·A(r, t) +

1

c2∂

∂tφ(r, t) +χ(r, t). (10.9)

If the gauge function is required to satisfy the wave equation

χ(r, t) = 0, (10.10)

the expression

K ≡ ∇ ·A(r, t) +1

c2∂

∂tφ(r, t) (10.11)

is gauge invariant. If one introduces a four-component potential

Aµ(x) = (A0,A) ≡(

φ

c,A

)

, (10.12)

it appears that Eq. (10.11) can be written as

K = ∂µAµ(x). (10.13)

Now, if K is independent of x ≡ (ct, r), Eq. (10.13) is the same in all inertial systems.The quantity Aµ(x) therefore is a contravariant four-vector (field). The first componentof the potential four-vector, A0 is (like φ = cA0) called the scalar potential. The covariantcounterpart (dual vector) to Aµ(x) is

Aµ(x) = (A0,A) = (−A0,A). (10.14)

Gauges for which the gauge function satisfies Eq. (10.10) are called covariant gauges. Withthe choice K = 0, we have

∂µAµ(x) = 0, (10.15)

a condition called the Lorenz condition, after the Danish physicist L. V. Lorenz [149, 122].The potentials are not completely determined by Eq. (10.15), because gauge transformationswhere the gauge function χ(r, t) satisfies Eq. (10.10) still lead to potentials obeying Eq.(10.15). Such transformations are said to be transformations within the Lorenz gauge.

Gauges where the gauge function does not satisfy Eq. (10.10) are called noncovariantgauges. Let me give three examples, which all are useful in electrodynamics.

If one imposes the condition

∇2χ(r, t) = 0 (10.16)

on the gauge function, it appears that

∇ ·A′(r, t) = ∇ ·A(r, t). (10.17)

The gauge invariance of ∇ ·A makes it possible to make the choice

∇ ·A(r, t) = 0. (10.18)

This choice characterizes the Coulomb gauge. The choice in Eq. (10.18) is possible becauseEq. (10.3) only specifies the curl of the vector potential.

Wave mechanics based on transverse vector potential 107

The requirement

∂

∂tχ(r, t) = 0 (10.19)

implies that the scalar potential is gauge invariant, i.e.,

φ′(r, t) = φ(r, t), (10.20)

and in the temporal gauge the constraint

φ(r, t) = 0 (10.21)

is used.If one requires that the derivative of the gauge function in a given direction (let us say

the z-direction or 3-direction) vanishes, that is

∂

∂zχ(r, t) = 0, (10.22)

the corresponding component of the vector potential is gauge invariant:

A′z(r, t) = Az(r, t). (10.23)

In the axial gauge the constraint

Az(r, t)[

= A3(r, t) = A3(r, t)]

= 0 (10.24)

is imposed.In the wave vector-frequency domain all the gauge conditions mentioned above can be

expressed in the compact form

nµAµ(q) = 0, (10.25)

where nµ = (1,0) [temporal gauge], nµ = (0,q) [Coulomb gauge], nµ = (0, 0, 0, 1)[axial gauge] and nµ = qµ [Lorenz gauge].

10.2 Tentative wave function and wave equation for transversephotons

A division of Eq. (10.4) into its transverse and longitudinal parts gives

ET (r, t) = − ∂

∂tAT (r, t), (10.26)

EL(r, t) = − ∂

∂tAL(r, t)− c∇A0(r, t). (10.27)

A similar division of the gauge transformation for the vector potential [Eq. (10.5)] showsthat

A′T (r, t) = AT (r, t), (10.28)

A′L(r, t) = AL(r, t) +∇χ(r, t). (10.29)


From Eq. (10.28) it follows that the transverse part of the vector potential, AT (r, t), thus isgauge invariant, a result which is in agreement with the fact that E and therefore ET , givenby Eq. (10.26), is gauge invariant. A gauge transformation therefore only affects AL andA0, and this in such a manner that the combination on the right-hand side of Eq. (10.27)always stays gauge independent (EL is gauge independent). In free space only Eq. (10.26)is of physical importance, and in photon physics this equation’s positive-frequency part,

E(+)T (r, t) = − ∂

∂tA

(+)T (r, t), (10.30)

plays a prominent role.In a first-quantized formulation we here use the positive-frequency part of the transverse

vector potential, A(+)T (r, t), to introduce a wave function for transverse photons (T-photons)

in free space in the manner described below. In free space, the electromagnetic field satisfiesEq. (2.2), the only Maxwell–Lorentz equation not yet considered in this chapter [Eqs. (2.1)and (2.4) automatically are satisfied in the potential description of electrodynamics, andEq. (2.3), ∇ · E = ∇ · ET = 0, identically, ensures that the electric field in free space isa genuine transverse vector field]. By inserting the relations B [= ∇×A] = ∇ ×AT andET = −∂AT/∂t into Eq. (2.2), and utilizing that ∇ ·AT = 0 (definition of transverse vectorfield), it appears that the transverse part of the vector potential satisfies the wave equation

AT (r, t) = 0. (10.31)

As we know from the analysis in Sec. 2.3, the analytical signal A(+)T (r, t) satisfies a form-

identical wave equation, viz.,

A(+)T (r, t) = 0. (10.32)

If one formally factorizes the d’Alembertian operator as follows:

=

(

√

−∇2 +i

c

∂

∂t

)(

i

c

∂

∂t−√

−∇2

)

, (10.33)

where the symbolic notation√−∇2 is defined in Eq. (2.127), it appears that all solutions

to(

√

−∇2 − i

c

∂

∂t

)

A(+)T (r, t) = 0 (10.34)

are also solutions to the wave equation in (10.32). This result of course is in agreementwith what was obtained from the general analysis of the propagation of analytical signals

[compare Eqs. (2.128) and (10.34)]. If A(+)T (r, t) is divided by a constant N (space and time

independent), A(+)T /N also satisfies Eq. (10.34). Using a proper normalization constant N ,

we tentatively consider

ψT (r, t) ≡ a(+)T (r, t) = N−1A

(+)T (r, t) (10.35)

to be a kind of vectorial wave function of the transverse photon. Multiplication of Eq.(10.34) by N−1~ [~ = h/(2π), where h is Planck’s constant] one obtains

c~√

−∇2a(+)T (r, t) = i~

∂

∂ta(+)T (r, t). (10.36)

We know from our analysis of the polarization of monochromatic and plane light waves


(Sec. 2.5) that the general transverse electric field can be resolved into two mutually or-thogonal components. It is expected therefore that one should be able to extract two inde-pendent wave equations from Eq. (10.36). In order to achieve this goal, we first transfer Eq.(10.36) to the wave-vector domain. Remembering the symbolic notation in Eq. (2.127), weimmediately obtain

c~qa(+)T (q; t) = i~

∂

∂ta(+)T (q; t). (10.37)

The ansatz

a(+)T (q; t) = a

(+)T (q, ω)e−iωt, ω > 0, (10.38)

gives us the time-independent wave equation

c~qa(+)T (q, ω) = ~ωa

(+)T (q, ω). (10.39)

We tentatively interpret H = c~q as the Hamilton operator of a monochromatic plane-wavetransverse photon, and E = ~ω its associated (eigen)energy. From Eq. (10.39) we have~ω = c~q, a result which is in agreement with the wave-particle duality inherent in theEinstein–de Broglie equations

E = ~ω, (10.40)

p = ~q, (10.41)

relating the particle energy (E) and momentum (p) to the wave angular frequency (ω) andwave vector (q). For a particle of rest mass m, the relativistic energy-momentum relationis given by

E = +

√

(cp)2+ (mc2)

2, (10.42)

and for a massless transverse photon Eq. (10.42) is reduced to E = cp = c~q, in agreementwith the conclusion reached from Eq. (10.39). A resolution of the transverse vector potential

a(+)T (q, ω) into two orthogonal components, i.e.,

a(+)T (q, ω) =

∑

s=1,2

a(+)T,s (q, ω)εs(κ), (10.43)

where εs(κ), s = 1, 2, are a pair of generally complex base vectors satisfying the conditionsin Eqs. (2.51) and (2.52), now gives us the two quantum mechanical photon wave equationsin q-space

c~qa(+)T,s (q; t) = i~

∂

∂ta(+)T,s (q; t), s = 1, 2, (10.44)

with

a(+)T,s (q; t) = a

(+)T,s (q, cq)εs(κ)e

−icqt. (10.45)

The transverse photon wave equations in Eq. (10.44), written here in Hamiltonian form(H = c~q), we shall from time to time in the following refer to as Schrodinger-like photonwave equations.

One may of course transfer the wave equations in (10.44) back to direct space following


the procedure outlined in Sec. 2.9. The dynamical evolution of the transverse photon wavefunction thus is governed by the integro-differential equation

∫ ∞

−∞H (|r− r′|)a(+)

T,s (r′; t)d3r′ = i~

∂

∂ta(+)T,s (r; t), (10.46)

where [R = r− r′]

H (|R|) =∫ ∞

−∞~cqeiq·R

d3q

(2π)3(10.47)

is a Hamiltonian density operator. The operator H is singular, and can be expressed in termsof the positive wave-number part of the Dirac delta function [123]. Instead of proceedingwith our analysis of transverse photon wave mechanics from Eq. (10.46), which holds forevery choice of polarization basis, we shall return to Eq. (10.44) and give priority to thecircularly polarized base states.

10.3 Transverse photon as a spin-1 particle

We have seen in Subsec. 2.5.2 that circularly polarized base states can be obtained fromEqs. (2.73)-(2.77) with the choice (∆R,∆I) = (0, 1). Denoting the right- and left-handedbase vectors by e+ and e−, respectively, we have

e+(κ) =1√2

(

εR1 + iεR2)

, (10.48)

e−(κ) =1√2

(

iεR1 + εR2)

eiδ. (10.49)

Note that the δ in Eq. (10.49) is π/2 larger than the δ used in Eq. (2.119). For reasons tobe given below these complex unit vectors are called helicity base vectors. As the readermay prove to herself, the helicity base vectors satisfy the following useful relations:

±iκ× e±(κ) = e±(κ), (10.50)

e∗±(κ) = −ie−iδe∓(κ), (10.51)

e± · e∗± = 1, (10.52)

e± · e± = e± · e∗∓ = 0. (10.53)

Let us now introduce a Cartesian spin-one operator

s = (sx, sy, sz), (10.54)

with components satisfying the angular momentum commutator rules

[si, sj ] = i~∑

k

εijk sk, (10.55)

where εijk is the Levi–Civita symbol [see Eq. (3.57)]. In compact notation the commutatorrelations can be written in the form

s× s = i~s. (10.56)


For what follows it is convenient to take as spin-state basis not the basis usually used inquantum mechanics, but a basis of so-called Cartesian spin states (see Sec. 13.2). In theCartesian basis the various components of s may be represented by 3 × 3 matrices withelements, e.g., given according to

(si)jk =~

iεijk. (10.57)

Written out explicitly one has

~−1sx =

0 0 00 0 −i0 i 0

, (10.58)

~−1sy =

0 0 i0 0 0−i 0 0

, (10.59)

~−1sz =

0 −i 0i 0 00 0 0

. (10.60)

The reader may check that the matrices in Eq. (10.58)-(10.60) do satisfy the angular mo-mentum commutation relations.

A photon helicity operator, h, now is introduced by the definition

h ≡ s · κ. (10.61)

The eigenvalues, λ, and the related eivenvectors, e, for h are obtained solving the matrixeigenvalue problem

he = λe. (10.62)

The solutions to Eq. (10.62) readily may be found by employing the relation

(s · a)F = i~a× F. (10.63)

The reader is encouraged to prove the correctness of Eq. (10.63) to herself. The matrix s · ais antisymmetric, and with the choices a = κ and F = e one has

(s · κ) e = i~κ× e. (10.64)

Since the quantity s ·κ is our helicity operator, the eigenvalue equation in (10.62) now canbe written in the form

i~κ× e = λe. (10.65)

Eq. (10.65) clearly has a solution e = κ with λ = 0. In the context of transverse photons

this solution is not of physical relevance since the vector potential a(+)T,s (q; t) must satisfy

the transversality condition κ · a(+)T,s (q; t) = 0. The two other solutions to the eigenvalue

equation for the helicity operator follow immediately from a comparison of Eqs. (10.50) and(10.65). The right- and left-handed base vectors e± thus are eigenvectors for the helicityoperator, the eigenvalues being ±~:

e = e±, λ = ±~. (10.66)


The quantum mechanical wave equations for the two helicity species are obtained fromEq. (10.44) and (10.45) setting ε1 = e+, andε2 = e−. Hence,

~cqa(+)T,±(q; t)e±(κ) = i~

∂

∂t

(

a(+)T,±(q; t)e±(κ)

)

, (10.67)

where

a(+)T,±(q; t) = e∗±(κ) · a(+)

T,±(q; t) (10.68)

in view of Eqs. (10.52). By combining Eq. (10.67) with the solutions to the transverseeigenvalue equation for the helicity operator, namely

(s · κ) e± = ±~e±, (10.69)

the Schrodinger-like wave equations for the two helicity species take the forms

H±(q)a(+)T,±(q; t)e±(κ) = i~

∂

∂t

(

a(+)T,±(q; t)e±(κ)

)

, (10.70)

with Hamilton operators given by

H±(q) = ±cq (s · κ) = ±c(

s

~· p)

, (10.71)

where p = ~qκ is the momentum operator in the q-representation.In order to obtain the dynamical equations for the transverse photons in direct space,

we use the Fourier integral transformations

a(+)T,±(r, t) =

∫ ∞

−∞a(+)T,±(q; t)e

iq·r d3q

(2π)3, (10.72)

and

~

i

∂

∂xa(+)T,±(r, t) =

∫ ∞

−∞~qxa

(+)T,±(q; t)

d3q

(2π)3, (10.73)

plus two analogous to the one in Eq. (10.73) for the derivatives with respect to y and z.With the particle momentum operator given by the usual expression

p =~

i∇ (10.74)

in the r-representation, the Hamiltonian forms of the quantum mechanical wave equationfor the two transverse photon helicity species read as follow:

±c(

s

~· ~i∇

)

a(+)T,±(r, t) = i~

∂

∂ta(+)T,±(r, t). (10.75)

Since the wave equation in Eq. (10.75) has been established entirely on the basis of classicalelectromagnetics, it is clear that Planck’s constant can be eliminated from the photon waveequations.


10.4 Neutrino wave mechanics. Massive eigenstate neutrinos

Although it has turned out that neutrinos have tiny but nonzero masses [46], usually onecan put the neutrino masses equal to zero when comparing theory and experiment. Theform of the wave equations for a massless neutrino is similar to that of a transverse photon,as we shall see below. A brief discussion of neutrino wave mechanics therefore may helpthe reader to obtain some further insight in photon wave mechanics. As we will understandlater on in this book (Chapt. 12), a transverse photon may acquire an effective mass in itsinteraction with massive charged particles, e.g., in a photon detection process. Theoreticalaspects and methods connected to so-called neutrino flavor mixing may therefore also be ofinterest in photon physics.

Neutrinos are spin-1/2 particles, and the starting point for our discussion thus is theDirac equation, which for a free particle may be written in the form

i~∂ψ

∂t=(

cα · p+ βmc2)

ψ. (10.76)

In order to satisfy the (squared) energy-momentum relation for a relativistic particle (restmass: m) the quantities αi (i = 1 − 3) and β must obey the anticommutator (· · · , · · · )constraints

αi, αj = 2δij1, (10.77)

αi, β = 0, (10.78)

and the requirement

β2 = 1. (10.79)

As already indicated by the presence of the unit tensor (1) on the right sides of Eqs. (10.77)and (10.79), the quantities αi and β cannot be simple numbers. The constraints in Eqs.(10.77)-(10.79), can be satisfied taking αi and β as (square) matrices of at least rank four.Representing αi and β by 4 × 4-matrices, the wave function ψ must be a four-componentspinor. In the so-called standard representation one takes

αi =

(

0 σiσi 0

)

, β =

(

1 00 −1

,

)

(10.80)

where

σ1 =

(

0 11 0

)

, σ2 =

(

0 −ii 0

)

, σ3 =

(

1 00 −1

)

(10.81)

are the well-known 2 × 2 Pauli spin matrices. The Pauli matrices are connected by therelations

σjσk = i∑

l

εjklσl + δjk1 (10.82)

as the reader may prove to herself. The quantity εjkl is the Levi–Civita symbol [Eq. (3.57)],and on this occasion the summation over l is displayed explicitly. The block structure of therepresentation in Eq. (10.80) allows one to write the wave function as a bispinor composedof two-component spinors, i.e.,

ψ =

ψ1

ψ2

ψ3

ψ4

≡(

χφ

)

. (10.83)


The free-particle Dirac equation has positive-energy (superscript (+)) and negative-energy(superscript (−)) plane-wave solutions

ψ(±)(r, t) = N(

χcσ·p

mc2±Eχ

)

exp

[

i

~(p · r∓ Et)

]

, (10.84)

where N is a normalization factor, and E(> 0) is given by Eq. (10.42).For a massless spin-1/2 particle the β-term is absent in the Dirac equation, and in order

to satisfy the constraint in Eq. (10.77) only a realization by 2 × 2-matrices is needed. Itfollows from Eq. (10.82) that the Pauli matrices have anticommutators given by

σj , σk = 2δjk1, (10.85)

and therefore these matrices may serve as a 2 × 2 realization of the αi’s, cf. Eqs. (10.77)and (10.85). In view of this, one arrives at the Weyl equation for a massless neutrino. Inthe r-representation it reads [246]

i~∂

∂tψ(r, t) = c

(

σ · ~i∇

)

ψ(r, t), (10.86)

where the wave function now is a two-component spinor. Remembering that the spin-1/2angular momentum (operator) s is related to the Pauli vector (operator) by s = (~/2)σ, aclose relation between the transverse photon wave equation(s) [Eq. (10.75)] and the Weylequation for a massless neutrino [Eq. (10.86)] starts to emerge.

In 1956, Lee and Yang pointed out that in the case of weak interactions the experimentalevidence for the validity of the particle-antiparticle (charge) conjugation C, the space re-flection (parity) P and the time reversal symmetry T operations was completely inadequate[142].

The discovery of parity violation in weak interactions in 1957 [254], and subsequentnuclear-decay experiments carried out in 1958 [79] indicated that the helicity of the assumedmassless neutrino is negative. The early history of the neutrino and the parity violationsymmetry is described, e.g., in Wolfgang Pauli’s book on physics and philosophy [186]. I urgethe reader to study also Eugene Paul Wigner’s Scientific American article on the violationsof symmetry in physics [252]. In the gauge theory of the electro-weak interaction [209, 242,46, 243, 156], leptonic interactions involving left-handed lepton fields play an importantrole. Such interactions involve chiral couplings in an unsymmetric manner [“chiral” comesfrom the Greek word for “hand,” kheir ]. To describe the wave mechanics of a left-handed(negative helicity; subscript − on the spinor) massless neutrino, one may thus use the two-component wave equation

i~∂

∂tψ(+)− = −cσ · pψ(+)

− . (10.87)

The structure of Eq. (10.87) is analogous to that of the wave equation for a left-handed(negative helicity) transverse photon, namely,

i~∂

∂ta(+)T,− = −cΣ · pa(+)

T,−. (10.88)

The form in Eq. (10.88) comes from Eq. (10.75) [lower sign], and for a direct comparison toEq. (10.87) a slight change in notation has been made: The three-component photon spinor

is written as a(+)T,− (not in boldface), and the Cartesian spin-1 matrix (operator) as Σ ≡ s/~.


The dynamics of a massless antineutrino, supposed to have positive (right-handed) helicity

[two-component wave function ψ(−)+ ], may be described by a wave equation

i~∂

∂tψ(−)+ = cσ · pψ(−)

+ (10.89)

having the same structure as the wave equation for a positive-helicity transverse photon,viz.,

i~∂

∂ta(+)T,+ = cΣ · pa(+)

T,+. (10.90)

A schematic overview of the various helicity eigenstates for photons, antiphotons, neutrinos,and antineutrinos is given in Fig. 10.1.

P N?AN=AP=

FIGURE 10.1Schematic pictures of the helicity eigenstates for assumed massless photons (P), antiphotons(AP), neutrinos (N), and antineutrinos (AN). The direction of the momentum (p) is indi-cated by a thick arrow, and the spin component along p is shown by a thin black arrow. Thetwo arrows belonging to a given particle (antiparticle) are colinear for a positive-helicitystate and anticolinear for a negative-helicity state. Only negative-helicity neutrinos andpositive-helicity antineutrinos exist. It is possible that the neutrino is a so-called Majoranaparticle, i.e., identical to its own antiparticle. Positive- and negative-helicity eigenstatesexist for both photons and antiphotons, and the photon is a Majorana particle.

Since it follows from the free Maxwell equations that the (transverse) photon and theantiphoton are identical, Eq. (10.90) may also be said to describe a positive-helicity antipho-

ton [a(+)T,+ ⇒ a

(−)T,+]. A particle which is its own antiparticle is called a Majorana particle

after the Italian physicist Ettore Majorana [153, 154]. Some theorists conjecture that the

neutrino is a Majorana particle, meaning that one may make the replacement ψ(−)+ ⇒ ψ

(+)+

in Eq. (10.89).In various interactions neutrinos can appear together with spin-1/2 particles which all

have a finite mass. To obtain a convenient description in such cases it is appropriate to usea bispinor (four-component) wave function for the neutrino. To keep in contact with thetwo-component Weyl formalism, we turn to a different representation of the Dirac matrices,viz.,

αi =

(

σi 0o −σi

)

, β =

(

0 11 0

)

. (10.91)

The reader may verify to herself that the choice in Eqs. (10.91) indeed satisfies the con-straints in Eqs. (10.77)-(10.79). [To prove Eq. (10.77) make use of the anticommutatorrelation between the Pauli matrices [Eq. (10.85)]. Let us now use the bispinor notation

ψ =

(

ψ(+)+

ψ(+)−

)

, (10.92)


where the upper (lower) component refers to a particle with positive (negative) helicity. Byinserting Eqs. (10.91) and (10.92) into the Dirac Eq. (10.75) one obtains the following twocoupled equations for the upper and lower components of the bispinor:

i~∂

∂tψ(+)+ = cσ · pψ(+)

+ +mc2ψ(+)− , (10.93)

i~∂

∂tψ(+)− = −cσ · pψ(+)

− +mc2ψ(+)+ . (10.94)

It appears from these equations that the coupling between them is associated with the as-sumed finite rest mass of the Dirac particle. In the limit m → 0 the equations decouple.Equation (10.94) then becomes the Weyl wave equation for a negative-helicity neutrino

(wave function: ψ(+)− ); cf. Eq. (10.87). If the neutrino is a massless Majorana particle Eq.

(10.93) describes the positive-helicity neutrino (wave function: ψ(+)+ ). If not, the replace-

ment ψ(+)+ ⇒ ψ

(−)+ turns Eq. (10.93) into the wave equation for a massless positive-helicity

antineutrino; cf. Eq. (10.89).In the photon case it is sometimes useful to introduce a bispinor (six-component) wave

function

ΨT =

(

a(+)T,+

a(+)T,−

)

(10.95)

for the transverse photon. In the absence of field-matter coupling the positive- and negative-helicity parts of the wave function are uncoupled. In the presence of matter (massive chargedparticles, here), it is possible in the so-called diamagnetic limit to describe the matter’sinfluence on the T-photon dynamics via an effective photon mass. Such a mass will leadto a coupling of the T-photon’s two helicity species, in analogy to the spin-1/2 case [Eqs.(10.93) and (10.94)].

The massless negative-helicity neutrino and the positive-helicity antineutrino can bedescribed in bispinor notation with the help of the projection operators

P± ≡ 1

2(1± γ5) , (10.96)

where

γ5 =

(

1 00 −1

)

. (10.97)

The projection operators, which also may be written in the block forms

P+ =

(

1 00 0

)

, P− =

(

0 00 1

)

, (10.98)

give when acting on the bispinor in Eq. (10.92)

P+ψ =

(

ψ(+)+

0

)

, P−ψ =

(

0

ψ(+)−

)

, (10.99)

as wanted. Note that P 2± = P± and P++ P− = 1, as one must demand. Projection operators

of the form given in Eqs. (10.96) and (10.97), yet with the block matrices having dimension3 × 3, can of course be used to project the positive- and negative-helicity parts out of thesix-component T-photon wave function in Eq. (10.95).


The leptonic electron (νe), muon (νµ), and tau (ντ ) neutrinos are linear combinations ofcertain eigenstate neutrinos νi (i = 1−3), which have mass eigenvaluesmi. Since it turns outthat the mass eigenvalues are different at least two of them must be nonzero. The conclusionthat neutrinos have nonzero masses comes from neutrino (flavor) mixing experiments. Theearliest evidence for neutrino oscillations came from the observation of solar neutrinos in1968. This observation and subsequent experiments on the mass of neutrinos are brieflydescribed in [46], for example.

For simplicity, let us consider the mixing of muon and electron neutrinos from just twoeigenstate neutrinos, say ν1 and ν2. Specified in terms of a mixing angle α, one may write

(

|νµ〉|νe〉

)

=

(

cosα sinα− sinα cosα

)(

|ν1〉|ν2〉

)

. (10.100)

If the orthogonal (〈ν1|ν2〉 = 0) mass eigenstates are normalized (〈ν1|ν1〉 = 〈ν2|ν2〉 = 1) theform in Eq. (10.100) ensures that the muon and electron states are orthonormalized, i.e.,〈νµ|νe〉 = 0 and 〈νµ|νµ〉 = 〈νe|νe〉 = 1. The stationary mass eigenstate develops in timeaccording to

|νi〉(t) = |νi〉(0) exp(

−iEit

~

)

= |νi〉(0) exp−imic

2t

~. (10.101)

A pure muon state created with momentum p at time t = 0, i.e.,

|νµ;p〉(0) = |ν1;p〉 cosα+ |ν2;p〉 sinα, (10.102)

after a time t will become

|?;p〉(t) = |ν1;p〉 exp(

−iE1t

~

)

cosα+ |ν2;p〉 exp(

−iE2t

~

)

sinα. (10.103)

In general, this state is not a pure muon state, but rather a superposition of the muon andelectron state. To understand this, one solves Eq. (10.100) for |ν1〉 and |ν2〉 in terms of |νµ〉and |νe〉 and inserts the result into Eq. (10.103). Hence, one obtains

|?;p〉(t) =|νµ;p〉(

cos2 α exp

(

−iE1t

~

)

+ sin2 α exp

(

−iE2t

~

))

+ |νe;p〉 sinα cosα

(

exp

(

−iE2t

~

)

− exp

(

−iE1t

~

))

. (10.104)

The probability of observing an electron neutrino at time t = t therefore is

|〈νe;p|?;p〉(t)|2 = sin2 (2α) sin2[

(E2 − E1) t

2~

]

. (10.105)

If the mixing angle is 0 or π/2, or if the two involved neutrino eigenstates have equal masses,the muon neutrino cannot turn into an electron neutrino. Since pc ≫ mic

2 (i = 1, 2), onehas approximately

E2 − E1 ≃(

m22 −m2

1

)

c4

2E, (10.106)

where E = pc. Neutrinos travel at a speed close to that of light in vacuum because oftheir small rest masses. If the distance between the source point, where the muon neutrino


is created at t = 0, and the detector, where the electron neutrino is observed at t = t isL ≃ ct, the transition probability in Eq. (10.105) may be written

|〈νe;p|?;p〉(t)|2 = sin2 (2α) sin2L

L0, (10.107)

where

L0 =4E~

(m22 −m2

1) c3

(10.108)

is the so-called oscillation length.

11

Longitudinal and scalar photons. Gauge and

near-field light quanta

11.1 L- and S-photons. Wave equations

It follows from Eqs. (10.44) and (10.45), and the orthonormality condition for the polariza-tion unit vectors [Eq. (2.52)] that the amplitude of the transverse vector potential belongingto index s, viz.,

a(+)T,s (q; t) = ε

∗s(κ) · a(+)

T,s (q; t), (11.1)

satisfies the wave equation (s = 1, 2)

i~∂

∂ta(+)T,s (q; t) = c~qa

(+)T,s (q; t). (11.2)

In free space, and in the Lorenz gauge, the analytical amplitudes of the longitudinal vectorpotential and the scalar potential satisfy form-identical Schrodinger-like dynamical equa-tions, as we now shall see. In free space the longitudinal part of the electric field vanisheseverywhere, so that [Eq. (10.27)]

∂

∂tAL(r, t) + c∇A0(r, t) = 0. (11.3)

In the Lorenz gauge, the potentials entering Eq. (11.3) are related by

∇ ·AL(r, t) +1

c

∂

∂tA0(r, t) = 0, (11.4)

see Eq. (10.15). By combining Eqs. (11.3) and (11.4) one can obtain the two wave equations

AL(r, t) = 0, (11.5)

A0(r, t) = 0. (11.6)

The reader may reach Eq. (11.5) by taking the gradient of Eq. (11.4), and then eliminating∇A0 using Eq. (11.3). Hence, she gets

∇∇ ·AL(r, t)−1

c2∂2

∂t2AL(r, t) = 0. (11.7)

Since 0 = ∇× (∇×AL) = ∇∇ ·AL −∇2AL, the first term in Eq. (11.7) can be replacedby ∇2AL, thus giving Eq. (11.5). By taking the divergence of Eq. (11.3), and eliminatingthereafter ∇ ·AL by means of Eq. (11.4), Eq. (11.6) is obtained.

Following the procedure leading from the wave equation in Eq. (10.31) to the dynamical

119


equation for the scalar wave functions of the transverse photons in the wave-vector domain,viz., Eq. (11.2), one obtains

i~∂

∂ta0(+)(q; t) = c~qa0(+)(q; t), (11.8)

and

i~∂

∂ta(+)L (q; t) = c~qa

(+)L (q; t), (11.9)

where

a(+)L (q; t) = κ · a(+)

L (q; t). (11.10)

It is easy to show that the wave equations in Eqs. (11.5) and (11.6) hold in all covariantgauges [replace the Lorenz condition in Eq. (10.15) by Eq. (10.13) to prove this]. In thesegauges, we consider Eqs. (11.8) and (11.9) as quantum mechanical wave equations for so-called scalar (S) and longitudinal (L) photons in the q-representation. The normalized wavefunction of the scalar photon is

a0(+)(q; t) = N−1(q)A0(+)(q; t), (11.11)

and that of the longitudinal photon

a(+)L (q; t) = N−1(q)A

(+)L (q; t). (11.12)

The normalization constants in Eqs. (11.11) and (11.12) are identical, and depend, as in-dicated, only on the magnitude of the wave vector. The explicit expression for N (q) willbe given in Sec. 11.2. Obviously the energy and momentum of the S- and L-photons satisfythe Einstein–de Broglie relations in Eqs. (10.40) and (10.41).

11.2 L- and S-photon neutralization in free space

In the absence of matter there is no net effect of the longitudinal and scalar photons, as weshall realize below. In the manifest covariant description of the electromagnetic interactionbetween charged particles they play an important role, however. To prove that the effect ofthe L-photon is compensated by that of the S-photon, we start from the Lorenz conditionin the (q; t)-domain, viz.,

1

c

∂

∂tA0(q; t) + iqAL(q; t) = 0, (11.13)

remembering that AL(q; t) = AL(q; t)κ. If Eq. (11.13) is transferred to the (q, ω)-domain,and an integration hereafter is carried out over all positive frequencies, one obtains thefollowing Lorenz condition between the normalized [Eqs. (11.11) and (11.12)] analyticalsignals:

i∂

∂ta0(+)(q; t) = cqa

(+)L (q; t). (11.14)

By combining Eqs.(11.8) and (11.14), it appears that

a0(+)(q; t) = a(+)L (q; t). (11.15)

Longitudinal and scalar photons. Gauge and near-field light quanta 121

The conclusion in Eq. (11.15) may be restated in a manner which connects to themanifestly covariant formulation of QED; see Chapt. 26. The fact that the longitudinalpart of the electric field is identically zero in free space implies that the positive-frequencypart of EL vanishes in the space-time domain, and therefore also in the wave-vector-timerepresentation:

E(+)L (q; t) = 0. (11.16)

In general, we have from Eq. (10.27)

E(+)L (q; t) = − ∂

∂tA

(+)L (q; t) − icqA0(+)(q; t), (11.17)

and with E(+)L (q; t) = E

(+)L (q; t)κ, one obtains via Eqs. (11.10)-(11.12)

E(+)L (q; t) = N (q)

(

− ∂

∂ta(+)L (q; t)− icqa0(+)(q; t)

)

. (11.18)

Using Eq. (11.15), formally one may write

E(+)L (q; t)

(

E(+)L (q; t)

)∗= c2q2|N (q)|2

(

|a(+)L (q; t)|2 − |a0(+)(q; t)|2

)

, (11.19)

since both sides of this equation are zero in free space. By integration over q-space, oneobtains the formal result

ǫ04

∫ ∞

−∞E

(+)L (q; t) ·

(

E(+)L (q; t)

)∗ d3q

(2π)3=

∫ ∞

−∞~cq

(

|a(+)L (q; t)|2 − |a0(+)(q; t)|2

)

d3q,

(11.20)

provided we take (up to a phase factor of modulus one)

N (q) = 2

[

(2π)3~

ǫ0cq

]1

2

. (11.21)

The right-hand side of Eq. (11.20) appears as a difference of the quantum mechanicalmean values of the photon energies (~cq) of the longitudinal and scalar photon states.The somewhat unusual appearance of a minus sign between the two contributions will bediscussed in Sec. 26.4. For a more detailed analysis the reader may consult Refs. [127, 53],for example. The left-hand side of Eq. (11.20) in a sense represents the total energy in thepositive-frequency part of the longitudinal electric field at time t = t, though zero in freespace. If one adds to this the energy in the negative-frequency part of the spectrum, onegets essentially the contribution

HL =ǫ02

∫ ∞

−∞EL(r, t) ·EL(r, t)d

3r (11.22)

of the longitudinal electric field to the total classical field energy, since the interference ofthe positive- and negative-frequencies is washed out when averaged over just a short time.

The real advantage in introducing the longitudinal- and scalar-photon concepts appearsin relation to studies of field-matter interaction in the manifest covariant formalism. Inthe presence of matter the wave amplitude a

(+)L (q; t) and a0(+)(q; t) are no longer identical

[123, 127, 156]. Without going into details here, the difference between the amplitudes are


obtained as follows: The inhomogeneous parts of the solutions to the positive-frequencywave equations in the Lorenz gauge

A(+)L (r, t) = −µ0J

(+)L (r, t), (11.23)

A0(+)(r, t) = −µ0J0(+)(r, t), (11.24)

are given by

A(+)L (r, t) = µ0

∫ ∞

−∞g (|r− r′|, t− t′)J(+)

L (r′, t′)d3r′dt′, (11.25)

A0(+)(r, t) = µ0

∫ ∞

−∞g (|r− r′|, t− t′)J0(+)(r′, t′)d3r′dt′, (11.26)

where (R = r− r′, τ = t− t′)

g(R, τ) = (4πR)−1δ

(

R

c− τ

)

(11.27)

is the scalar (Huygens) propagator. In the (q; t)-domain this propagator is given by

g(q; τ) =c

qθ(τ) sin (cqτ) , (11.28)

θ(τ) being the unit step function. A transformation of Eqs. (11.25) and (11.26) from directspace to wave-vector space subsequently leads to the following difference between the L-and S-wave functions:

a(+)L (q; t)− a0(+)(q; t) = µ0

∫ ∞

−∞g(q, τ)

(

κ ·J (+)L (q; t′)−J

0(+)(q; t′))

dt′. (11.29)

The calligraphic notation is meant to indicate that the four-current density (J 0(+),J(+)L )

is properly normalized. The presence of the unit vector κ stems from the transition from avectorial to a scalar amplitude for the L-photon; see Eq. (11.10). The equation of continuityfor the charge,

κ ·J (+)L (q; t) =

i

cq

∂

∂tJ

0(+)(q; t), (11.30)

allows one to eliminate κ · J(+)L in favor of the time derivative of J 0(+) in Eq. (11.29), if

wished.

11.3 NF- and G-photons

The appearance of the difference between the longitudinal and scalar photon wave functions

in the expression for the longitudinal electric field, E(+)L (q; t) [Eq. (11.18)], and the equality

of these wave functions in free space, makes it fruitful to introduce two new dynamicalvariables by the unitary transformation [125]

a(+)NF (q; t) ≡

i√2

(

a(+)L (q; t)− a0(+)(q; t)

)

, (11.31)

a(+)G (q; t) ≡ 1√

2

(

a(+)L (q; t) + a0(+)(q; t)

)

. (11.32)


As we shall understand soon, the quantities a(+)NF (q; t) and a

(+)G (q; t) may be considered

as the wave functions of two new types of photons. For reasons to be given below, thesephotons are named near-field (NF) and gauge (G) photons. The inverse transformation tothe one in Eqs. (11.31) and (11.32), viz.,

a(+)L (q; t) =

1√2

(

a(+)G (q; t) − ia

(+)NF (q; t)

)

, (11.33)

a0(+)(q; t) =1√2

(

a(+)G (q; t) + ia

(+)NF (q; t)

)

, (11.34)

gives when inserted in Eq. (11.18) the following expression for the positive-frequency longi-tudinal electric scalar field:

E(+)L (q; t) =

N (q)√2

[(

i∂

∂t+ cq

)

a(+)NF (q; t) + i

(

i∂

∂t− cq

)

a(+)G (q; t)

]

. (11.35)

Although E(+)L (q; t) in Eq. (11.35) has been expressed in terms of both a

(+)NF (q; t) and

a(+)G (q; t), it appears from the Lorenz gauge condition, which can be written in the form

(

cq − i∂

∂t

)

a(+)G (q; t) = i

(

cq + i∂

∂t

)

a(+)NF (q; t), (11.36)

as the reader may verify by combining Eqs. (11.14), (11.33), and (11.34), that the electric

field can be given solely in terms of a(+)NF (q; t) or a

(+)G (q; t):

E(+)L (q; t) =

√2N (q)

(

i∂

∂t+ cq

)

a(+)NF (q; t)

=i√2N (q)

(

i∂

∂t− cq

)

a(+)G (q; t). (11.37)

In free space the L- and S-photons satisfy the dynamical equations in (11.8) and (11.9).The corresponding wave equations for the NF- and G-photons are readily obtained bysubtraction and addition of these equations, respectively. Hence,

i~∂

∂ta(+)NF (q; t) = c~qa

(+)NF (q; t), (11.38)

and

i~∂

∂ta(+)G (q; t) = c~qa

(+)G (q; t). (11.39)

The forms of Eqs. (11.38) and (11.39) in a sense justify that a(+)NF (q; t) and a

(+)G (q; t) are

photon wave functions (in q-space). In free space, where the longitudinal part of the electric

field is identically zero, another dynamical equation appears for a(+)NF (q; t) from the first

member of Eq. (11.37), viz.,

i∂

∂ta(+)NF (q; t) = −cqa(+)

NF (q; t). (11.40)

To satisfy simultaneously Eqs. (11.38) and (11.40) one must have

a(+)NF (q; t) = 0. (11.41)

This means that near-field photons only exist when E(+)L (q; t) 6= 0. Since E

(+)L (r, t) is

nonvanishing only in the near-field (rim) zone of matter, it makes sense to use the namenear-field (NF) photon (wave function) for the quantity obtained by the linear combinationin Eq. (11.31).


11.4 Gauge transformation within the Lorenz gauge

One cannot conclude that the wave function for the G-photon, a(+)G (q; t), vanishes in free

space, because the second member of Eq. (11.37) for E(+)L (q; t) = 0 is identical to Eq.

(11.39). However, since there is no net effect of the L- and S-photons in free space, as wehave seen in Sec. 11.2, and because of Eq. (11.41), one should somehow be able to eliminatethe G-photon in free space. As we shall realize below, this elimination can be achieved bya suitable gauge transformation within the Lorenz gauge.

The reader may recall from the analysis in Sec. 10.1 that given the Lorenz condition inEq. (10.15), gauge transformations for which the gauge function χ(r, t) satisfies Eq. (10.10)still lead to sets of potentials obeying Eq. (10.15). Since Eq. (10.10) takes the form

(cq + ω) (cq − ω)χ(q, ω) = 0 (11.42)

in the (ω,q)-domain, it follows that the positive-frequency part of the gauge function,χ(+)(q, ω), satisfies

(cq − ω)χ(+)(q, ω) = 0, (11.43)

a requirement which is tantamount to(

i∂

∂t− cq

)

χ(+)(q; t) = 0 (11.44)

in the wave-vector-time domain.From Eqs. (10.5) and (10.6) the general gauge transformation for the positive-frequency

part of the normalized (with N (q)) potentials can be given. Thus, with a superscript NEWon the new potentials, one has

a(+)NEWL (q; t) = a

(+)L (q; t) + iqχ(+)(q; t), (11.45)

a0(+)NEW (q; t) = a0(+)(q; t)− 1

c

∂

∂tχ(+)(q; t). (11.46)

Utilizing Eqs. (11.31) and (11.32), it appears that the transformations of the NF- andG-potentials are given by

a(+)NEWNF (q; t) = a

(+)NF (q; t) +

1

c√2

(

i∂

∂t− cq

)

χ(+)(q; t), (11.47)

a(+)NEWG (q; t) = a

(+)G (q; t) +

i

c√2

(

i∂

∂t+ cq

)

χ(+)(q; t). (11.48)

Since the positive-frequency gauge function satisfies Eq. (11.44) in the Lorenz gauges, itfollows that the near-field potential is invariant against transformations within the Lorenzgauge, i.e.,

a(+)NEWNF (q; t) = a

(+)NF (q; t). (11.49)

This result was to be expected due to the fact that a(+)NF (q; t) = 0 in free space, where the

gauge-independent quantity E(+)L (q; t) = 0, always. If the time derivative of χ(+)(q; t) in

Eq. (11.48) is eliminated by means of Eq. (11.44), it is realized that the G-photon variabletransforms according to

a(+)NEWG (q; t) = a

(+)G (q; t) + i

√2qχ(+)(q; t). (11.50)


The wave function a(+)G (q; t) therefore is not gauge invariant, and this is the reason that

the associated photon type is named the gauge (G)-photon. Although the G-photon wave

function, a(+)G (q; t), does not necessarily vanish in free space, the NEW G-photon wave

function will vanish, i.e.,

a(+)NEWG (q; t) = 0, (11.51)

provided the choice

χ(+)(q; t) =i

q√2a(+)G (q; t) (11.52)

is made for the gauge function, a choice which of course satisfies the Lorenz condition, cf.Eqs. (11.39) and (11.44).

12

Massive photon field

12.1 Proca equation

In relation to our study of the potential description of electrodynamics in curved-space time(Sec. 8.4) I promised to show that the potential in Minkowski space satisfies the covariantfield equation in (8.116). In preparation for this we first note that the relations between(E,B) and (A, φ = cA0) given in Eqs. (8.103) and (8.104) imply that the Maxwell–Lorentzequations [Eqs. (8.49) and (8.50)] automatically are met. Expressed in terms of the four-potential, the inhomogeneous field equation given in Eq. (8.52) may be written in the form

A−∇

(

∇ ·A+1

c

∂

∂tA0

)

= −µ0J (12.1)

because ∇×(∇×A) = ∇∇·A−∇2A. With J0 = cρ the inhomogeneous Maxwell–Lorentzequation in Eq. (8.51) becomes in the potential version

∇2A0 +1

c

∂

∂t∇ ·A = −µ0J

0 (12.2)

remembering that ǫ0µ0 = c−2. By adding and subtracting a term c−2∂2A0/∂t2, Eq. (12.2)can be given a form closely resembling the one in Eq. (12.1), viz.,

A0 +1

c

∂

∂t

(

∇ ·A+1

c

∂

∂tA0

)

= −µ0J0. (12.3)

In general, Eqs. (12.1) and (12.3) constitute a set of coupled equations for A and A0.However, the equations decouple in the Lorenz gauge, where the potentials satisfy theconstraint in Eq. (10.11) [with K = 0]. Thus, in this gauge we have

A(r, t) = −µ0J(r, t), (12.4)

A0(r, t) = −µ0J0(r, t). (12.5)

Using covariant notation [∂µ =(

1c

∂∂t ,∇

)

, ∂µ =(

− 1c

∂∂t ,∇

)

, Aµ =(

A0,A)

, Jµ =(

J0,J)

, ∂µ∂µ = and ∂µA

µ = ∇ ·A+ c−1 ∂∂tA

0], Eqs. (12.1) and (12.3) can be written inthe compact form

(∂ν∂ν)Aµ(x) − ∂µ (∂νA

ν(x)) = −µ0Jµ(x), µ = 0− 3, (12.6)

with x = xµ = (ct, r). the result in Eq. (12.6) is just the contravariant form of the fieldequation given in covariant form in Eq. (8.116).

Let us now temporarily focus the attention on the homogeneous equation

(∂ν∂ν)Aµ(x) − ∂µ (∂νA

ν(x)) = 0, (12.7)

127


describing a free electromagnetic field, and let us add to this equation a mass term by thereplacement

∂ν∂ν = ⇒ −Q2

C , (12.8)

where [see Eq. (8.138)]

QC =Mc

~(12.9)

is the Compton wave number of a particle with mass M . In a bold fashion we take M as a(hypothetical) photon mass. In the context of the covariant four-photon description givenin Chapt. 11, it is thus assumed that all the photons (two T’s, L and S) possess the samemass. The replacement in Eq. (12.8) gives us the so-called Proca equation(s)

(

−Q2C

)

Aµ(x) − ∂µ (∂νAν(x)) = 0. (12.10)

On taking the four-divergence of this equation, namely,(

−Q2C

)

∂µAµ(x)− (∂µ∂

µ) (∂νAν(x)) = 0 (12.11)

one obtains

Q2C (∂µA

µ(x)) = 0. (12.12)

For M 6= 0, it thus appears that the Lorenz condition

∂µAµ(x) = 0 (12.13)

necessarily must be satisfied. This is in contrast to the usual (correct) electromagnetic theory[Eq. (12.7)], where the Lorenz condition is a subsidiary condition. (Taking the divergenceof Eq. (12.7) gives the identity 0 = 0.) If one wants the freedom of gauge invariance for thefree (Jµ = 0) electromagnetic field the related photon(s) necessarily must be massless.

In view of Eq. (12.13), the Proca equation reduces to(

−Q2C

)

Aµ(x) = 0. (12.14)

Remembering that the squared energy-momentum relation for a relativistic particle of restmass m, i.e.,

E2 =(

mc2)2

+ (pc)2, (12.15)

upon the operator (ˆ) translations

E → E = i~∂

∂t, (12.16)

p → p =~

i∇, (12.17)

leads to the Klein–Gordon equation[

−(mc

~

)2]

ψ(x) = 0 (12.18)

for the scalar boson field ψ(x), a comparison of Eqs. (12.14) and (12.18) indicates thatthe Proca equation is the quantum mechanical Klein–Gordon wave equation for a spin-oneparticle. Although the index µ in Eq. (12.14) runs over four numbers, the Lorenz conditionshows that one has three independent potential components, only.

Massive photon field 129

12.2 Dynamical equations for E and A

In order to keep the hope of saving the gauge invariance freedom and simultaneously alsohave a massive photon field one must turn to the inhomogeneous field equations for thepotentials, written in covariant notation in Eq. (12.6). Before embarking on the generalproblem, let us consider the situation in the Lorenz gauge where

Aµ(x) = −µ0Jµ(x). (12.19)

Compatibility of Eqs. (12.14) and (12.19) requires that the four-current density and thefour-potential are related by

Jµ(x) = −µ−10 Q2

CAµ(x). (12.20)

This equation is definitely not gauge invariant, and this fact appears as a serious obstaclefor a further development. Let us forget this for the moment, and concentrate on setting updynamical field equations (Maxwell–Lorentz equations) in the case where Jµ(x) is given byEq. (12.20).

Using standard (noncovariant) notation, the current and charge densities become

J(r, t) = −Q2C

µ0A(r, t), (12.21)

ρ(r, t) = −Q2C

cµ0A0(r, t). (12.22)

The explicit presence of the vector potential in Eq. (12.21) suggests that A(r, t) servesas one of the field variables. In this case, B(r, t) = ∇ × A(r, t) cannot be another fieldvariable. From the Maxwell–Lorentz equation given in Eq. (8.52) it then appears that thetime development of the electric field is given by the dynamical (first order in time) equation

1

c2∂

∂tE(r, t) =

[(

Q2C −∇2

)

U+∇∇]

·A(r, t). (12.23)

The dynamical equation for A(r, t) is derived starting from Eq. (10.4), which here is re-peated:

∂

∂tA(r, t) = −E(r, t)− c∇A0(r, t). (12.24)

As we have seen earlier, the connection in Eq. (12.24) implies that the Maxwell–Lorentzequation in Eq. (8.49) automatically is satisfied. To have the dynamical equation for A(r, t)related solely to the other dynamical variable, E(r, t), the scalar potential in Eq. (12.24)must be eliminated in favor of E(r, t). This is done by means of the Maxwell–Lorentzequation in Eq. (8.51), remembering that the charge density is given by Eq. (12.22). Hence,one obtains

A0(r, t) = − 1

cQ2C

∇ · E(r, t), (12.25)

as the reader readily may show. Combining Eqs. (12.24) and (12.25), the dynamical equationfor the vector potential follows:

∂

∂tA(r, t) =

(

1

Q2C

∇∇−U

)

·E(r, t). (12.26)

In the Lorenz gauge, and with a constitutive equation given by Eq. (12.20), Eqs. (12.23)and (12.26) constitute a basic set of coupled dynamical equations for the electric field andthe vector potential.


12.3 Diamagnetic interaction: Transverse photon mass

It appears from Eq. (12.1) that the transverse part of the vector potential, AT (r, t), satisfiesthe inhomogeneous wave equation

AT (r, t) = −µ0JT (r, t), (12.27)

where JT (r, t) is the transverse part of the current density. From the analysis in Sec. 10.2we know that AT (r, t) is gauge invariant [Eq. (10.28)], and that ET (r, t) = −∂AT (r, t)/∂t[Eq. (10.26)]. The positive-frequency part of Eq. (12.27), i.e.,

A(+)T (r, t) = −µ0J

(+)T (r, t) (12.28)

relates to the wave mechanics of the transverse photon: In free space NA(+)T (r, t) may be

considered as the vectorial wave function of a transverse photon; cf. the description in Sec.

10.2. The analytical transverse current density, J(+)T (r, t), hence is taken as the source of

the T-photon. To attach a mass,MTR, to a transverse photon (in a given inertial frame) wemust investigate whether it is possible, and under which conditions, to have a constitutiveequation of the form

JT (r, t) = −µ−10

(

QTRC

)2AT (r, t) (12.29)

with

~cQTRC

(

= ~ωTRC

)

=MTRc2. (12.30)

A positive answer to this question, in case, might help us to establish a physical modelleading to the covariant demand in Eq. (12.20).

The linear relation between JT and AT in Eq. (12.29) encourage us to seek the possibleanswer in the framework of linear many-body response theory. I cannot here present thistheory in detail, but the interested reader may find an account suitable for what follows inmy book Quantum Theory of Near-Field Electrodynamics [127]. The starting point for usis the linear and space-time nonlocal microscopic constitutive equation

J(r, t) =

∫ ∞

−∞Σ(r, r′, t, t′) ·ET (r

′, t′)dt′d3r′, (12.31)

where Σ(r, r′, t, t′) is the many-body conductivity tensor. For a medium possessing trans-lational invariance in time, so that Σ(r, r′, t, t′) = Σ(r, r′, t − t′), Eq. (12.31) takes thespatially nonlocal form

J(r;ω) =

∫ ∞

−∞Σ(r, r′;ω) · ET (r

′, ω)d3r′ (12.32)

in the space-frequency domain, because Eq. (12.31) is reduced to a folding integral. Since

ET (r;ω) = iωAT (r;ω), (12.33)

according to Eq. (10.26), we have the J−AT relation

J(r;ω) = iω

∫ ∞

−∞Σ(r, r′;ω) ·AT (r

′;ω)d3r′. (12.34)


To see whether Eq. (12.34) can be reduced to Eq. (12.29), which in the (r;ω)-domain reads

JT (r;ω) = −µ−10

(

QTRC

)2AT (r;ω), (12.35)

we must at least (i) get rid of the spatial nonlocality, (ii) remove the frequency dependencein ωΣ, and (iii) have replaced J by JT . To address the first two questions we need theexplicit expression for the many-body conductivity tensor. For the present purpose only theorbital part of Σ(r, r′;ω) is needed, and in [127], this is given in the form

Σ(r, r′;ω) =i

ω

∑

I,J

~ω

EI − EJ

PJ − PI

~ω + EJ − EIJI→J (r)JJ→I (r

′). (12.36)

In this equation the double summation is over the various many-body stationary states(I, J). The energies of these states are (EI , EJ), and the probabilities that the states areoccupied are denoted by (PI , PJ). The conductivity is a sum of terms in which tensorproducts JI→J (r)JJ→I (r

′) appear. Here JI→J (r) [JJ→I(r′)] is the transition current density

from I to J [J to I]. The spatially nonlocal structure of Σ(r, r′;ω) is reflected in thecircumstance that the transition current densities are to be evaluated at different spacepoints, viz., r and r′.

Inspired by the fact that the particle-wave duality of the transverse photon in Planck’sthermal black-body radiation spectrum is dominated by the photon’s particle properties athigh frequencies (see [60, 126, 189, 190, 64]), let us consider the high-frequency limit (ω →∞) of iωΣ(r, r′;ω). From Eq. (12.36) one immediately obtains the asymptotic expression

limω→∞

(iωΣ(r, r′;ω)) =∑

I,J

PI − PJ

EI − EJJI→J (r)JJ→I (r

′). (12.37)

This certainly is a step in the right direction because the factor relating J(r;ω) and AT (r;ω)has no frequency dependence. However, the various terms in the double summation relateJ and AT in a nonlocal manner, and at first sight this seems incompatible with the localstructure of Eq. (12.29). Despite this, it is possible to prove that the sum of all the termsin Eq. (12.37) reduces to a spatially local form [127]. Hence, for electrons of mass m andcharge −e

limω→∞

(iωΣ(r, r′;ω)) = −e2

mN0(r)δ(r − r′)U, (12.38)

where

N0(r) =∑

I

PIN0,I(r) (12.39)

is the many-body particle density at r. As it emerges, N0(r) is synthesized from the particledensities N0,I(r) in the various eigenstates I, and the probabilities PI that these states areoccupied. By inserting the expression in Eq. (12.38) into Eq. (12.34) we find that

J(r;ω) = −e2

mN0(r)AT (r;ω) (12.40)

at high frequencies. The transverse part of the current density can be filtered out from Eq.(12.40) by means of the transverse delta function, δT (r − r′), a spatially nonlocal dyadicquantity [127]. Thus,

JT (r;ω) = −e2

m

∫ ∞

−∞δT (r − r′) · [N0(r

′)AT (r′;ω)] d3r′. (12.41)


If one assumes that the electron density is homogeneous, N0(r) = N0, Eq. (12.41) is reducedto

JT (r;ω) = −e2

mN0AT (r;ω), (12.42)

but this is precisely the form needed for introducing a mass concept for transverse photons.A comparison of Eqs. (12.35) and (12.42) shows that the T-photon Compton wave numberis given by

QTRC =

ωp

c, (12.43)

where

ωp =

(

N0e2

mǫ0

)1

2

(12.44)

is the cyclic plasma frequency of the assumed homogeneous electron system. The transversephoton mass therefore can be calculated as

MTR =~ωp

c2. (12.45)

Provided the particle density is space independent, or almost so, it is thus meaningful toclaim that the transverse photon in its high-frequency interaction with matter acquires amass. In general it is not possible to take the step leading from Eq. (12.41) to (12.42).

The current density in Eq. (12.40) is called the diamagnetic current density [127, 212].Since the transverse part of the vector potential is gauge invariant [Eq. (10.28)], in con-trast to Eq. (12.20), the relation in Eq. (12.40) is gauge invariant. The diamagnetic currentdensity plays a crucial role in superconductivity, because the many-body state of a su-perconductor exhibits rigidity against transverse excitations [212]. In a sense a transversephoton wave packet composed of frequencies lying below the superconducting gap frequencyacquires a mass in its interaction with a superconductor.

It is important to emphasize that the transverse photon here acquires its mass in theemission or detection interaction with matter.

12.4 Massive vector boson (photon) field

In this section we shall study the interaction between the electromagnetic field and a rela-tivistic spinless boson particle. In a certain approximation this interaction makes the photonfield massive, and in a sense the electrodynamics properties of the coupled system becomesanalogous to that of a relativistic superconductor. The analysis shows that it is possible tomake the photon field massive but the freedom of gauge invariance is lost. In Chapt. 30we shall extend the photon wave mechanical description to the field-quantized level, andrealize that even in so-called vacuum the photon can become massive provided the groundstate of the vacuum (the state of minimum energy) is not a state in which the quantum fieldhas zero average value. Notwithstanding the circumstance that this situation probably doesnot occur for the electromagnetic field, the concept of diamagnetic screening currents inthe electromagnetic vacuum nevertheless is of some interest, for instance in preparation forapplication to weak interactions. Here, the vector bosons seem to acquire their masses via


interaction with the Higgs field, a scalar boson field of zero spin, which have nonvanishingquantum expectation value in “vacuum.” Spontaneous symmetry breaking in the vacuumallows the vector boson to be massive and in a manner where the gauge freedom is kept,although hidden.

To describe the interaction between the electromagnetic field and a spinless chargedboson particle, we start from the contravariant form of the Maxwell–Lorentz equations,given in Eq. (12.6). The first step is a determination of the contravariant current densityof the boson particle. For an isolated particle, the relevant wave equation is the free scalarKlein–Gordon equation, viz.,

(

∂µ∂µ − q2C

)

ψ(x) = 0, (12.46)

where

qC =mc

~(12.47)

is the Compton wave number, assuming the particle to have a mass m. The particle’scoupling to the electromagnetic field is obtained making the minimal coupling substitution

∂µ ⇒ Dµ = ∂µ − iq

~Aµ (12.48)

in Eq. (12.46). Thus,[(

∂µ − iq

~Aµ(x)

)(

∂µ − iq

~Aµ(x)

)

− q2C

]

ψ(x) = 0. (12.49)

In order to determine the four-current density, Jµ(x), one multiplies Eq. (12.49) with thecomplex conjugate wave function from the left. From the resulting equation one subtractsthe complex conjugate Klein–Gordon equation multiplied by the wave function itself fromthe left. Altogether, one obtains

ψ∗DµDµψ − ψD∗

µDµ∗ψ∗ = 0. (12.50)

With the help of the easily derived relation

ψ∗∂µ∂µψ − ψ∂µ∂

µψ∗ = ∂µ (ψ∗∂µψ − ψ∂µψ∗) , (12.51)

the reader may show that Eq. (12.50) goes into

∂µ (ψ∗∂µψ − ψ∂µψ∗)− iq

~[ψ∗∂µ (A

µψ) +Aµψ∂µψ∗ + ψ∂µ (A

µψ∗) +Aµψ∗∂µψ] = 0,

(12.52)

remembering that Aµ∂µ = Aµ∂µ. From Eq. (12.52), we obtain next

∂µ (ψ∗∂µψ − ψ∂µψ∗)− 2iq

~∂µ(

Aµ|ψ|2)

= 0. (12.53)

This equation has the form of a four-divergence of a vector field. If, for a reason to beexplained below, one multiplies Eq. (12.53) by the factor q~/(2im), we finally have

∂µJµ(x) = 0, (12.54)

with

Jµ(x) =q~

2im(ψ∗(x)∂µψ(x)− ψ(x)∂µψ∗(x)) − q2

m|ψ(x)|2Aµ(x). (12.55)


Equation (12.54) may be interpreted as the equation of continuity for the charge, herewritten in covariant notation, and with the components of the four-current density givenby Eq. (12.55).

In a quantum electrodynamic context a gauge transformation requires a simultaneoustransformation of the field, according to [cf. Eqs. (10.5) and (10.6)]

Aµ(x) ⇒ AµNEW (x) = Aµ(x) + ∂µχ(x), (12.56)

and of the (phase of the) wave function as follows

ψ(x) ⇒ ψNEW (x) = exp

(

iq

~χ(x)

)

ψ(x), (12.57)

see e.g. [6]. If one requires that local phase changes of the wave function in space-timeof the form given in Eq. (12.57) do not change any physical properties, the presence ofelectrodynamic couplings between particles follows as a consequence [6, 114] The currentdensity Jµ(x) is readily proved to be gauge invariant. Hence,

JµNEW =

q~

2im(ψ∗

NEW∂µψNEW − ψNEW∂µψ∗NEW )− q2

m|ψNEW |2Aµ

NEW

=q~

2im

(

ψ∗∂µψ − ψ∂µψ∗ +2iq

~|ψ|2∂µχ

)

− q2

m|ψ|2 (Aµ + ∂µχ) = Jµ. (12.58)

As the calculation above shows, the two parts of Jµ(x) in Eq. (12.55) are only gaugeinvariant together. The (often neglected) part containing Aµ(x) thus is needed to ensurethe gauge invariance. In standard notation the probability current and charge densities aregiven by

J =q~

2im(ψ∗∇ψ − ψ∇ψ∗)− q2

m|ψ|2A, (12.59)

and

ρ = − q~

2imc2

(

ψ∗ ∂ψ∂t

− ψ∂ψ∗

∂t

)

− q2

mc2|ψ|2φ, (12.60)

respectively. The need of the factor q~/(2im) can be justified referring to a free particle(A = 0, φ = 0) of energy E, and in an eigenstate for the momentum (p). With box (volume:V ) normalization the wave function

ψ(r, t) = V − 1

2 exp

[

i

~(p · r− Et)

]

(12.61)

leads to a current density

J =q

V

p

m, (12.62)

as expected. In our first-quantized theory the expression given in Eq. (12.60) for the chargedensity is not without problems. Referring again to the free-particle state [Eq. (12.61)], theprobability density (?) calculated from Eq. (12.60) is

ρ

q=

1

V

E

mc2, (12.63)

a quantity which becomes negative for E < 0. Since negative energy solution cannot be


neglected, the first-quantized theory can be only partially successful. When extended to thesecond-quantized level all states defined will have positive definite norms, and the negativeenergy states can be reinterpreted as positive energy states for antiparticles [209, 156].

With the four-current density given by Eq. (12.55) the Maxwell–Lorentz equation forAµ given in Eq. (12.6) becomes

(∂ν∂ν)Aµ − ∂µ (∂νA

ν) = −µ0

[

q~

2im(ψ∗∂µψ − ψ∂µψ∗)− q2

m|ψ|2Aµ

]

, µ = 0− 3.

(12.64)

In Sec. 12.3, we were able to introduce a transverse photon mass in the diamagnetic limitassuming that the many-body electron density was uniform; cf. Eqs. (12.35) and (12.41). Auniform density in the present case corresponds to a Klein–Gordon wave function

ψ(x) = |ψ(x)|eiα(x), (12.65)

where the norm, |ψ(x)|, is robust (constant in space-time). Let us therefore see where theassumption |ψ(x)|=constant takes us. For the current density we obtain, when only thephase α(x) is allowed to vary in space-time, the expression

Jµ(x) =q2

m|ψ|2

(

~

q∂µα(x) −Aµ(x)

)

, (12.66)

and with this Jµ(x) the µth component of the contravariant Maxwell–Lorentz equationtakes the form

(∂ν∂ν)Aµ(x) − ∂µ (∂νA

ν(x)) =µ0q

2

m|ψ(x)|2

(

Aµ(x)− ~

q∂µα(x)

)

. (12.67)

We know from the analysis leading up to Eq. (12.64) that the left side of Eq. (12.67) is gaugeinvariant, and we have shown that the general current density appearing on the right sideof Eq. (12.64) is gauge invariant, too [see Eq. (12.58)], when the phase of the wave functiontransforms as given by Eq. (12.57). It is obvious then that a current density associated witha wave function with robust norm [|ψ|2=constant] also must be gauge invariant. The readermay convince herself of this explicitly by applying the transformation given in Eqs. (12.56)and (12.57) to the quantity Aµ − (~/q)∂µα. The inhomogeneous equation given for Aµ(x)in Eq. (12.67) hence is gauge invariant.

Let us now make a particular choice for the gauge function, viz.,

χ(x) = −~

qα(x). (12.68)

With this choice the vector potential is changed to

AµNEW (x) = Aµ(x)− ~

q∂µα(x), (12.69)

and the wave function to

ψNEW (x) = exp (−iα(x)) exp (iα(x)) |ψ(x)| = |ψ(x)|. (12.70)

In the new gauge our robust wave function therefore is real, and the wave equation for thenew potential takes the form

(∂ν∂ν)Aµ

NEW (x)− ∂µ (∂νAνNEW (x)) =

µ0q2

m|ψ|2Aµ

NEW (x). (12.71)


A comparison to Eq. (12.11) shows that Eq. (12.71) is precisely the Proca equation for amassive free vector boson (photon) field, i.e.,

(

−Q2C

)

AµNEW (x)− ∂µ (∂νA

νNEW (x)) = 0, (12.72)

with Compton wave number

QC =1

c

(

q2|ψ|2mǫ0

)1

2

≡ Ωp

c. (12.73)

The reader may note the close relationship to the massive transverse photon case, whereQTR

C is given by Eq. (12.43) with Eq. (12.44) inserted. For a single particle the particledensity N0 is replaced by |ψ|2. It does not matter here whether the particles are fermionsor bosons. The mass of the vector boson (photon) field is given by the usual relation

M =~

cQC =

~

c2Ωp. (12.74)

By taking the four-divergence of Eq. (12.71) analogous considerations to those leading fromEq. (12.10) to Eqs. (12.13) and (12.14) result in the Proca equations

[

−(

Mc

~

)2]

AµNEW (x) = 0, µ = 0− 3, (12.75)

with constraint

∂µAµNEW (x) = 0. (12.76)

From Eq. (10.9) written in covariant notation (and with A′µ ≡ AµNEW ), i.e.,

∂µAµNEW (x) = ∂µA

µ(x) +χ(x), (12.77)

and Eq. (12.68), it appears that our original Aµ(x) can be regarded as being in a gaugespecified by

∂µAµ(x) = −~

qα(x). (12.78)

In the framework of a field-quantized description we shall examine in Chapt. 30 the situationwhere the particle wave function is not robust, and the gauge choice not specified by Eq.(12.78). We extend this part of the analysis to the field-quantized level to incorporatethe possibility that the screening current stems from a vacuum exhibiting spontaneoussymmetry breaking. We do this despite the fact that no photon experiments have indicatedthe presence of a photon mass in vacuum. As we shall realize in Chapt. 30, the theoreticalsituation is not so obvious because photons only can be studied when interacting withmatter, i.e., in their birth and death processes [127].

12.5 Massive photon propagator

Let us return to the Proca equation for the massive free photon field, viz.,

(

−Q2C

)

Aµ(x) − ∂µ (∂νAν(x)) = 0, (12.79)


cf. Eq. (12.72) with the now unnecessary subscript NEW left out. For reasons to be givenshortly, we do not use the Lorenz constraint in Eq. (12.76) to reduce the Proca equationto the simpler form given in Eq. (12.75). Since we in the remaining part of the book shallbe dealing mainly with electrodynamics in Minkowski space, we from now on will use theconventional notation gµν for the metric tensor of flat space-time, i.e.,

ηµν ≡ gµν. (12.80)

Utilizing that Aµ = gµνAν and ∂νAν = ∂νAν , one may write Eq. (12.79) in the form

[(

−Q2C

)

gµν − ∂µ∂ν]

Aν(x) = 0. (12.81)

By inserting the plane-wave expansion

Aν(x) = (2π)−4∫ ∞

−∞Aν(q)e

iqαxα

d4q (12.82)

for the four-vector potential into Eq. (12.81), one obtains the following algebraic equationsamong the Fourier amplitudes Aν(q):

[(

−q2 −Q2C

)

gµν + qµqν]

Aν(q) = 0, µ = 0− 3, (12.83)

with the abbreviation

q2 ≡ qµqµ = q · q−

(ω

c

)2

. (12.84)

The reader may easily prove Eq. (12.83) by means of the contravariant relation

∂µ (qαxα) = qµ, (12.85)

the corresponding covariant relation, and Eq. (3.45).To the 4× 4 matrix operator O, which contravariant element

Oµν =(

−q2 −Q2C

)

gµν + qµqν , (12.86)

which appears in Eq. (12.83) [OµνAν = 0], we seek to associate a propagator G in the usualmanner [127]:

G = O−1. (12.87)

The inverse operator to O is defined by the requirement

OG = U, (12.88)

where U is the 4× 4 unit matrix. The form in Eq. (12.86) indicates that the µνth elementof G must have the (covariant) form

Gµν = Agµν +Bqµqν , (12.89)

where A and B are constants. We determine these constants from the requirement in Eq.(12.88) that is

[(

−q2 −Q2C

)

gµη + qµqη]

[Agην +Bqηqν ] = δµν . (12.90)


Remembering that gµηgην = δµν , qηqη = q2, and gµηqη = qµ (qηgην = qν), Eq. (12.90) may

be written as follows:

[

1 +(

q2 +Q2C

)

A]

δµν =(

A−BQ2C

)

qµqν , (12.91)

and the factors to δµν and qµqν must therefore both be zero. Thus,

A = − 1

q2 +Q2C

, (12.92)

B = − 1

Q2C (q2 +Q2

C). (12.93)

Written in contravariant form the µνth element of the so-called massive photon propagatorhence is given by

Gµν = − 1

q2 +Q2C

(

gµν +qµqν

Q2C

)

. (12.94)

Remembering that the expression for G was derived from the Proca equation (12.79) forwhich the Lorenz condition is a constraint, we here face a general fact: In all gauge theories achoice of gauge must be made to define the propagators of the gauge quanta. The subsequentanalysis of the massive photon propagator in Eq. (12.94) shall underline the aforementionedfact.

According to standard photon physics free photons are massless, and it is therefore ofinterest to investigate the relation between the massive propagator, G, and the ones used,e.g., in near-field electrodynamics. Naively, one might think the relation would emerge takenthe limit QC → 0 (M → 0) in Eq. (12.94). However, Gµν does not exist in this limit [itdiverges as −qµqν/(q2Q2

C)]. After all, we know that the Proca equation (12.81) has theLorenz condition as a constraint. In the four-wave vector (four-momentum) representationof the Proca equation given in Eq. (12.81), where the Lorenz condition reads qνAν(q) = 0,the term qµqνAν can be omitted, and the elements Oµν [Eq. (12.86)] hence replaced byOµν = −

(

q2 +Q2C

)

gµν . The propagator (elements) related to this, namely,

GµνF (q;M) =

−gµνq2 +Q2

C

(12.95)

is the so-called massive Feynman (subscript F) propagator (in the four-momentum repre-

sentation). The scalar −(

q2 +Q2C

)−1=[

(ω/c)2 − q · q−Q2C

]−1

often is referred to as the

Feynman meson propagator, a propagator of relevance for the mesons of the Klein–Gordonfield. In Eq. (12.95) one may take the limit QC → 0, and hence obtain the Feynman (mass-less) photon propagator [156], with elements

GµνF (q;M → 0) = −g

µν

q2. (12.96)

The locations of the poles in the massive Feynman propagator are given by the dispersionrelations

ω = ±c(

q · q+Q2C

)1

2 , (12.97)

which, when multiplied by ~, are just the particle energy-momentum relations for positiveand negative energies; cf. Eq. (12.15). In the massless limit Eqs. (12.97) reduce to the

free-space dispersion relations of light, ω = ±c (q · q) 1

2 .


Further insight in the structure of Gµν can be obtained by expanding the metric tensor(elements) as follows:

gµν =

3∑

r=0

ζrεµr (q)ε

νr (q), (12.98)

where ζ0 = −1 and ζ1 = ζ2 = ζ3 = +1. The decomposition in Eq. (12.98) is a completenessrelation for the four linearly independent unit polarization vectors εµr (q), r = 0−3, whichhere are chosen to be real, and to satisfy the orthonormality condition

εrµ(q)εµs (q) = ζrδrs. (12.99)

We now make the choice that the polarization vector ǫµ0 is parallel to the four-wave vectorthat is

ǫµ0 = Q−1C qµ. (12.100)

By means of Eqs. (12.84) and (12.97) [squared] the reader may check that the vector inEq. (12.100) satisfies the normality condition ε0µǫ

µ0 = −1. The Lorenz constraint on the

four-potential,

qµAµ(q) = 0, (12.101)

which at a certain stage must be invoked in the analysis, implies that the general four-component Fourier amplitude of the potential can be expressed via a superposition of con-tributions from three independent polarization states. With the choice in Eq. (12.100) theseare ε µ

r with r = 1, 2, 3, and from the Lorenz constraint it follows that

qµεµr (q) = 0, r = 1− 3. (12.102)

With the choice for ǫ µ0 given in Eq. (12.100), the orthogonality of the polarization vec-

tors of course requires that Eq. (12.102) is obeyed. With ǫ µ0 given by Eq. (12.100) a

completeness relation of the form

3∑

r=1

ε µr (q)ε ν

r (q) = gµν +Q−2C qµqν (12.103)

follows for the three independent unit polarization vectors. By combining Eqs. (12.94) and(12.103) the element µν of the massive photon propagator can be written

Gµν(q) = − 1

q2 +Q2C

3∑

r=1

εµr (q)ενr (q). (12.104)

For the subsequent discussion it is useful to work in a frame in which

qµ = (ω/c, 0, 0, |q|) , (12.105)

remembering that ωq ≡ ω = c(

q · q+Q2C

)1

2 . In this frame a suitable choice for the three(r = 1− 3) polarization vectors is

ε µ1 = (0, 1, 0, 0), (12.106)

ε µ2 = (0, 0, 1, 0), (12.107)

ε µ3 = (|q|, 0, 0, ω/c)Q−1

C . (12.108)


These vectors are clearly orthonormal, and the reader may verify that they satisfy thecompleteness relation in Eq. (12.103). In the rest (REST) frame of the free massive photon,the photon momentum is zero; that is, q = 0. The ǫ µ

0 -vector then becomes purely time-like, i.e.,

ǫ µ0 (REST ) = (1, 0, 0, 0). (12.109)

The two vectors ε µ1 [Eq. (12.106)] and ε µ

2 [Eq. (12.107)], which are purely space-likeand perpendicular to the q-direction (and thus called transverse polarization vectors) areindependent of |q|. Hence they are not altered when we go to the rest frame:

ε µ1 = ε µ

1 (REST ) = (0, 1, 0, 0), (12.110)

ε µ2 = ε µ

2 (REST ) = (0, 0, 1, 0). (12.111)

The longitudinal polarization vector ε µ3 [Eq. (12.108)], which spatial part is parallel to

the q-direction, also has a time-like part. In the rest frame this part disappears, i.e.,

ε µ3 (REST ) = (0, 0, 0, 1). (12.112)

The fact that we have just three independent polarization vectors is what we expect for amassive spin-1 particle. One may replace the set ε µ

1 and ε µ2 by the complex helicity

unit vectors

ε µ+ =

1√2(0, 1, i, 0), (12.113)

ε µ− =

1√2(0, 1,−i, 0), (12.114)

using the prescription in Eqs. (10.48) and (10.49) [with δ = −π/2]. According to the analysisin Sec. 10.3, these eigenstates have the eigenvalues λ = +1 and −1 for the helicity. Thethird state with polarization ε µ

3 (REST ) has the eigenvalue λ = 0. For a massless freephoton the longitudinal state plays no physical role as noted previously.

Let us briefly consider the frame velocity (V) needed to make a Lorentz transformationfrom the frame in which qµ is given by Eq. (12.105) to the massive photon’s rest frame.From Eq. (3.76), we have

q+ κ[

(γ − 1) |q| − γVω

c2

]

= q(REST ) = 0, (12.115)

since the boost is parallel to κ = q/|q| [V ‖ κ, q‖ = |q|]. Simple manipulations of Eq.(12.115) lead to a frame velocity

V =c2q

ω. (12.116)

This velocity must equal the particle’s group velocity in the original frame. The group ve-locity can be calculated from the positive-frequency dispersion relation given in Eq. (12.97),and one obtains

∇qω(q) =c2q

ω, (12.117)

in agreement with the result for V.


In the inertial frame where qµ is given by Eq. (12.105) one may define a transverse(subscript T) propagator by

GµνT (q) ≡ − 1

q2 +Q2C

2∑

r=1

εµr (q)ενr (q), [SF ]. (12.118)

After Eq. (12.118) I have added [SF] to underline that the definition belongs to a SpecialFrame. Since the transverse propagator is purely space-like its elements can be written inthe form

GµνT (q) = Gij

T (q)δµiδνj , (12.119)

where

GijT (q) = −U− κκ

q2 +Q2C

(12.120)

is the effective transverse propagator in the subspace of three-vectors. This propagatoris identical to the transverse part of the massive Feynman propagator in the space-likesubspace; see Eq. (12.95). In the massless limit one obtains from Eq. (12.120) the well-known transverse photon propagator [127]

GT (q;M → 0) = − U− κκ(

ωc

)2 − q · q. (12.121)

The remaining part of the massive photon propagator, viz.,

GµνL (q) ≡ Gµν(q)− Gµν

T (q) = − ε µ3 ε

µ3

q2 +Q2C

, [SF ], (12.122)

which we (loosely speaking) call the longitudinal (subscript L) propagator, has both time-and space-like components in our special frame. This propagator relates in a physicallyinteresting manner to the spatial localization problem for transverse photons, as we shallrealize in Part VI; see also [123, 127, 128].

13

Photon energy wave function formalism

13.1 The Oppenheimer light quantum theory

In Chapt. 10, photon wave mechanics was based on the transverse part of the vector po-tential. As we shall realize soon, it is quite possible to claim rightfully that other quantitiesconstructed from electromagnetic field vectors may be considered as the photon wave func-tion in free space. The crucial point is that whatever choice one makes for the photonwave function the subsequent photon-matter interactions must all result in the same ob-servable consequences. In this chapter an important and interesting choice based on theRiemann–Silberstein vectors, which were introduced in Sec. 2.8, will be described.

Before we relate photon wave mechanics to the free space Maxwell equations it is fruit-ful to establish some general features for a light quantum theory following the approachoriginally suggested by Oppenheimer [178]. In this approach the angular momentum of thephoton takes a central position. Bearing in mind the Weyl equation for a massless neutrino,viz.,

(

σ ·∇+1

c

∂

∂t

)

ψ(r, t) = 0, (13.1)

[Eq. (10.86)], it appears natural to replace the Pauli vector σ, related to the angular mo-mentum of a spin-1/2 particle (here a massless neutrino), by an (dimensionless) angularmomentum operator Σ = (Σ1,Σ2,Σ3) which components may be represented by 3× 3 Her-mitian matrices, relevant for a spin-1 particle. (Here, and below the operator symbol ˆ isleft out from the notation.) With Oppenheimer we thus assume that

(

Σ ·∇+1

c

∂

∂t

)

Φ+(r, t) = 0 (13.2)

is a wave equation for a massless light quantum in space-time. In the 3 × 3 matrix repre-sentation the photon wave function Φ+ = (Φ+,1,Φ+,2,Φ+,3) is a three-component spinor.As in the neutrino case, a unit tensor in front of c−1∂/∂t is left out from the notationin Eq. (13.2), and in subsequent equations. A subscript + appears on the wave functionto indicate that Eq. (13.2) describes, as we soon shall see, a photon composed of positivehelicity components, cf. Eq. (10.89) for the positive-helicity antineutrino. If one makes theidentification

Φ+(r, t) ⇔ a(+)T,+(r, t), (13.3)

Eq. (13.2) becomes the wave equation for a photon composed of positive-helicity speciesand described by a transverse vector potential wave function [see Eq. (10.75)]. The (dimen-sionless) angular momentum operator of course must satisfy the fundamental commutatorrelation

Σ×Σ = iΣ, (13.4)

143


cf. Eq. (10.56), with (s ≡)s = ~Σ. The Oppenheimer wave equation for a light quantumcomposed of negative-helicity species is

(

−Σ ·∇+1

c

∂

∂t

)

Φ−(r, t) = 0, (13.5)

where Φ−(r, t) is the associated three-component spinor. The connection of Eq. (13.5) tonegative helicities is proved soon. Together we write Eqs. (13.2) and (13.5) in the compactform

(

±Σ ·∇+1

c

∂

∂t

)

Φ±(r, t) = 0, Σ = Σ†. (13.6)

In the absence of interaction with matter, a photon cannot be created nor destroyed.In consequence, there exists a photon conservation law, and one must be able to derive itfrom Eqs. (13.6). To this end, we need the Hermitian conjugate of Eq. (13.6), i.e., since theangular momentum operator is Hermitian (Σ = Σ†)

∇Φ†± ·Σ+

1

c

∂

∂tΦ†

± = 0, (13.7)

If one multiplies Eqs. (13.6) by Φ†± from the left, and Eqs. (13.7) by Φ± from the right, and

thereafter adds the resulting equations, one obtains

∂

∂t

(

Φ†±Φ±

)

+∇ ·[

Φ†± (±cΣ)Φ±

]

= 0, (13.8)

as the reader may show. By identifying

ρ±(r, t) = Φ†±(r, t)Φ±(r, t) (13.9)

as probability densities for the two photon types (+,−), and

j±(r, t) = ±cΦ†±(r, t)ΣΦ±(r, t), (13.10)

as the related photon probability current densities, Eq. (13.8) is just the law expressing thephoton conservation:

∇ · j±(r, t) +∂

∂tρ±(r, t) = 0. (13.11)

It appears that separate conservation laws hold for the two helicity species. Only by inter-action with matter can the photon helicity be changed. In view of Eq. (13.10) the quantities

V± = ±cΣ (13.12)

in some respect play the role as photon velocity operators. For massless antineutrinos andneutrinos the velocity operators are cσ and −cσ, respectively, cf. Eqs. (10.89) and (10.87).For a massive Dirac spin-1/2 particle (e.g., an electron) the quantity cα apparently servesas a velocity operator [see Eq. (10.75)]. The components of the Hermitian photon velocityoperator do not commute, and are therefore not precisely measurable simultaneously. As weshall see below, the eigenvalues of a given velocity operator component are 0 and ±c. For amassless antineutrino (neutrino) the eigenvalues of the components of the velocity operatoris +c (−c). The eigenvalues of the velocity components of a Dirac particle (electron) arealso ±c. At first sight this seems to contradict the fact that a massive particle alwayspropagates with a speed less than the speed of light in vacuum. An extended analysis due

Photon energy wave function formalism 145

to Foldy and Wouthuysen [74] shows that the true velocity operator for the electron isc2p/[(cp)2 + (mc2)2]1/2. The components of this commute, and the numerical eigenvaluesare always less than c. The change from the original Dirac representation to the Foldy–Wouthuysen representation of relativistic wave mechanics is discussed in [86], for example.The introduction of the photon probabilities ρ±(r, t) does not allow one to conclude that aphoton can be completely localized in space-time. In Parts V and VI, we shall discuss thephoton localizability problem as this appears in relation to emission processes from variousphoton source domains. The Foldy–Wouthuysen analysis shows that the problem with Diracvelocity operator, cα, is related to the lack of complete localizability of the electron. Thus,the r-vector operator in the Dirac equation turns out to be a sort of mean position operatorfor the spin-1/2 particle.

By considering the transformation of a 3-component vector field (here Φ+ or Φ−) underinfinitesimal rotations in space one obtains the expression

J = −i~r×∇+ ~Σ (13.13)

for the generator of infinitesimal rotations (multiplied by ~) [127, 178, 131]. The quantityJ is the sum of the orbital angular momentum operator,

L = −i~r×∇, (13.14)

and the spin operator, S = ~Σ:

J = L+ S. (13.15)

The total angular momentum of the photon is a constant of the motion. In order to provethis we note that the photon wave equations in Eq. (13.6) are written in Hamiltonian form

H±Φ± = i~∂

∂tΦ±, (13.16)

where the Hamilton operators are

H± = ∓i~cΣ ·∇ = ±cΣ · p, (13.17)

p(≡ p) = −i~∇ being the particle momentum operator. The angular momentum has noexplicit time dependence and therefore it will be a conserved quantity provided it commuteswith the Hamilton operator(s); that is,

[J, H±] = 0, (13.18)

or equivalently

[L, H±] = − [S, H±] . (13.19)

To demonstrate the correctness of Eq. (13.19) one utilizes the commutator relation

[r,p] = i~U, (13.20)

and the last expressions given for H± in Eq. (13.17), to determine the [L, H±] commutators.Denoting vector components by indices x, y, z, one obtains for the x-component

[Lx,Σ · p] = [ypz − zpy,Σxpx +Σypy +Σzpz]

= [ypz,Σypy]− [zpy,Σzpz]

= Σypz [y, py]− Σzpy [z, pz]

= −i~ (pyΣz − pzΣy) = −i~ (p×Σ)x , (13.21)


and consequently

[L, H±] = ∓i~cp×Σ. (13.22)

To calculate the [S, H±] = [~Σ, H±] commutators we just need the commutator relationsbetween the components of Σ, viz., Eq. (13.4). Since

[Sx,Σ · p] = ~ [Σx,Σxpx +Σypy +Σzpz]

= ~py [Σx,Σy] + ~pz [Σx,Σz]

= i~ (pyΣz − pzΣy) = i~ (p×Σ)x , (13.23)

one obviously obtains

[S, H±] = ±i~cp×Σ. (13.24)

The results in Eqs. (13.22) and (13.24) show that Eq. (13.19) is correct, and the totalangular momentum of the photon therefore is conserved in free space.

The association of Φ+ and Φ− with photons composed of positive (+) and negative(−)−helicity species follows upon the insertion of the plane-wave states

Φ± = N± exp [i (q · r− ωt)] (13.25)

into the dynamical equations given in Eq. (13.6). As the reader may convince herself of,this leads to the matrix eigenvalue problems

(±cqh− ~ω)N± = 0 (13.26)

for the helicity matrix h = ~κ ·Σ (= κ · J). As shown in Sec. 10.3, the eigenvalues of thisequation are

λ± = ±~ω

cq=

±1∓10

. (13.27)

We know that the eigenvalues λ± = 0 must be abandoned because the wave functions offree photons shall obey the transversality condition

κ · (Φ±,1,Φ±,2,Φ±,3) = 0. (13.28)

Since particles have ω > 0, Eq. (13.27) shows that λ+ = +1 corresponds to positive helicityfor the components of Φ+, and λ− = −1 corresponds to negative helicity of the componentssuperimposed in Φ−.

13.2 Interlude: From spherical to Cartesian representation

In Sec. 10.3 a Cartesian representation was employed for the components of the dimension-less spin-one operator Σ; see Eqs. (10.58)-(10.60). The standard theory [52, 210] expressesthe components of a general angular momentum operator J in the so-called spherical rep-resentation, and it might therefore be of interest briefly to study the transition from thespherical to the Cartesian representation for Σ.


In the spherical representation a basis is used in which the operators J2 and Jz arediagonal matrices, and with eigenvalues (ev)

ev(J2) = j (j + 1) ~2, (13.29)

ev(Jz) = m~, m = −j,−j + 1, · · · , j − 1, j. (13.30)

The matrix elements of Jz hence are given by

〈j,m|Jz|j′,m′〉 = m~δjj′δmm′ . (13.31)

The quantum number j is integral for bosons and half-integral for fermions. To determinethe matrix elements of Jx and Jy it is convenient to express these operators in terms of two

Hermitian conjugate operators J+ and J− = J†+ as follows

Jx =1

2(J+ + J−) , (13.32)

Jy =1

2i(J+ − J−) . (13.33)

The matrix elements of J+ and J− are given by [52]

〈j,m|J±|j′,m′〉 = ~ [j (j + 1)−m′ (m′ ± 1)]1

2 δjj′δm,m′±1. (13.34)

For a spin-one particle, where j = 1 and m = −1, 0, 1, Eqs. (13.31)-(13.34) lead to thefollowing spherical representation for the components of the dimensionless operator Σ:

Σx =1√2

0 1 01 0 10 1 0

, (13.35)

Σy =1√2

0 −i 0i 0 −i0 i 0

, (13.36)

Σz =1√2

1 0 00 0 00 0 −1

. (13.37)

The eigenvectors in the spherical representation are easily calculated from the eigenvalueproblem for Σz. With the abbreviated notation |1, 1〉 = |+1〉, |1, 0〉 = |0〉 and |1,−1〉 = |−1〉the normalized column vectors are

|+ 1〉 =

100

, |0〉 =

010

, | − 1〉 =

001

. (13.38)

The relations between the Cartesian (|x〉, |y〉, |z〉) and spherical (| + 1〉, |0〉, | − 1〉) basevectors we here take as follows:

|+ 1〉 = 1√2(|x〉+ i|y〉) , (13.39)

| − 1〉 = eiβ√2(−|x〉+ i|y〉) , (13.40)

|0〉 = |z〉. (13.41)


In Eq. (13.40) we have included an arbitrary phase factor exp(iβ) to obtain a certainflexibility in the Cartesian representation of the components of Σ. One may generalize theconnection between the two sets of base vectors by including arbitrary phase factors also inEqs. (13.39) and (13.41). Most often one takes β = 0 or β = π. The flexibility introducedby means of β is similar to the one we have used in the choice of the helicity unit vectors;see Eqs. (10.48) and (10.49). The notations in Eqs. (10.48) and (10.49) and in Eqs. (13.39)and (13.40) match for β = δ − π/2.

From the matrix elements of an operator O in the spherical representation (|m〉, |n〉)the matrix elements of O in the Cartesian representation (|i〉, |j〉) can be obtained from theformula

〈i|O|j〉 =∑

m,n

〈i|m〉〈m|O|n〉〈n|j〉, (13.42)

since O = 1O1, where 1 =∑

m |m〉〈m|(=∑n |n〉〈n|) is the unit operator. The relations inEqs. (13.39)-(13.41) give one the table

〈x| + 1〉〈x| − 1〉〈x|0〉〈y|+ 1〉〈y| − 1〉〈y|0〉〈z|+ 1〉〈z| − 1〉〈z|0〉

=1√2

1 −eiβ 0i ieiβ 0

0 0√2

. (13.43)

Utilizing this table, and the spherical representation given in Eqs. (13.35)-(13.37), in Eq.(13.42), the Cartesian Hermitian matrix representation for the components of Σ can bedetermined. As the reader may show, one obtains

〈i|Σx|j〉 =1

2

0 0 1− eiβ

0 0 i(

1 + eiβ)

1− e−iβ −i(

1 + e−iβ)

0

, (13.44)

〈i|Σy|j〉 =1

2

0 0 −i(

1 + eiβ)

0 0 1− eiβ

i(

1 + e−iβ)

1− e−iβ 0

, (13.45)

〈i|Σz|j〉 =1

2

0 −i 0i 0 00 0 0

. (13.46)

Since a commutator relation is independent of the representation used, the matrices in Eqs.(13.44)-(13.46) satisfy the commutator relation Σ×Σ = iΣ [Eq. (13.4)], as one may checkby a direct calculation. The representation independence of a commutator relation [a, b] = cfollows directly from Eq. (13.42) taking O = [a, b] − c. The eigenvalues of the operatorsΣx, Σy, and Σz are independent of the representation used, and thus always λ = +1, 0,−1.The eigenfunctions belonging to these eigenvalues do depend on the representation. In theCartesian representation, the normalized eigenvectors of Σz are (each up to an overall phasefactor)

| ± 1;Cz〉 =1√2

1±i0

; |0;Cz〉 =

001

. (13.47)

Inside the kets a “Cz” has been added to remember that the results belong to the z-component of Σ in the Cartesian (C) representation. The normalized eigenvectors of Σx


and Σy depend on β, and they are

| ± 1;Cx〉 =1√2

∓ieiβ/2 sin β2

±ieiβ/2 cos β2

1

; |0;Cx〉 =

cos β2

sin β2

0

, (13.48)

and

| ± 1;Cy〉 =1√2

∓ieiβ/2 cos β2

∓ieiβ/2 sin β2

1

; |0;Cy〉 =

− sin β2

cos β2

0

, (13.49)

as the reader is encouraged to show. She may check also that the set of eigenvectors belongingto a given Cartesian component of Σ are mutually orthogonal.

To make the bridge to the treatment of the transverse photon as a spin-1 particle (Sec.10.3), we set the phase β = 0. With this choice the Cartesian representations of Σx and Σy

become

〈i|Σx|j〉 =

0 0 00 0 i0 −i 0

. (13.50)

〈i|Σy|j〉 =

0 0 −i0 0 0i 0 0

. (13.51)

The associated eigenvectors, obtained from Eqs. (13.48) and (13.49), become

| ± 1;Cx〉 =1√2

0±i1

; |0;Cx〉 =

100

, (13.52)

| ± 1;Cy〉 =1√2

∓i01

; |0;Cy〉 =

010

, (13.53)

The states | ± 1;Cx〉 in Eq. (13.52) are circularly polarized in a plane perpendicular to thex-axis, and with respect to the positive x-direction the state |+1;Cx〉 has negative helicity,and the state | − 1;Cx〉 has positive helicity. This interchange of helicity for the two statescompared to the result in Sec. 10.3, stems from the fact that the matrix representationsfor Σx in Sec. 10.3 [Eq. (10.58)] and above [Eq. (13.50)] are the complex conjugate of eachother. The helicity interchange also appears for the states | ± 1;Cy〉, and the reason is thesame: The matrix representations given for Σy in Eqs. (10.60) and (13.51) are each other’scomplex conjugate. The two matrix representations for Σz are the same [compare Eqs.(10.60) and (13.46)].

In order to obtain the quantum mechanical wave equation in Eq. (10.75) it is necessarythat the relation in Eq. (10.63) holds in a Cartesian representation. Denoting the dimen-sionless spin vector in Sec. 10.3 by ΣOLD, and the one used in this section by ΣNEW , weknow that both satisfy the angular momentum commutator rule, that is

ΣOLD ×ΣOLD = iΣOLD, ΣNEW ×ΣNEW = iΣNEW . (13.54)

Since only purely imaginary components appear in the matrix representations of ΣOLDx and

ΣOLDy it is clear that

ΣNEW =(

−ΣOLDx ,−ΣOLD

y ,ΣOLDz

)

. (13.55)


A relation similar to the one in Eq. (10.63), viz.,

(

ΣOLD · a)

F = ia× F, (13.56)

does not hold for ΣNEW in general. The Cartesian representation given for Σ = s/~ in Eqs.(10.58)-(10.60) therefore is the relevant one for photon physics.

13.3 Photons and antiphotons: Bispinor wave functions

In order to relate the Oppenheimer theory to the free-space Maxwell equations let us writethe photon wave equations in Eqs. (13.16) in a slightly different notation, viz.,

∓i~cΣ ·∇f(+)± (r, t) = i~

∂

∂tf(+)± (r, t), (13.57)

where f(+)± (r, t) are the vectorial wave functions for photons composed of positive and

negative-helicity species. A superscript (+) has been added on the f±’s to remind the readerthat only positive frequencies enter the frequency composition for a particle state. By means

of Eq. (10.63) [with a = ∇ and F = f(+)± ], Eqs. (13.57) are transformed to the forms

i~∂

∂tf(+)± (r, t) = ±~c∇× f

(+)± (r, t). (13.58)

At this stage it appears that Eqs. (13.58) are form-identical to the dynamical equations forthe two Riemann–Silberstein vectors F±(r, t); see Eqs. (2.110). It is obvious from the anal-

ysis in Secs. 2.4 and 2.8, that also the positive-frequency parts of these vectors, F(+)± (r, t),

satisfy dynamical equations of the same forms; that is,

i∂

∂tF

(+)± (r, t) = ±c∇× F

(+)± (r, t). (13.59)

The considerations above suggest that one may take

f(+)± (r, t) =

√

ǫ02

(

e(+)T (r, t)± icb(+)(r, t)

)

(13.60)

as the wave functions for the two photon wave packets composed of respectively positive and

negative-helicity species. In Eqs. (13.60), e(+)T (r, t) and b(+)(r, t) are the positive-frequency

parts of the transverse electric and magnetic fields. Small letters have been used for thesefields to indicate that their magnitude have been so chosen that the photon wave functionsare properly normalized (see below). The objects in Eqs. (13.60) are called photon energy

wave functions because the related “probability densities” f(+)± · (f (+)

± )∗ have the dimensionof energy density. To the extent that this concept is physically meaningful it relates to theprobability of having the photon energy localized at a given space-time point. In analogyto what we suggested when we related photon wave mechanics to the transverse vectorpotential in Chapt. 10, it is for systematic reasons useful to take the six-component bispinor

Φ(r, t) =

(

Φ+(r, t)Φ−(r, t)

)

≡(

f(+)+ (r, t)

f(+)− (r, t)

)

(13.61)


as the energy wave function of the photon. Despite the fact that no T-subscript has beenadded to Φ it must be remembered that the bispinor here relates to free (transverse) photons.

The negative-frequency parts of the properly normalized Riemann–Silberstein vectors

f(−)± (r, t) =

√

ǫ02

(

e(−)T (r, t)± icb(−)(r, t)

)

(13.62)

likewise as above may be synthesized into a six-component bispinor

ΦA(r, t) =

(

f(−)+ (r, t)

f(−)− (r, t)

)

, (13.63)

which we consider as the energy wave function of the antiphoton (subscript A). Since theinformation carried by an arbitrary combination of negative frequencies in the Maxwellequations is completely contained in the corresponding spectrum of positive frequencies,the antiphoton is identical to the photon. In the framework of the Maxwell theory we thusare allowed to say that photons have no antiparticles, or equivalently, that the photon isits own antiparticle. The photon thus is a spin-1 Majorana boson. The real character of theelectromagnetic field vectors implies that

f(−)± (r, t) =

[

f(+)∓ (r, t)

]∗, (13.64)

cf. Eq. (2.121). On the basis of Eq. (13.64) it appears that the particle-antiparticle conju-gation is given by

ΦA = σ1Φ∗, (13.65)

where σ1 is the first of the Pauli spin matrices in Eqs. (10.81). Since σ1 is Hermitian [σ†1 = σ1]

one obtains

Φ†A = (Φ∗)† σ1, (13.66)

and then

Φ†AΦA = (Φ∗)† σ1σ1Φ

∗ = (Φ∗)† Φ∗ =(

Φ†Φ)∗

= Φ†Φ. (13.67)

As expected, because the photon is its own antiparticle, the photon and antiphoton proba-bility densities are identical.

Our inability to localize a photon completely in space means that the photon probabilitydensity is not an observable and hence of limited value from a fundamental point of view.The integral of the probability density over the entire space gives a definite meaning becausea photon once introduced in empty space never disappears. To calculate the entire energyin the one-photon field we first give a useful expression for the energy density, WT , in thefree field. From the relations

f±(r, t) =

√

ǫ02(eT (r, t)± icb(r, t)) (13.68)

one easily realizes that

WT =1

2(Φ + ΦA)

†(Φ + ΦA) . (13.69)


An integration of WT over the entire space leads to the expression

HT =1

2

∫ ∞

−∞

(

Φ†Φ+ Φ†AΦA

)

d3r (13.70)

for the total field energy because the photon-antiphoton interference energy vanishes, i.e.,

1

2

∫ ∞

−∞

(

Φ†AΦ + Φ†ΦA

)

d3r = 0. (13.71)

The two contributions in Eq. (13.71) vanish separately. To prove this one first realizes that

1

2

∫ ∞

−∞Φ†

AΦd3r =

1

2

∫ ∞

−∞(Φ∗)† σ1Φd

3r =

∫ ∞

−∞f(+)+ (r, t) · f (+)

− (r, t)d3r

=

∫ ∞

−∞f(+)+ (q; t) · f (+)

− (−q; t)d3q

(2π)3. (13.72)

The last member of Eq. (13.72) is obtained by inserting the Fourier integrals of f(+)+ (r, t)

and f(+)− (r, t) and remembering the plane wave expansion of the Dirac delta function. Since

f(+)± (±q; t) = f

(+)± (±q; t)e±(±κ), (13.73)

and

e+(κ) · e−(−κ) = 0, (13.74)

it follows that∫ ∞

−∞Φ†

AΦd3r = 0. (13.75)

I urge the reader to convince herself that the scalar product of the helicity unit vectors e+(κ)and e−(−κ) indeed is zero. [Hint: Use a real basis decomposition of the helicity unit vectors[Eqs. (2.118) and (2.119)] and choose an arbitrary angle between εR1 (κ) and εR1 (−κ)]. InEq. (13.70) the field energy is divided equally between the photon and antiphoton; cf. Eq.(13.67). It is of course correct also to ascribe the entire energy (now named E) to the photonalone. To determine the one-photon energy,

E =

∫ ∞

−∞Φ†(r, t)Φ(r, t)d3r, (13.76)

it is necessary to study the photon emission process from the given source. To guaranteethat precisely one (transverse) photon is emitted, the magnitude of the source currentdensity must be correctly adjusted [120]. We shall study one-photon wave packets and theirsources in detail in Parts IV and V. We already know from the analysis in Sec. 2.7 that theone-photon energy in Eq. (13.76) must be a conserved quantity that is

d

dt

∫ ∞

−∞Φ†(r, t)Φ(r, t)d3r =

d

dt

∫ ∞

−∞f+(r, t) · f−(r, t)d3r = 0. (13.77)

An explicit proof of Eq. (13.77) may be obtained using the dynamical equations for f+ andf−, namely

∂

∂tf±(r, t) = ∓ic∇× f±(r, t). (13.78)


Hence,

d

dt

∫ ∞

−∞f+ · f−d3r =

∫ ∞

−∞

(

∂f+∂t

· f− + f+ · ∂f−∂t

)

d3r

= −ic∫ ∞

−∞(f− ·∇× f+ − f+ ·∇× f−) d

3r

= −ic∫ ∞

−∞∇ · (f+ × f−) d

3r = 0. (13.79)

The last equality in Eq. (13.79) follows upon transformation of the volume integral to asurface integral (Gauss’ theorem) which is zero because the fields have finite support inspace-time and thus vanish at infinity.

13.4 Four-momentum and spin of photon wave packet

If the expression for E given in Eq. (13.76) is to represent the energy of a single photon,the wave function Φ(r, t) must be properly normalized. The normalization condition canbe obtained using the fact that a monochromatic photon carries an energy ~cq. A spatialFourier-integral transformation of the wave function, i.e.,

Φ(r, t) =√~c

∫ ∞

−∞Φ(q; t)eiq·r

d3q

(2π)3, (13.80)

allows us to express the one-photon energy in terms of an integral over wave-vector space.For convenience a factor

√~c has been introduced in front of the integral in Eq. (13.80). We

consider Φ(q; t) as the spinorial photon wave function in the wave-vector representation.By means of the Parseval–Plancherel identity [53] one now obtains

E =

∫ ∞

−∞~cqΦ†(q; t)Φ(q; t)

d3q

(2π)3q. (13.81)

In view of this the one-photon normalization condition must take the form∫ ∞

−∞Φ†(q; t)Φ(q; t)

d3q

(2π)3q= 1. (13.82)

The form of the normalization condition in Eq. (13.82) relates to the relativistically invariantscalar product in quantum mechanics, as we shall see in Sec. 13.5. This is satisfactorybecause photon wave mechanics necessarily must be relativistic. If the reader prefers to workwith the nonrelativistic form of the scalar product in the momentum (p) representation, ascaled spinorial photon wave function

Γ(p; t) ≡ (2π~)−3

2

(

~

p

)1

2

Φ(p

~; t)

(13.83)

has to be introduced. The scaling factor must be proportional to |p|−1/2(= p−1/2) to com-pensate for the presence of the q−1-factor in Eq. (13.82). In terms of Γ(p; t), the integralfor the one-photon energy now is given by

E =

∫ ∞

−∞cpΓ†(p; t)Γ(p; t)d3p, (13.84)


with Γ(p; t) normalized according to

∫ ∞

−∞Γ†(p; t)Γ(p; t)d3p = 1. (13.85)

It can be shown [16, 123] starting from Eq. (2.124) that the one-photon momentum (P) isgiven by

P =

∫ ∞

−∞~qΦ†(q; t)Φ(q; t)

d3q

(2π)3q, (13.86)

as the reader might have expected since a monochromatic plane-wave photon has a momen-tum ~q. In terms of Γ(p; t) one gets

P =

∫ ∞

−∞~qΓ†(p; t)Γ(p; t)d3p. (13.87)

The results in Eqs. (13.81) and (13.86) can be expressed compactly in four-vector notation.Hence,

Pµ =

∫ ∞

−∞pµΦ†(q; t)Φ(q; t)

d3q

(2π)3q, (13.88)

where pµ = ~(q,q) is the photon four-momentum operator in q-space, and Pµ =(E/c,P) the one-photon four-momentum. Since we are using spinorial notation, it is froma formal point of view reasonable to introduce the supermatrix operator

[pµ] ≡(

U− κκ 00 U− κκ

)

pµ, (13.89)

because the expression for the photon four-momentum then takes the familiar quantummechanical mean value form, viz.,

Pµ =

∫ ∞

−∞Φ†(q; t) [pµ] Φ(q; t) d3q

(2π)3q. (13.90)

The introduction of the transverse projection operator U−κκ [Eq. (2.95)] in Eq. (13.89) ismeant to underline the transversality of the photon helicity species Φ+ and Φ− [Eq. (13.28)].

Let us briefly study the spins, (S+,S−), of the two photon helicity species. On the basisof the photon spin operator, ~Σ, these are given by [16, 123, 127]

S± =

∫ ∞

−∞Φ†

±(q; t)~ΣΦ±(q; t)d3q

(2π)3q. (13.91)

The mean values in Eqs. (13.91) can be expressed in an illuminating manner, as we nowshall see. Since

Φ±(q; t) =1√~cf(+)± (q; t)e±(κ), (13.92)

as the reader easily may convince herself of comparing the upper and lower spinor parts ofEq. (13.80) with Eq. (13.73), we get

S± = c−1

∫ ∞

−∞

∣

∣

∣f(+)± (q; t)

∣

∣

∣

2

e†±(κ) ·Σe±(κ)d3q

(2π)3q. (13.93)


The quantity e†± ·Σe± in the integrand of Eq. (13.93) is simplified as follows:

e†± ·Σe± = −Σ · e∗±e± = −ie∗± × e± = ±κ. (13.94)

The first member of Eq. (13.94) comes from the antisymmetry of the Cartesian componentsof Σ [see Eqs. (10.58)-(10.60)], the second member follows by use of Eq. (10.63) [witha = e∗± and F = e±], and the last member is obtained by means of Eqs. (10.48) and (10.49)[remembering that εR1 , ε

R2 and κ form a right-handed set of orthonormal real vectors]. For

the S±’s one hence reaches the expressions

S± = (c~)−1

∫ ∞

−∞(±~κ)

∣

∣

∣f(+)± (q; t)

∣

∣

∣

2 d3q

(2π)3q. (13.95)

From Eq. (13.92) one realizes that

Φ†±(q; t)Φ±(q; t) = (~c)−1

∣

∣

∣f(+)± (q; t)

∣

∣

∣

2

, (13.96)

since e∗± · e± = 1 (Eq. (10.52)). The two spins therefore are given by the integral formulae

S± = ±∫ ∞

−∞~κΦ†

±(q; t)Φ±(q; t)d3q

(2π)3q. (13.97)

These results are the expected ones since the helicities of the underlying plane-wave statesare just ±~κ · κ = ±~. The total spin of the photon wave packet finally is given by

S = S+ + S− =

∫ ∞

−∞~κ[

Φ†+(q; t)Φ+(q; t)− Φ†

−(q; t)Φ−(q; t)] d3q

(2π)3q. (13.98)

It is of course possible to rewrite Eq. (13.98) in familiar quantum mechanical mean valueform, using an appropriate supermatrix for the photon spin operator. I leave this as anexercise for the reader.

13.5 Relativistic scalar product. Lorentz-invariant integration onthe energy shell

When expressing the one-photon normalization condition [Eq. (13.82)], the wave packetfour-momentum [Eq. (13.88)], and the photon spin [Eq. (13.98)] by integral (mean value)expressions over wave-vector (q) space, the usual volume element known from nonrelativisticwave mechanics, d3q, was replaced by d3q/q. The root to this difference lies in the needfor generalizing the nonrelativistic scalar product of two quantum states to the relativisticdomain in such a manner that the product is Lorentz invariant. For a massless photon,certainly a relativistic object, such a generalization is necessary, and, as we shall realizebelow, the relativistic scalar product relates to Lorentz-invariant integration on the lightcone.

In q-space, the nonrelativistic scalar product of two spinorial quantum states |Ψ1〉 and|Ψ2〉 is defined by

〈Ψ1|Ψ2〉 =∫ ∞

−∞Ψ†

1(q; t)Ψ2(q; t)d3q

(2π)3. (13.99)


The volume element d3q is not invariant under a general Lorentz transformation, and anobvious generalization of the nonrelativistic definition is to take

〈Ψ1|Ψ2〉 =∫ ∞

−∞K(q)Ψ†

1(q; t)Ψ2(q; t)d3q

(2π)3, (13.100)

and then try to fix the function K(q) so that Eq. (13.100) attains a Lorentz-invariantform. Rotational invariance requires that K(q) = K(q) [q = |q|], and to determine thefunctional form of K(q) it is sufficient to consider boosts along the 3-axis, say. In terms ofthe rapidity (boost parameter) θ, which relates to the parameterization β(= v/c) = tanh θ,γ[= (1− β2)−1/2] = cosh θ, βγ = sinh θ, the transformation of the wave four-vector is givenby

q0 →(

q0)′

= q0 cosh θ − q3 sinh θ, (13.101)

q3 →(

q3)′

= −q0 sinh θ + q3 cosh θ, (13.102)

and q1 →(

q1)′

= q1, q2 →(

q2)′

= q2. With the help of the relativistic dispersion relation

q0 =(

q2 +Q2C

)1

2 , (13.103)

one obtains from Eq. (13.102)

d(

q3)′

dq3= cosh θ − q3

(q2 +Q2C)

1

2

sinh θ =

(

q0)′

q0, (13.104)

using in the last step Eqs. (13.101) and (13.103). The volume element hence transforms asfollows:

d3q → d3q′ =

(

q0)′

q0d3q. (13.105)

The quantity d3q/q0 therefore is a Lorentz invariant. If one chooses a K(q) proportional to(q0)−1 the scalar product in Eq. (13.100) takes a Lorentz-invariant form. Let us here takethe proportionality factor as unity, so that the relativistically invariant scalar product inq-space is defined as

〈Ψ1|Ψ2〉 =∫ ∞

−∞Ψ†

1(q; t)Ψ2(q; t)d3q

(2π)3 (q2 +Q2C)

1

2

. (13.106)

In the nonrelativistic limit (q → 0), which of course is of no relevance for a massless photon,the definition in Eq. (13.106) coincides with the nonrelativistic scalar product in Eq. (13.99)up to a factor Q−1

C = ~/(mc), m being the particle rest mass. For a massless photon withthe general spinorial wave function denoted by Φ(q; t), as in Sec. 13.4 , the scalar producttakes the relativistically invariant form

〈Φ1|Φ2〉 =∫ ∞

−∞Φ†

1(q; t)Φ2(q; t)d3q

(2π)3q, (13.107)

in agreement with the one-photon normalization condition in Eq. (13.82), and the quantummechanical mean values of the photon four-momentum [Eq. (13.90)] and spin [Eq. (13.98)]operators.


The form of the scalar product in the coordinate representation may be obtained in-serting the inverse of the Fourier-integral transformation in Eq. (13.80) into Eq. (13.107).Interchanging the q-integration and the two integrations over space, we need to evaluatethe integral

f(r− r′) = (2π)−3∫ ∞

−∞eiq·(r−r′) d

3q

q. (13.108)

This can be done in spherical coordinates (polar axis along the R = r− r′ direction) but aregularization of the radial integration at infinity is needed [123]. The regularization in theq → ∞ limit does not affect any of the physical properties of a photon wave packet. As thereader may show, one finds

f(r− r′) =(

2π2R2)−1

, (13.109)

so that the relativistic scalar product takes the form

〈Φ1|Φ2〉 =1

2π2~c

∫ ∞

−∞Φ†

1(r, t)Φ2(r′, t)

d3r′d3r

|r− r′|2 , (13.110)

in the r-representation.The form of the relativistic scalar product in Eq. (13.107) appears in a broader perspec-

tive as an example of Lorentz-invariant four-dimensional integration in qµ-space restrictedto the three-dimensional surface given by the relativistic dispersion relation [Eq. (13.103)].Thus let us consider the Lorentz-invariant integral formula

I = 2

∫ ∞

−∞θ(q0)δ

[

(q0)2 − q2 −Q2C

]

H(q, q0)dq0d3q

(2π)3, (13.111)

where the presence of the unit step function θ(q0) indicates that the integration is limitedto the branch of the dispersion relation for which q0 > 0.

The delta function identity

2θ(q0)δ[

(q0)2 − q2 −Q2C

]

=δ(

q0 −√

q2 +Q2C

)

√

q2 +Q2C

(13.112)

enables us to write the integral in Eq. (13.111) as follows:

I =

∫ ∞

−∞H

(

q,√

q2 +Q2C

)

d3q

(2π)3 (q2 +Q2c)

1

2

. (13.113)

Since H(q,√

q2 +Q2C) is an arbitrary function of q, the scalar product in Eq. (13.106)

constitutes a special case of Lorentz-invariant integration on the q0 > 0 branch of thedispersion relation (energy shell, mass shell). For the massless photon the integration

I(QC = 0) =

∫ ∞

−∞H(q, q)

d3q

(2π)3q(13.114)

is on the light cone. When we later on [Chapt. 15] study single photon wave packets andthe position-operator problem for transverse photons (resting on photon wave mechanicsbased on the transverse vector potential), the importance of Lorentz-invariant integrationon the light cone is further strengthened.

Part IV

Single-photon quantum opticsin Minkowskian space

14

The photon of the quantized electromagnetic field

According to modern quantum field theory all elementary particles are quantum excita-tions of an underlying quantum field. In the case of the photon the underlying field isthe electromagnetic field. In photon wave mechanics the photon concept was reached by areinterpretation of the physical meaning of the free classical electromagnetic field. Such areinterpretation does not give us a quantum theory of light quanta because the elementaryquantum of action is foreign to classical physics. Although Planck’s constant does occurin the various Schrodinger-like wave equations we have presented in Part III it just playsthe role of a multiplicative factor, see e.g., Eq. (10.36). The “justification” for bringing ~

into the formalism is the wish to make contact to the Einstein–de Broglie relations [Eqs.(10.40) and (10.41)]. In a sense one may call the light particle appearing from classicalelectrodynamics the classical photon.

Quantum physics appears when pairs of classical conjugate dynamical variables arereplaced by noncommuting operators satisfying a commutator relation proportional toPlanck’s constant. The quantum photon emerges upon quantization of the electromagneticfield. In the covariant quantization scheme the central equal-time commutator relation be-tween the four-potential operator, Aµ, and its conjugate, Πµ = −ǫ0∂Aµ/∂t, thus isgiven by [209, 127, 156]

[

Aµ(r, t), Πν(r′, t)]

=~

igµνδ(r − r′), (14.1)

where gµν(= ηµν) is the Minkowski metric tensor in Eq. (3.4). It is possible to introducea photon wave function concept also in the field-quantized formalism, as we shall see inChapt. 15.

In Sec. 15.1 we describe the canonical quantization of the transverse electromagnetic free-space field [155, 53]. The central issue is the plane-wave quantization of the transverse vector

potential, AT (r, t). From the plane-wave expansion of AT (r, t) corresponding expressions

for the transverse electric (ET (r, t)) and magnetic (B(r, t)) field operators readily can beobtained. In Sec. 15.2, explicit expressions for the energy, momentum, and spin operatorsare presented. All operators are build as a sum of operators associated to the individualplane-wave modes (wave vector: q, polarization index s = 1, 2). For the plane-wave mode

i = (qi, si), the number operator Ni = a†i ai plays a particular role. As indicated, it is formed

by the product of the mode annihilation (ai) and creation (a†i ) operators. The eigenvalues,

ni, for Ni are zero and all positive integers, ni = 0, 1, 2, · · · , and the related eigenvectors,|ni〉, are the so-called number or Fock states (see Sec. 15.3). The state |ni〉 holds ni quanta ofenergy ~ωi = ~cqi and momentum ~qi. These elementary quanta (“particles”) are our plane-wave photons. The single-photon state belonging to mode (q, s), |1qs〉, may be generatedfrom the global vacuum state, |0〉, with the help of the mode creation operator. Thus,|1qs〉 = a†qs|0〉.

Elementary excitations (photons) of the quantized electromagnetic field need not be

161


monochromatic and plane. The superposition (in a cubic box of volume L3)

|Φ〉 = L− 3

2

∑

q,s

φqsa†qs|0〉 (14.2)

also qualifies as a single-photon state. The state is a so-called wave-packet state, and it isin general polychromatic. Some of the properties of single-photon wave-packet states arediscussed in Sec. 15.4. Lorentz-invariant (relativistic) normalization shows that the quantity

Φqs = q1

2φqs (14.3)

must be identified as the transverse scalar photon wave function in the wave-vector repre-sentation. Since one is unable to localize a photon completely in space one cannot ob-tain a direct-space photon wave function associated to Eq. (14.3) by a simple Fouriertransformation. Every attempt to provide a precise definition of a position coordinate forthe free transverse photon stands in contradiction with (special) relativity (see Sec. 15.5)[123, 155, 188, 127, 74, 102, 7, 187, 94, 2, 18, 171, 250]. Physically, I do not consider thiscircumstance as problematic, because one has to remember that a photon as a phenomenonis only a phenomenon when it is a registered (observed) phenomenon. Observation requirescoupling to matter (a detector). By introduction of some non-sharp position state, |R〉,the vectorial photon wave function in direct space, which consequently is associated to theone-photon Hilbert state in Eq. (14.2), is

Φ(r, t) = 〈R|Φ〉(r, t). (14.4)

I have held the point of view in recent years that the choice (introduced in Sec. 15.5)

|R〉(r, t) ≡(

2ǫ0c

~

)1

2

A(−)T (r, t)|0〉, (14.5)

representing a sort of mean position state for the transverse photon in Hilbert space,physically is the best and most comprehensive choice one can make. The bridge betweenEqs. (14.3) and (14.4) [with |R〉 given by Eq. (14.5)] is made in Sec. 15.6, and the non-orthogonality of |R〉-states belonging to two different positions is discussed in Sec. 15.7.

A set of single-photon wave-packet states, |Wi〉, numbered by the index i, may be formedby superposition of the single-photon states |1j〉 ≡ |1qs〉, i.e.,

|Wi〉 =∑

j

tij a†j|0〉. (14.6)

If the coefficient matrix, tij, is unitary the members of the set are orthonormalized, andsatisfy in the one-photon subspace a completeness relation, as shown in Sec. 16.1. Insteadof expanding the general single-photon state |Φ〉 in a monochromatic plane-wave basis [Eq.(14.2)] one thus may expand |Φ〉 after an appropriate wave-packet basis. Notwithstandingthe fact that it, from a mathematical point of view, often will be most convenient to expandthe one-photon field emerging from a given emission process in a monochromatic plane-wave basis, single photons radiated from a given source nevertheless are polychromatic.From a physical point of view, the wave-packet picture of single-photon states appearsconceptually satisfactory. Moreover, it may happen in a given problem that the sought-forsingle-photon state to a good approximation can be expanded in terms of a few wave-packetbasis modes. Although a photon emitted from a given source may be calculated to be insome polychromatic Hilbert state, |Φ〉, we must not forget that the photon is a phenomenononly when registered in space-time. In a local photodetection process by an atom it is the

The photon of the quantized electromagnetic field 163

wave packet Φ(r, t) = 〈R|Φ〉(r, t) which most naturally may be conceived as a propagatingphoton in the space of an inertial observer.

The single-photon wave-packet basis state |Wi〉, given in Eq. (14.6), can be obtained by

acting on the vacuum state with the creation operator b†i =∑

j tij a†j . The operators bi and

b†i satisfy the usual boson operator algebra, and multi-photon wave packets belonging tomode i hence can be constructed in a manner analogous to the one used for a monochromaticplane-wave mode; see Sec. 16.2.

Let us assume that some single-photon field has been generated by emission from asource. Once the field-matter interaction process responsible for emission of the transversephoton has come to an end, the weight factors φqs characterizing the state vector of the field,Φ [Eq. (14.2)], have become time independent. The quantum state of the emitted one-photonwave-packet field is unknown in general, and even though it may be possible to calculateit (approximately) in the simplest cases, it certainly is important to seek experimentalinformation on |Φ〉. This requires that the field interacts with a detector. Although alldetection processes in the final stage involves observations in the macroscopic classicaldomain of physics [33, 34], it is sufficient for our purpose to let the detector be a singleone-electron atom. Single-atom field detectors are of fundamental interest in their own right[214], here, because they may serve as local detectors helping us in probing the single-photonwave function Φ(r, t) = 〈R|Φ〉(r, t).

In Sec. 16.3, we briefly review how an integral equation for the time evolution operatorin the Interaction Picture can be derived for an arbitrary physical system. The expressionone obtains to first order in the interaction Hamiltonian subsequently is applied to studythe interaction between a single-electron atom and a single-photon field (Sec. 16.4). Tak-ing as a starting point the nonrelativistic interaction Hamiltonian as this appears in theCoulomb gauge, and assuming that the spatial variation in the transverse part of the vectorpotential across the atom (position of the nucleus: r0) is negligible, a calculation of thetotal probability per unit time (Pi→exc(t)) that the system after a certain time (t) has gonefrom the initial (uncoupled) state (i) to any arbitrary final (uncoupled) excited state (exc)is carried out. Under the assumption that the atom initially (t = 0) is in the ground state,and the electromagnetic field in a pure free state |Φ〉, here the single-photon wave-packetstate given in Eq. (14.2), one obtains

Pi→exc(t) = (2ǫ0c~)−1∫ t

0

∫ t

0

GA(t′, t′′) : GF (r0, t

′, t′′)dt′′dt′, (14.7)

where GA and GF are the atomic (A) and field (F) correlation matrices, and q and m theelectron charge and mass. The field correlation matrix appearing in Eq. (14.7) is a specialcase (r = r′ = r0) of the first-order [superscript (1)] correlation matrix relating to twospace-time points [x = (ct, r) and x′ = (ct′, r′)],

G(1)(x, x′) =2ǫ0c

~〈Φ|AT (x)AT (x

′)|Φ〉. (14.8)

The G(1)(x, x′)-matrix is of central importance in photon physics since it is a quantitywhich depends solely on the state of the quantum field, which is not necessarily a one-photon field. In Sec. 16.5 the physical structure of G(1)(x, x′) is discussed in the case where|Φ〉 represents a single-photon state. In the framework of the rotating-wave-approximationit turns out that

G(1)(x, x′) =2ǫ0c

~〈Φ|A(−)

T (x)A(+)T (x′)|Φ〉

=Φ∗(x)Φ(x′). (14.9)


The last member of Eq. (14.9) is the dyadic product of the single-photon wave functionstaken at the space-time points x and x′. It is our use of the Hamiltonian in the minimalcoupling form, and the definition of the photon wave function given in Eqs. (14.4) and(14.5), which together have led to the interesting result in the equation above. The trace ofG(1)(x, x) equals the probability density for detecting the transverse polychromatic photonin the space point r at time t.

In Chapt. 17, interference phenomena with single-photon states are studied on the basisof the approximation given for the first-order correlation matrix in Eq. (14.9). It is con-venient to resolve the single-photon wave function into a complete set of orthonormalizedwave-packet modes, and from a general point of view the observation of interference thenrelates to the sum of the terms in G(1)(x, x′) which contain a product of different modes.The basic principle of single-photon interference is illustrated by means of a Young-typedouble-source interference experiment in which the usual opaque screen with two holes isreplaced by two two-level atoms; see Secs. 17.2 and 17.3.

Among the various single-photon states, the three which relate to the photon meanposition state |R(x0)〉 ≡ |R(r0, t0)〉 are of particular importance from a fundamental pointof view. In the Interaction Picture one has |R(x0)〉 = |R(r0, 0)〉. The information in thethree first-order correlation matrices is conveniently gathered in a correlation supermatrixG(x, x′|r0), which can be written as a tensorial product involving two transverse photonpropagators:

G(x, x′|r0) =(

2

c

)2

G∗T (r− r0, t)GT (r

′ − r0, t′). (14.10)

The field correlations in photon mean position states are studied in Sec. 17.4.In Chapt. 18, the time evolution and commutation relations of various free-field operators

are discussed. After having established the free-space Maxwell operator equations, it isshown that the quantum mechanical mean values of the Maxwell operator equations in theglobally coherent state, |αi〉, coincide with the classical Maxwell equations.

The fundamental importance of the Landau–Peierls–Sudarshan (LPS) equation for com-plex analytical signals in free-space optics [155, 226] makes it interesting to generalize thisdynamical equation to the field-quantized level. This is done for the positive-frequency part

of the transverse vector potential operator A(+)T (r, t) in Sec. 18.2. The matrix element of the

LPS-operator equation between the global vacuum state, |0〉, and the single-photon wave-packet state, |Φ〉, leads directly to a LPS-equation for the vectorial photon wave functionin space-time. If the LPS-operator equation is applied to the coherent state it appears that

the eigenvalue of A(+)T (r, t), viz., A

(+)T (αi; r, t), satisfies the classical LPS-equation.

15

Polychromatic photons

15.1 Canonical quantization of the transverse electromagnetic field

In this section we shall take up the problem of quantizing the electromagnetic field in freespace. In the absence of charges and currents the classical electric and magnetic fields satisfythe free-space Maxwell equations given in Eqs. (2.1)-(2.4). The free fields are transverse, andfor reasons which will become clear as we proceed, conveniently represented by the gaugeinvariant transverse vector potential AT (r, t). In terms of AT (r, t) the electric (ET (r, t))and magnetic (B(r, t)) fields are given by

ET (r, t) = − ∂

∂tAT (r, t), (15.1)

B(r, t) = ∇×AT (r, t). (15.2)

The transverse vector potential satisfies the homogeneous wave equation

AT (r, t) = 0 (15.3)

in free space; cf. the analysis in Sec. 10.2, resulting in Eq. (10.31).In order to quantize the electromagnetic field in a manner which leads up to the

monochromatic plane-wave photon concept, we imagine the field to be contained in a largecubic box of side L. The transverse vector potential can then be presented as a Fourier series,and hence specified in terms of an infinite but denumerable number of (vectorial) Fourierexpansion coefficients. The procedure above is not essential for the quantization of the freefield. Thus, no physically meaningful result will depend on L when L is much larger thanall relevant wavelengths entering the given problem. In the limit L→ ∞ the Fourier seriesrepresentation of the field goes into a Fourier integral representation. From a formal pointof view it might seem more natural to base the free-field quantization on the continuousmode representation of the field in an infinite space domain. For the study of certain prob-lems in quantum electrodynamics it is an advantage to make use of the continuous moderepresentation. For simplicity, not necessity, we impose the well-known periodic boundaryconditions on the field [155, 211, 53]. With these, the plane-wave Fourier expansion of thetransverse vector potential takes the form

AT (r, t) = L− 3

2

∑

q

(

AT,q(t)eiq·r + c.c.

)

, (15.4)

where the wave vectors q are given by

q =2π

L(n1, n2, n3), n1, n2, n3 = 0,±1,±2, · · · . (15.5)

The transversality of the vector potential implies that

q ·AT,q(t) = 0, (15.6)

165


and the vectorial Fourier amplitudeAT,q(t) therefore can be resolved after a pair of generallycomplex orthonormal base vectors, cf. the discussion in Sec. 2.5. For later convenience, thedecomposition is written in the form

AT,q(t) =

(

~

2ǫ0ωq

)1

2 ∑

s=1,2

αqs(t)εqs, (15.7)

where ωq = c|q|(= cq). In order for the expansion

AT (r, t) =∑

q,s

(

~

2ǫ0L3ωq

)1

2(

αqs(t)εqseiq·r + c.c.

)

(15.8)

to be a solution to the wave equation in Eq. (15.3), the time dependent amplitude αqs(t)must satisfy the harmonic oscillator equation

(

d2

dt2+ ω2

q

)

αqs(t) = 0, ∀q, s. (15.9)

It will prove convenient to utilize the solution

αqs(t) = αqs(0)e−iωqt (15.10)

to the oscillator equation in what follows. With this choice we reach the following monochro-matic plane-wave expansion of the transverse vector potential:

AT (r, t) =∑

q,s

(

~

2ǫ0V ωq

)1

2 [

αqs(0)εqsei(q·r−ωqt) + c.c.

]

, (15.11)

where V = L3 is the field confinement volume.It can be shown [53] that the decomposition in Eq. (15.11) leads to a compact expression

for the total energy of the transverse electromagnetic field, viz.,

HT =ǫ02

∫

V

(

E2T (r, t) + c2B2(r, t)

)

d3r

=∑

q,s

~ωqα∗qs(t)αqs(t)

=∑

q,s

~ωqα∗qs(0)αqs(0). (15.12)

It appears from this result that the energy can be expressed as a sum of energies for theindividual (qs)-modes, ~ωq|αqs|2. On physical grounds, and from the analysis in Sec. 2.7 weknow that the energy of the free field is time independent, in agreement with what appearsexplicitly from the last member of Eq. (15.12). When the transverse field interacts withcharged matter, the time dependence of αqs(t) is not so simple as in Eq. (15.10). The lastbut one member of Eq. (15.12) still holds, but HT will depend on time, in general. For thepurpose of field quantization it is useful to write the energy in the symmetrized form

HT =1

2

∑

q,s

~ωq

(

α∗qs(t)αqs(t) + αqs(t)α

∗qs(t)

)

, (15.13)

Polychromatic photons 167

or in an equivalent form with αqs(0) and α∗qs(0). In preparation of the canonical quantization

procedure, αqs(t) and its complex conjugate are replaced by a pair of real canonical variablesQqs(t) and Pqs(t):

αqs(t) = (2~ωq)− 1

2 (ωqQqs(t) + iPqs(t)) , (15.14)

α∗qs(t) = (2~ωq)

− 1

2 (ωqQqs(t)− iPqs(t)) . (15.15)

Expressed in terms of Qqs(t) and Pqs(t), the energy in the free field takes the form

HT =1

2

∑

qs

(

P 2qs(t) + ω2

qQ2qs(t)

)

. (15.16)

Formally, this will be recognized as the energy of a system of independent one-dimensionalharmonic oscillators one for each (q, s)-mode. The quantities Qqs(t) and Pqs(t) take therole of scalar “coordinate” and “momentum” for the given mode. The expression for αqs(t)in Eq. (15.10) is the solution to the differential equation

(

d

dt+ iωq

)

αqs(t) = 0, (15.17)

and if one inserts Eq. (15.14) for αqs(t) into Eq. (15.17), and divides the resulting equationinto its real and imaginary parts one obtains the relations

d

dtQqs(t) = Pqs(t), (15.18)

d

dtPqs(t) = −ω2

qQqs(t). (15.19)

With the help of these, one immediately obtains the “usual” set of Hamilton equations ofmotions for the canonical variables, viz.,

∂HT

∂Pqs=

d

dtQqs(t), (15.20)

∂HT

∂Qqs= − d

dtPqs(t). (15.21)

The field quantization is accomplished by (i) replacing the classical canonical variablesby Hermitian operators (operator symbol:ˆ) that is

Qqs(t) ⇒ Qqs(t) = Q†qs(t), (15.22)

Pqs(t) ⇒ Pqs(t) = P †qs(t), (15.23)

where † stands for Hermitian conjugate, and (ii) assuming that these operators satisfy thefollowing equal-time commutation relations:

[

Qqs(t), Pq′s′(t)]

= i~δqq′δss′ (15.24)[

Qqs(t), Qq′s′(t)]

=[

Pqs(t), Pq′s′(t)]

= 0. (15.25)

The amplitude αqs(t) and its complex conjugate α∗qs(t) are elevated to the operator level

inserting the replacements in (15.22) and (15.23) into Eqs. (15.14) and (15.15). Hence,

αqs(t) ⇒ aqs(t) = (2~ωq)− 1

2

(

ωqQqs(t) + iPqs(t))

, (15.26)

α∗qs(t) ⇒ a†qs(t) = (2~ωq)

− 1

2

(

ωqQqs(t)− iPqs(t))

, (15.27)


The Hermiticity of Qqs(t) and Pqs(t) implies that the operators associated to αqs(t) andα∗qs(t) are each other’s Hermitian conjugate, as indicated in the notation above. The opera-

tor aqs(t) is manifest non-Hermitian. With the help of Eqs. (15.24) and (15.25), the readermay show that the equal-time commutation relations for aqs(t) and a

†qs(t) are

[

aqs(t), a†q′s′(t)

]

= δqq′δss′ , (15.28)

[aqs(t), aq′s′(t)] =[

a†qs(t), a†q′s′(t)

]

= 0. (15.29)

For reasons to be given in Sec. 15.3, aqs(t) and a†qs(t) are called the annihilation and creation

operator, respectively.The canonical quantization procedure implies that the classical transverse vector poten-

tial is replaced by a field operator, i.e.,

AT (r, t) ⇒ AT (r, t) =∑

q,s

(

~

2ǫ0V ωq

)1

2(

aqs(t)εqseiq·r + h.c.

)

=∑

q,s

(

~

2ǫ0V ωq

)1

2 [

aqs(0)εqsei(q·r−ωqt) + h.c.

]

, (15.30)

where h.c. means the Hermitian conjugate of the preceding term. The last member of Eq.(15.30) follows because the annihilation and creation operators for free fields have the timedependencies

aqs(t) = aqs(0)e−iωqt, (15.31)

a†qs(t) = a†qs(0)eiωqt. (15.32)

From the expression for AT (r, t) explicit formulas for the transverse electric (ET (r, t)) and

magnetic (B(r, t)) field operators can immedeately be written down utilizing that

ET (r, t) = − ∂

∂tAT (r, t), (15.33)

B(r, t) = ∇× AT (r, t). (15.34)

15.2 Energy, momentum, and spin operators of the transverse field

The canonical quantization procedure implies that the classical Hamiltonian of the trans-verse field [Eq. (15.16)] turns into the Hamilton operator

HT =1

2

∑

qs

(

P 2qs(t) + ω2

qQ2qs(t)

)

. (15.35)

The inverse transformation to the one given by Eq. (15.26) and (15.27), namely,

Qqs(t) =

(

~

2ωq

)1

2(

aqs(t) + a†qs(t))

, (15.36)

Pqs(t) = i

(

~ωq

2

)1

2(

a†qs(t)− aqs(t))

, (15.37)


allows one to express HT in terms of the annihilation and creation operators. Respectingthe order of these operators we find

HT =1

2

∑

q,s

~ωq

(

a†qs(t)aqs(t) + aqs(t)a†qs(t)

)

. (15.38)

This expression also follows directly from Eq. (15.13), by the replacements in Eqs. (15.26)and (15.27), because the order of the commuting classical amplitudes αqs(t) and α∗

qs(t)has been respected in this equation. Using the commutator relation in Eq. (15.28) for(qs) = (q′s′), viz.,

[

aqs(t), a†qs(t)

]

= 1, (15.39)

the Hamilton operator of the transverse electromagnetic field finally can be written in thevery important form

HT =∑

q,s

~ωq

(

a†qs(t)aqs(t) +1

2

)

. (15.40)

The terms in this equation which contain a†qsaqs-factors relate to the photon concept in

quantum optics, as we soon shall see. The terms containing 12 -factors relate to the so-called

photon vacuum, a concept to be studied in Part VII. From many observations it has beenconcluded that the photon vacuum possesses physical properties. The quantized theoryhence has demonstrated that the electromagnetic fields are more than “just” the photons.

The classical momentum of the transverse field, namely [206, 155, 127, 53]

PT = ǫ0

∫

V

ET (r, t)×B(r, t)d3r (15.41)

can, like the energy, be expressed in terms of the amplitudes αqs(t). Hence [127, 53]

PT =∑

q,s

~q

2

(


∗qs(t)

)

. (15.42)

By elevation to the quantized level one obtains the following expression for the transversefield momentum operator:

PT =∑

q,s

~q

2

(


)

. (15.43)

By means of the commutator relation in Eq. (15.39), the above expression for PT can besimplified to

PT =∑

q,s

~qa†qs(t)aqs(t). (15.44)

There is no vacuum contribution to the field momentum since

~

2

∑

q

q = 0, (15.45)

a result following readily from the form of the wave-vector spectrum given in Eq. (15.5).


It is known [155, 127, 53] that the angular momentum of the transverse field [Eq. (2.102)]is the sum of two parts,

JT = L+ S, (15.46)

where L is the so-called orbital angular momentum of the field. This part of JT dependson the point of reference. The other part S, which is independent of the choice of referencepoint, is called the spin of the transverse field. In Sec. 2.7 it was stated that JT is a conservedquantity in free space. It can be proved [127] that L and S are separately conserved, that is

dL

dt=

dS

dt= 0 (15.47)

in the absence of field-matter interaction. The classical spin of the transverse field, which isgiven by [155, 127, 53]

S(≡ ST ) = ǫ0

∫

V

ET (r, t)×AT (r, t)d3r, (15.48)

can be expressed in terms of the αqs’s. Thus [155, 127],

S =~

2

∑

q,s,s′

(

iεqs × ε∗qs′αqs(t)α∗qs′ (t) + c.c.

)

. (15.49)

It appears from Eq. (15.49) that the spin besides the contributions from the individual (qs)-modes in general contains terms coupling the two polarization eigenstates (s, s′) belongingto a given wave vector. This crosscoupling can be avoided if helicity unit vectors are usedas polarization basis. Hence, one obtains from Eqs. (2.118) and (2.119)

eqs × e∗qs′ = −iλsκδss′ , (15.50)

where s and s′ equal + or −, and λ+ = +1 and λ− = −1. In the helicity basis the spin ofthe transverse field therefore takes the form

S =~

2

∑

q,s

λsκ(


∗qs(t)

)

, s = +,−. (15.51)

Canonical quantization next gives a spin operator

S =~

2

∑

q,s

λsκ(


)

= ~∑

q,s

λsκ

(

a†qs(t)aqs(t) +1

2

)

, s = +,−. (15.52)

Since∑

s=+,−λs = 0, (15.53)

Eq. (15.52) simplifies to

S =∑

q

Sq =∑

q

~κ(

a†q+(t)aq+(t)− a†q−(t)aq−(t))

. (15.54)


The description given here of the spin (operator) of the transverse electromagnetic fieldis intimately related to the spin-one particle formalism for transverse photons (see Sec.10.3). The connecting link is the formula

e†±(κ)sie±(κ) = ±~(κ)i, i = x, y, z. (15.55)

The number on the right side of this equation is obtained from the matrix product of thecolumn vector e±(κ), the Cartesian spin-one matrix si, and the Hermitian conjugate (row)

vector e†±(κ) to e±(κ). The reader is urged to check the correctness of Eq. (15.55). In viewof Eq. (15.55) the spin of the classical transverse field can be written as follows in thehelicity basis:

S =∑

q,s

λsα∗qs(t)e

†s(κ)ses(κ)αqs(t). (15.56)

When extended to the quantum level one obtains the spin operator

S =∑

q,s

λse†s(κ)ses(κ)a

†qs(t)aqs(t). (15.57)

The canonical quantization of the transverse electromagnetic field allows one to describemulti-photon states. In Secs. 15.4 and 15.5 the connection between one-photon quantumoptics and single-particle photon wave mechanics based on the transverse vector potentialwill be firmly established. Once the connection is made the interpretation of the formulasfor the classical spin [Eq. (15.56)] and the spin operator [Eq. (15.57)] becomes completelytransparent.

15.3 Monochromatic plane-wave photons. Fock states

A polychromatic photon state can be synthesized by suitable superposition of monochro-matic plane-wave states. These last states, and the related elementary excitations (photons),are discussed in detail in numerous books, e.g., in [155, 211, 53]. Below we review only as-pects (results) of particular relevance for the introduction and analysis of polychromatictransverse photons.

Consider first a single mode i ≡ (q, s), and the eigenvalues and eigenstates for theoperator

Ni = a†i (t)ai(t) = a†i (0)ai(0). (15.58)

In the Dirac notation we write the eigenvalue equation for Ni as

Ni|ni〉 = ni|ni〉. (15.59)

The eigenvalues of Ni are the integers

ni = 0, 1, 2, · · · , (15.60)

and Ni is called the number operator for the mode i. The eigenvectors in the transversesubspace of Hilbert space, |ni〉, which are orthonormalized, i.e.,

〈ni|nj〉 = δij , (15.61)


satisfy the relations

a†i |ni〉 = (ni + 1)1

2 |ni + 1〉, (15.62)

ai|ni〉 = n1

2

i |ni − 1〉, (15.63)

with the mode ground state |0i〉 given by

ai|0i〉 = 0. (15.64)

The various eigenvectors can be generated from the ground state as follows:

|ni〉 =

(

a†i

)ni

√ni!

|0i〉. (15.65)

Since the mode Hamilton and momentum operators relate to the number operator via

HT,i = ~ωi

(

Ni +1

2

)

, (15.66)

PT,i = ~qiNi, (15.67)

the number states |ni〉 are eigenstates for these operators, and the eigenvalues are

Ei =

(

ni +1

2

)

~ωi, (15.68)

Pi = ni~qi. (15.69)

With respect to the ground state it appears as if the state |ni〉 consists of ni “particles”with energy ~ωi and momentum ~qi. These “particles,” called (transverse) photons, describeelementary excitations of mode i of the quantized electromagnetic field. We have realizedby the analysis in Chapts. 10 and 13 that the helicity basis plays a particularly importantrole in photon wave mechanics. From Eq. (15.54) it appears that the eigenvalues for thespin in the two helicity states [|ni〉 = |nq+〉, |nq−〉] are

Si(±) = ±ni~κ, (15.70)

in an obvious notation. The two photon types hence have spins ±~κ, and helicities ±~.Let us now turn our attention toward a multi-mode (global) field. Obviously, the number

operator N for such a field is

N =∑

i

Ni =∑

i

a†i ai, (15.71)

and the eigenstates for N therefore are tensor products of eigenstates for the various i-modes, i.e.,

|n1, n2, · · · , ni, · · · 〉 ≡ |ni〉 = |n1〉|n2〉 · · · |ni〉 · · · ≡∏

i

|ni〉. (15.72)

The global number states in Eq. (15.72) are eigenstates of the total Hamilton [Eq. (15.40)]and momentum [Eq. (15.44)] operators:

HT |ni〉 = E|ni〉 (15.73)

PT |ni〉 = P|ni〉, (15.74)


with eigenvalues

E =∑

i

(

ni +1

2

)

~ωi, (15.75)

P =∑

i

ni~qi. (15.76)

The global ground state

|0〉 ≡ |01, 02, · · · , 0i, · · · 〉 = |0i〉 (15.77)

contains no photons, and it is clear from Eq. (15.65) that all number eigenstates, oftencalled Fock states, can be generated from the global ground state as follows:

|ni〉 =

∏

i

(

a†i

)ni

√ni!

|0〉. (15.78)

The photons introduced in this section are of the monochromatic transverse plane-wavetype. Instead of expanding the transverse field in monochromatic plane waves one may,e.g., expand the field in monochromatic multipole waves [225, 101, 53, 40, 164]. In this casequantization leads to elementary excitations (photons) characterized by well-defined valuesof the energy, the square of the angular momentum, one of the Cartesian components ofthe angular momentum, and of the parity [52, 12].

15.4 Single-photon wave packets

Elementary excitations of the quantized field (photons) need not be monochromatic, andin the present section we shall discuss the polychromatic photon concept. In what followsthe words polychromatic photon and single-photon wave packet will be used synonymously,although a partly localized polychromatic photon does not necessarily form a wave packetof the conventional (textbook) type. As the reader shall gradually understand as she readsthe remaining part of this book single-photon wave packets are of the utmost importance inbasic photon physics. To some extent this is already clear from previous parts of the bookbecause a polychromatic photon concept must be introduced to make the bridge to single-photon wave mechanics. To the best of my knowledge, polychromatic single-photon statesare not discussed in detail in existing books, and it is commonly (but wrongly) believed intextbooks that a photon must be monochromatic.

It appears from Eq. (15.62) that a single-photon state belonging to the mode i = (q, s),|1qs〉, can be generated by acting with the creation operator a†qs on the vacuum state |0qs〉:

|1qs〉 = a†qs(0)|0qs〉. (15.79)

By linear superposition of such monochromatic (ωq = c|q|) plane-wave states a polychro-matic single-photon state

|Φ〉 = L− 3

2

∑

q,s

φqsa†qs(0)|0〉 (15.80)

can be formed. The quantities φqs are amplitude weight factors, and their physical role will


be discussed in Sec. 15.5. In free space the weight factors are time independent becausephotons cannot be destroyed nor created in the various (qs)-modes. In what follows weonly deal with free fields and our |Φ〉 hence is a time independent state vector. In a broaderframework we therefore essentially work in the so-called Heisenberg Picture [53].

In the presence of field-matter interaction the number of photons in the modes will varyin time, and the amplitude weight factors become time dependent. During an interactionprocess the normalization condition [given below in Eq. (15.85)] cannot be maintained, anda genuine single-photon state does not exist. Nevertheless, the quantity

|Ψ〉(t) = L− 3

2

∑

q,s

φqs(t)a†qs(0)|0〉, (15.81)

which I have called a photon embryo state in connection with single-photon emission from asource [123, 127], is an extremely useful quantity, as exemplified in relation to spontaneousemission from a two-level atom (see Sec. 34.2). In a one-photon scattering process theinitial, |Ψ〉(t → −∞) ≡ |Φi〉, and final, |Ψ〉(t → ∞) ≡ |Φf 〉 states satisfy a normalizationcondition given soon. There are of course special superpositions for which |Ψ〉(t) describesa free photon. Thus, if for instance all nonvanishing terms in Eq. (15.81) have the samemagnitude of q, and the common time dependence

φqs(t) = φqs(0)e−iωqt, (15.82)

one obtains

|Ψ〉(t) = e−iωqt|Ψ〉(0), (15.83)

but this is just the corresponding |Φ〉 state transferred to the Schrodinger Picture. The statein Eq. (15.83) is a monochromatic (ω = ωq) multipole one-photon state.

To qualify as a genuine single-photon state |Φ〉 must be an eigenfunction for the globalnumber operator [Eq. (15.71)] with eigenvalue 1:

N |Φ〉 = |Φ〉. (15.84)

Although it is obvious from the construction in Eq. (15.80) that |Φ〉 is a one-photon state,the reader may check the correctness of Eq. (15.84) by an explicit calculation, using thecommutator relation in Eq. (15.28), and Eq. (15.64) for the ground states of the modes.

Normalization of the polychromatic one-photon state, viz.,

〈Φ|Φ〉 = 1, (15.85)

requires that the weight functions φqs satisfy a certain condition which we now shall es-tablish and comment upon. For this purpose, it is important to understand that the |Φ〉in Eq. (15.80) is a state vector in the wave-vector (q) [or equivalent to the momentum(p)] representation of the Hilbert space of the transverse photons. Using the completenessrelation for the one-photon subspace, viz.,

∑

q,s

|1qs〉〈1qs| = 1, (15.86)

where 1 is the unit operator, the identity

|Φ〉 =∑

q,s

|1qs〉〈1qs|Φ〉, (15.87)


upon comparison to Eq. (15.80), shows that

L− 3

2φqs = 〈1qs|Φ〉. (15.88)

Since the individual one-photon mode states are orthonormalized,

〈1q′s′ |1qs〉 = δqq′δss′ , (15.89)

the state Φ is normalized, i.e.,

〈Φ|Φ〉 = 1, (15.90)

provided the weight factors satisfy the condition

L−3∑

q,s

|φqs|2[

→∑

s

∫ ∞

−∞|φs(q)|2

d3q

(2π)3

]

= 1. (15.91)

If the photon was a nonrelativistic object (which it is not), the quantity 〈1qs|Φ〉 would bethe photon wave function in the q-representation, and the weight function φqs → φs(q)

[divided by L3

2 → (2π)3/2] therefore the wave function in wave-vector space. Since thephoton is a relativistic object, it is the quantity

Φqs ≡ q1

2φqs

[

⇒ Φs(q) ≡ q1

2φs(q)]

(15.92)

which must be identified as the (scalar) photon wave function in the wave-vector represen-tation. In terms of Φqs[→ Φs(q)] the normalization condition in Eq. (15.91) reads

L−3∑

q,s

q−1|Φqs|2[

→∑

s

∫ ∞

−∞|Φs(q)|2

d3q

(2π)3q

]

= 1, (15.93)

in agreement with the Lorentz-invariant form of the scalar product given in Eq. (13.107).The single-photon state in Eq. (15.80) is a nonstationary state in general, since it is

not an eigenstate of the Hamilton operator for the transverse field, HT [Eq. (15.40)]. Inthe present context, where we are dealing only with free fields, the vacuum contribution toHT plays no physical role, and it is sufficient to relate the calculations below solely to thetransverse photon Hamiltonian

Hph ≡ HT −∑

q,s

1

2~ωq =

∑

q,s

~ωqa†q,saqs. (15.94)

As already said, the one-photon state in Eq. (15.80) is a nonstationary state in general.This follows explicitly from the fact that when Hph acts on |Φ〉 we obtain

Hph|Φ〉 = L− 3

2

∑

q,s

~ωqφqsa†qs|0〉, (15.95)

and only for superpositions involving q’s of equal magnitude will |Φ〉 be an eigenstate ofHph (with eigenvalue ~ωq), cf. the remarks given to Eq. (15.83). Repeated operation with

Hph gives

Hnph|Φ〉 = L− 3

2

∑

q,s

(~ωq)nφqsa

†qs|0〉. (15.96)


Although our one-photon wave packet is not an eigenstate of Hph, the mean value of Hph,namely

〈Φ|Hph|Φ〉 = L−3∑

q,s

~ωq|Φqs|2q−1 → ~c∑

s

∫ ∞

−∞|Φs(q)|2

d3q

(2π)3, (15.97)

is time independent, as it must be in free space. If needed, the reader may calculate thetime-independent variance of Hph, i.e.,

∆Hph =

[

〈Φ|H2ph|Φ〉 −

(

〈Φ|Hph|Φ〉)2]

1

2

(15.98)

using Eq. (15.90). When the photon momentum operator, Pph = PT , given in Eq. (15.44),acts on the one-photon state one obtains

Pph|Φ〉 = L− 3

2

∑

q,s

~qφqsa†qs|0〉, (15.99)

a result which shows that |Φ〉 is not an eigenstate of the field momentum, unless only a singleq contributes to the superposition. With only one q the superposition contains no more thanthe related two polarization states. In the case of a single wave vector q, the eigenvalue ofthe photon momentum operator is ~q. The mean value of the photon momentum is givenby

〈Φ|Pph|Φ〉 = L−3∑

q,s

~κ|Φqs|2 →∑

s

∫ ∞

−∞~κ|Φs(q)|2

d3q

(2π)3, (15.100)

as the reader may show.Let us finally discuss the spin and helicity of a polychromatic single-photon state in the

helicity basis for the polarization. With |Φ〉 given by

|Φ〉 = L− 3

2

∑

q

(

φq+a†q+ + φq−a

†q−)

|0〉, (15.101)

the action of the photon spin operator, Sph = S, given by Eq. (15.54) in the helicity basis,leads to

S|Φ〉 = L− 3

2

∑

q

~κ(

φq+a†q+ − φq−a

†q−)

|0〉. (15.102)

The state |Φ〉 thus is not an eigenstate for the spin operator. The Lorentz-invariant mean

value of S is

〈Φ|S|Φ〉 = L−3∑

q

~κ(

|Φq+|2 − |Φq−|2)

q−1 →∫ ∞

−∞~κ(

|Φ+(q)|2 − |Φ−(q)|2) d3q

(2π)3q.

(15.103)

Polychromatic one-photon states composed of positive or negative helicity species only,namely

|Φ±〉 = L− 3

2

∑

q

φq±a†q±|0〉, (15.104)


are eigenstates for the global photon helicity operator, h, defined by [cf. Eq. (10.61)]

h =∑

q

κ · Sq = ~∑

q

(

a†q+aq+ − a†q−aq−)

, (15.105)

remembering that Sq is given in Eq. (15.54). Hence,

h|Φ±〉 = ±~|Φ±〉, (15.106)

with the expected eigenvalues ±1 for h/~.

15.5 New T-photon “mean” position state

The single-photon state |Φ〉 given in Eq. (15.80) is an object in Hilbert space, and since allmeasurements are carried out in space-time it is at some stage of a calculation (at least inthe final stage) necessary to transform quantum states to direct space. With a knowledge ofthe photon wave function in q-space [Eq. (15.92)], one might be tempted to believe that thewave function in direct space can be obtained by a usual Fourier (integral) transformation.For fundamental reasons this is not the case, however.

To approach the problem, let us consider the quantum description of a spinless non-relativistic particle. The direct space and wave-vector space representations we denote by|r〉 and |q〉, respectively, and we work in the continuum model. A general state vectorin Hilbert space, Ψ, is in the q-representation given by

|Ψ〉 =∫ ∞

−∞|q〉〈q|Ψ〉d3q, (15.107)

where 〈q|Ψ〉 = Ψ(q) is the wave function in q-space. The wave function in the r-representation, Ψ(r) = 〈r|Ψ〉, thus may be expressed in the form

〈r|Ψ〉 =∫ ∞

−∞〈r|q〉〈q|Ψ〉d3q. (15.108)

A comparison to the Fourier integral

Ψ(r) =

∫ ∞

−∞Ψ(q)eiq·r

d3q

(2π)3(15.109)

shows that the change in the representation (basis) is given by the number

〈r|q〉 = (2π)−3eiq·r. (15.110)

Any wave function Ψ(r0) in configuration space (r) can be expanded in terms of the “bases”∆r(r0) = δ(r0 − r) and ωq(r0) = (2π)−3 exp(iq · r0), which are “orthonormalized” in theDirac sense. The association of the delta function and plane-wave bases to the kets |r〉 and|q〉 is as follows:

|r〉 ⇔∆r(r0) = δ(r0 − r), (15.111)

|q〉 ⇔ωq(r0) = (2π)−3eiq·r0 . (15.112)


The associations in the relations (15.111) and (15.112) give the inner product

〈r|q〉 =∫ ∞

−∞δ(r0 − r)eiq·r0

d3r0(2π)3

= (2π)−3eiq·r, (15.113)

i.e., precisely Eq. (15.110). The use of the simple Fourier integral transformation thus pre-supposes that the particle is completely localizable in space [localization at r described bythe delta function δ(r0 − r) in configuration space (r0)]. In nonrelativistic quantum me-chanics there is no problem in finding a (Hermitian) position operator r for a necessarilyspinless and massive particle. From the well-known definition in wave function space

〈r|r|Ψ〉 = r〈r|Ψ〉, (15.114)

follows the eigenvalue equation for the position operator in state space

r|r〉 = r|r〉. (15.115)

Now the question arises whether it is possible to introduce a position (pos) operator

(observable), Rpos, for the photon satisfying the eigenvalue equation

Rpos|pos〉 = Rpos|pos〉, ?, (15.116)

for a photon sharply localized at Rpos. In case, the sharply localized position state, |pos〉,gives us a completely localized photon wave function according to

Φ(r, t) = 〈pos|Φ〉(r, t). (15.117)

It turns out that a position operator for the photon does not exist in the strict sense[123, 155, 188, 127, 102, 7, 187, 94, 2, 18, 171, 250]: From a physical point of view it isimpossible to localize a photon completely in space.

If one again remembers with Niels Bohr [29, 30, 31], that “No elementary phenomenon isa phenomenon until it is a registered (observed) phenomenon,” the free-photon concept is atbest a useful algorithm. The introduction of a photon wave function in space-time, Φ(r, t),related to the polychromatic single-photon state |Φ〉 in Hilbert space [Eq. (15.80)] thereforecannot be separated from the photon detection process, as this appears in space-time. Froma study of the field-matter interaction associated with detection process, various kinds ofnon-sharp photon position states, |R〉, have been suggested over the years [155, 214, 77, 78].A given choice implies a related photon wave function

Φ(r, t) = 〈R|Φ〉(r, t) (15.118)

in space-time. Below we shall discuss a choice which only recently has come to the front[128], and which, from the point of view of the present author, in its fundamental aspectsappears superior to other choices. The full potential of our choice only emerges in connectionto studies of field-matter interaction, e.g., in investigations of the one-photon interactionwith a single atom.

Let us first divide the Hermitian transverse vector potential operator into its positive[superscript (+)] and negative [superscript (−)]-frequency parts, i.e.,

AT (r, t) = A(+)T (r, t) + A

(−)T (r, t). (15.119)

It appears from Eq. (15.30) that the two parts in free space are given by

A(+)T (r, t) =

[

A(−)T (r, t)

]†= L− 3

2

∑

q,s

(

~

2ǫ0ωq

)1

2

aqs(0)eqsei(q·r−ωqt) (15.120)


in the helicity basis for the polarization. In Eq. (15.30), the polarization basis can be ar-bitrary, and the basis entering the plane-wave expansions of the positive and negative-frequency parts of AT consequently can be arbitrary, if wished. A sort of mean positionstate for the transverse photon in Hilbert space now is introduced by the definition

|R〉(r, t) ≡(

2ǫ0c

~

)1

2

A(−)T (r, t)|0〉, (15.121)

which in the helicity basis (s = +,−) takes the explicit form

|R〉(r, t) = L− 3

2

∑

q,s

q−1

2 e∗s(κ)e−i(q·r−ωqt)a†qs(0)|0〉. (15.122)

15.6 T-photon wave function and related dynamical equation

With the polychromatic single-photon state |Φ〉 resolved in the helicity basis [Eq. (15.101)],I now define a vectorial photon wave function in space-time, Φ(r, t), as follows:

Φ(r, t) ≡ 〈R|Φ〉(r, t). (15.123)

A physical equivalent description can be obtained letting the photon wave function be thespinor

Φ(r, t) ≡(

Φ+(r, t)Φ−(r, t)

)

=

(

〈R|Φ+〉(r, t)〈R|Φ−〉(r, t)

)

. (15.124)

In the continuum limit the explicit expression for the photon wave function in space-timeis

Φ(r, t) =∑

s=+,−

∫ ∞

−∞Φs(q)es(κ)e

i(q·r−ωqt)d3q

(2π)3q. (15.125)

In full agreement with the principle of Lorentz-invariant integration on the light cone [rel-ativistic scalar product]

Φs(q) ≡ Φs(q)es(κ), s = +,−, (15.126)

is the photon wave function in q-space for the helicity species s. The bridge to the scalarwave function in wave-vector space [Eq. (15.92)] thus has been established.

In order to determine the normalization condition in direct space we take the timederivative of the photon wave function in Eq. (15.125) multiplied by i/c, i.e.,

i

c

∂

∂tΦ(r, t) =

∑

s=+,−

∫ ∞

−∞Φs(q)es(κ)e

i(q·r−ωqt)d3q

(2π)3, (15.127)

and thereafter we integrate the scalar product of Φ†(r, t) and (i/c)∂Φ(r, t)/∂t over the


entire r-space. This gives∫ ∞

−∞Φ†(r, t) ·

[

i

c

∂

∂tΦ(r, t)

]

d3r

=∑

s,s′

∫ ∞

−∞Φ∗

s′(q′)Φs(q)e

†s′ (κ

′) · es(κ)q−1ei(q−q′)·rei(ωq′−ωq)td3rd3q′

(2π)3d3q

(2π)3

=∑

s,s′

∫ ∞

−∞Φ∗

s′(q)Φs(q)e†s′ (κ) · e(κ)

d3q

(2π)3q, (15.128)

where the last member comes from the fact that the integral of (2π)−3 exp[i(q−q′) · r] overr-space equals the delta function δ(q− q′), so that the integration over q′-space readily iscarried out. The orthonormality of the helicity vectors belonging to a given κ [Eq. (2.52)]next allows us to eliminate the summation over s′ in Eq. (15.128). In view of Eq. (15.93),we finally obtain

∫ ∞

−∞Φ†(r, t) ·

[

i

c

∂

∂tΦ(r, t)

]

d3r =∑

s

∫ ∞

−∞|Φs(q)|2

d3q

(2π)3q= 1. (15.129)

The result above expresses the normalization conditions for the photon wave function indirect space and in wave-vector space.

The normalization condition in r-space can be given in an important alternative formutilizing the dynamical (Schrodinger-like) equation for the wave function Φ(r, t). In orderto derive this form one takes the inner product of

Hph|Φ〉 = L− 3

2 ~c∑

q,s

q1

2Φqsa†qs|0〉 (15.130)

and |R〉(r, t). Hence

〈R|Hph|Φ〉(r, t) = ~cL−3∑

q′,s′

∑

q,s

q−1

2 es(κ)ei(q·r−ωqt)(q′)

1

2Φq′s′〈0|aqs(0)a†q′s′(0)|0〉

= ~cL−3∑

q,s

Φqses(κ)ei(q·r−ωqt)

→ ~cL−3∑

s

∫ ∞

−∞Φs(q)es(κ)e

i(q·r−ωqt)d3q

(2π)3, (15.131)

because

〈0|aqs(0)a†q′s′(0)|0〉 = δqq′δss′ , (15.132)

as the reader may verify using the fundamental commutation relation in Eq. (15.28) and themode ground-state condition aqs|0〉 = 0. A comparison of the result in Eq. (15.131) withEq. (15.127), gives one the Hamiltonian form of the dynamical equation for the photon wavefunction in direct space, viz.,

i~∂

∂t〈R|Φ〉(r, t) = 〈R|Hph|Φ〉(r, t). (15.133)

This important wave equation allows one to rewrite the normalization condition in Eq.(15.129) as follows:

1

~c

∫ ∞

−∞〈Φ|R〉〈R|Hph|Φ〉d3r = 1. (15.134)


15.7 The non-orthogonality of T-photon position states

Since a T-photon cannot be completely localized in space-time, the |R〉 position statesbelonging at a given time t to different positions, r and r′, say, are not orthogonal, i.e.,〈R(r, t)|R(r′, t)〉 6= δ(r− r′). By means of the definition of the position state [Eq. (15.122)]one has

〈R(r, t)|R(r′, t)〉 = L−3∑

q,s

∑

q′,s′

(qq′)−1

2 es(κ) · e∗s′(κ′)ei(q·r−cqt)e−i(q′·r′−cq′t)

× 〈0|aqs(0)a†q′s′(0)|0〉. (15.135)

Eq. (15.135) is simplified to an integral over q-space utilizing Eq. (15.132). Hence

〈R(r, t)|R(r′, t)〉 = L−3∑

q,s

q−1eiq·(r−r′) → 2

∫ ∞

−∞eiq·(r−r′) d3q

(2π)3q, (15.136)

the factor of 2 coming from the summation over the two polarization states. We now seek tomake the integration in spherical coordinates, with the polar axis along the (r−r′)-direction.After integration over the azimuth and polar angles we have

〈R(r, t)|R(r′, t)〉 = 1

πiR

[

1

2π

∫ ∞

0

eiqRdq − c.c

]

, (15.137)

with R = |r− r′|. By means of the positive wave-number delta function

δ(+)(R) ≡ 1

2π

∫ ∞

0

eiqRdq =1

2

[

δ(R) +i

πP (

1

R)

]

, (15.138)

where P (· · · ) stands for the Principal Part of (· · · ), we may write

〈R(r, t)|R(r′, t)〉 = 1

π2|r− r′|P(

1

|r− r′|

)

. (15.139)

Outside the singularity we thus have

〈R(r, t)|R(r′, t)〉 = 1

π2|r− r′|2 , (15.140)

corresponding to the regularization limk→∞ cos(kR) = 0.

16

Single-photon wave packet correlations

16.1 Wave-packet basis for one-photon states

In Sec. 15.4 we introduced polychromatic single-photon states in Hilbert space [Eq. (15.80)],

|Φ〉 = L− 3

2

∑

i

φi|Pi〉, i = (q, s), (16.1)

by linear superposition of monochromatic plane (P) one-photon states

|Pi〉 ≡ |1i〉 = a†i |0〉. (16.2)

In the one-photon subspace the |Pi〉 states satisfy the orthonormality,

〈Pj |Pi〉 = δij , (16.3)

and completeness

∑

i

|Pi〉〈Pi| = 1 (16.4)

relations. Instead of taking the |Pi〉’s as a basis set for the expansion of |Φ〉, one may usedifferent sets of one-photon wave-packet states, as we now shall see. The generalizationto wave-packet expansion is of importance from a fundamental point of view, and in thischapter we shall realize the usefulness of such an expansion for single-photon correlationstudies.

A set (number index i) of one-photon wave-packet (W ) states, |Wi〉, is introduced by

|Wi〉 =∑

j

tij |Pj〉 =∑

j

tij a†j(0)|0〉. (16.5)

The coefficient matrix t = tij must be unitary if one requires the wave-packet set to beorthonormalized. To prove this one forms the inner product

〈Wj |Wi〉 =∑

m,n

t∗jntim〈0|an(0)a†m(0)|0〉

=∑

m,n

t∗jntimδnm =∑

m

t∗jmtim. (16.6)

When t is unitary the last sum over m equals δij [see Eq. (2.88)], and the one-photon wavepackets states hence satisfy the orthonormality condition

〈Wj |Wi〉 = δij . (16.7)

183


In the one-photon subspace the |Wi〉’s also satisfy the completeness relation

∑

i

|Wi〉〈Wi| = 1, (16.8)

since∑

i

|Wi〉〈Wi| =∑

i,m,n

timt∗ina

†m(0)|0〉〈0|an(0)

=∑

m,n

δmna†m(0)|0〉〈0|an(0) =

∑

m

|Pm〉〈Pm|, (16.9)

and the last member of Eq. (16.9) is just the identity operator 1, see Eq. (16.4). The generalpolychromatic single-photon state hence can be expanded after a wave-packet basis:

|Φ〉 =∑

i

|Wi〉〈Wi|Φ〉. (16.10)

In analogy to Eq. (16.1), Eq. (16.10) may be written in the form

|Φ〉 = L− 3

2

∑

i

ci|Wi〉, (16.11)

with ci = L3

2 〈Wi|Φ〉. The state |Φ〉 is normalized when L−3∑

i |ci|2 = 1.

It is possible and convenient to introduce annihilation (bi(0)) and creation (b†i (0)) op-erators which relate to elementary excitations of the individual wave-packet modes, i. Bycombining Eqs. (16.5) and (16.11) one immediately obtains

|Φ〉 = L− 3

2

∑

i

cib†i (0)|0〉, (16.12)

where

b†i (0) =∑

j

tij a†j(0). (16.13)

The creation operator b†i (0) hence generates the one-photon state |Wi〉 by acting on the(global) vacuum state, that is

|Wi〉 = b†i (0)|0〉. (16.14)

The creation operator b†i (0) is a linear superposition of the a†j(0)-operators, the coefficientsbeing elements of the unitary tij-matrix. Hermitian conjugation of Eq. (16.13) gives usthe related annihilation operator

bi(0) =∑

j

t∗ij aj(0). (16.15)

16.2 Wave-packet photons related to a given t-matrix

Before turning our attention toward the study of single-photon correlation effects let usbriefly indicate how the one-photon wave packet formalism described in the previous sec-tion can be extended to multi-photon wave packets. We start by deriving a fundamental

Single-photon wave packet correlations 185

commutation relation between the wave-packet mode operators bi(0) and b†i (0). Utilizingthe superpositions in Eqs. (16.13) and (16.15) one obtains

[

bi(0), b†j(0)

]

=∑

m,n

t∗imtjn[

am(0), a†n(0)]

=∑

m,n

t∗imtjnδmn =∑

m

t∗imtjm, (16.16)

where the third member of Eq. (16.16) follows from Eq. (15.28) [for t = 0]. Since thet-matrix is unitary, one finally has

[

bi(0), b†j(0)

]

= δij . (16.17)

The reader may show to herself that[

bi(0), bj(0)]

=[

b†i (0), bj(0)]

= 0. (16.18)

The commutation relations in Eqs. (16.17) and (16.18) refer, as indicated, to time zero. Infree space the elements of the unitary t-matrix are time independent. This implies that thetime dependencies of bi and b

†i are given by

bi(t) =∑

j

t∗ij aj(0)e−iωjt, (16.19)

and

b†i (t) =∑

j

tij a†j(0)e

iωjt, (16.20)

in the Heisenberg picture, remembering the simple time evolution for ai [Eq. (15.31)] and

a†i [Eq. (15.32)]. It appears from the equations above that bi(t) and b†i (t) do not have simple

harmonic time evolutions in free space. This is not unexpected since these operators referto wave packets. Using the commutation relations in Eqs. (15.28) and (15.29) one obtainsthe following equal-time commutation relations:

[

bi(t), b†j(t)]

= δij , (16.21)[

bi(t), bj(t)]

=[

b†i (t), b†j(t)]

= 0. (16.22)

Let us now consider a single wave-packet mode, i. The boson operator algebra in Eqs.(16.21) and (16.22) implies that results analogous to those derived up to Eq. (15.65) inSec. 15.3 can be obtained in the wave-packet case. We start by defining a time dependentnumber operator

Ni(t) ≡ b†i (t)bi(t). (16.23)

The eigenvalue problem for Ni(t), i.e.,

Ni(t)|Ni〉(t) = Ni|Ni〉(t) (16.24)

can be solved in the usual manner [155, 53, 214], and the eigenvalues Ni are the integers

Ni = 0, 1, 2, · · · . (16.25)

Acting on the global ground state with the wave-packet annihilation operator one obtains

bi(t)|0〉 = 0, (16.26)


as one readily verifies using Eq. (16.19). The various number state eigenvectors can begenerated from the global ground state as follows:

|Ni〉(t) =

(

b†i (t))Ni

√Ni!|0i〉. (16.27)

The denominator (Ni!)1/2 ensures that orthogonal eigenstates are normalized, so that

〈Ni(t)|Ni(t)〉 = 1 (16.28)

at all times. The state |Ni〉 thus appears to consist of Ni particles (wave-packet photons).The one-photon wave-packet state denoted by |Wi〉 earlier [Eq. (16.14)], in the notationabove usually is written as |1i〉.

In Sec. 15.4, we discussed a number of properties of a general polychromatic single-photon state, |Φ〉 [Eq. (15.80)]. It is obvious that the one-photon wave-packet state |Wi〉 [Eq.(16.5)] is just a particular kind of polychromatic single-photon state. With the identification

tij ⇔ L− 3

2φj , (16.29)

the results obtained in Sec. 15.4 can be transferred from |Φ〉 to |Wi〉.

16.3 Integral equation for the time evolution operator in the in-teraction picture

Let us consider an arbitrary physical system characterized at time t by the state vector|ψ(1)(t)〉 in a given representation 1 [denoted by superscript (1)]. In another representation2, the same system is described by a different state vector |ψ(2)(t)〉 at the same time. Aunitary transformation operator T (t),

T−1(t) = T †(t), (16.30)

which in general depends on time, connects the state vectors in the two representations:

|ψ(2)(t)〉 = T (t)|ψ(1)(t)〉. (16.31)

How is an arbitrary operator (O(t)) [e.g., an observable] changed when one goes fromrepresentation 1 (O(1)(t)) to 2 (O(2)(t))? This question can be answered on the basis ofthe physical requirement that the matrix element of O(t) between two arbitrary states(subscripts α and β) must be the same at all times, that is

〈ψ(1)β (t)|O(1)(t)|ψ(1)

α (t)〉 = 〈ψ(2)β (t)|O(2)(t)|ψ(2)

α (t)〉. (16.32)

Inserting the inverse transformation to the one in Eq. (16.31), viz.,

|ψ(1)(t)〉 = T †(t)|ψ(2)(t)〉, (16.33)

and the associated bra relation into the left side Eq. (16.32) we obtain

〈ψ(1)β (t)|O(1)(t)|ψ(1)

α (t)〉 = 〈ψ(2)β (t)|T (t)O(1)(t)T †(t)|ψ(2)

α (t)〉, (16.34)


and therefore

O(2)(t) = T (t)O(1)(t)T †(t). (16.35)

The identity operator, 1, is the same in both representations because T is unitary, and thetransition matrix elements in turn must be the same in (1) and (2):

〈ψ(1)β (t)|ψ(1)

α (t)〉 = 〈ψ(2)β (t)|ψ(2)

α (t)〉. (16.36)

The probability per unit time

Pα→β(t) = |〈ψβ(t)|ψα(t)〉|2 (16.37)

that a physical system in the state |ψα(t)〉 by observation will be found in the state |ψβ(t)〉hence is the same in all representations.

The general considerations concerning change of representation now are applied to reachan important integral equation for the time evolution operator. A first-order approximationto the solution of the integral equation in turn is used to establish a free-field correlationmatrix (Sec. 16.4) which in Sec. 16.5 will be employed in our study of single-photon inter-ference effects. We begin in the so-called Schrodinger Picture, a representation in which alloperators O ≡ O(1) are time independent so that the time evolution of the physical systemis carried exclusively by the state vector |ψ(t)〉 ≡ |ψ(1)(t)〉. In the Schrodinger Picture thedynamical evolution of the state vector is given by

i~d

dt|ψ(t)〉 = H|ψ(t)〉, (16.38)

where H is the time-independent Hamilton operator of the system. The general solution toEq. (16.38) has the form

|ψ(t)〉 = U(t− t0)|ψ(t0)〉, (16.39)

where

U(t− t0) = exp

[

− i

~H(t− t0)

]

(16.40)

is the time evolution operator in the Schrodinger Picture, and |ψ(t0)〉 is the state vectorat a fixed time t = t0. As indicated, the evolution operator U(t − t0) is a function of thetime difference, t − t0. This is necessarily so because the Hamilton operator (H) [energyoperator] describing the dynamical evolution of the system is constant in time.

In studies of field-matter interactions it is most often convenient to divide the Hamiltonoperator into two parts, i.e.,

H = H0 + HI , (16.41)

where H0 refers to the uncoupled dynamics of the electromagnetic field and the matterparticles, and HI relates to the field-matter interaction (subscript I). In the SchrodingerPicture H0 and HI are both time independent. The change from the Schrodinger Picture tothe so-called Interaction Picture (representation) is defined by the specific transformationoperator

T (t) = exp

(

i

~H0t

)

, (16.42)


which is manifestly unitary, In the remaining part of this section, and in Sec. 16.4, quantities

in the Interaction Picture will be assigned a tilde (˜) on top [|ψ(t)〉 ≡ |ψ(2)(t)〉, Ô ≡ O(2)].In the Interaction Picture one has

ˆH0 = exp

(

i

~H0t

)

H0 exp

(

− i

~H0t

)

= H0, (16.43)

since H0 commutes with exp(iH0t/~). The Hamilton operator describing uncoupled fieldand matter-particle dynamics hence is the same (and time independent) in the Schrodingerand Interaction Pictures. The Interaction Hamiltonian, given by

ˆHI(t) = exp

(

i

~H0t

)

HI exp

(

− i

~H0t

)

. (16.44)

in the Interaction Picture, as indicated will be time dependent, since H0 and HI do notcommute in general. A determination of the time evolution operator in the InteractionPicture cannot start from Eq. (16.35) because U(t − t0) relates state vectors taken atdifferent times. Instead we transform Eq. (16.39), using Eq. (16.33) for times t and t0.Hence,

T †(t)|ψ(t)〉 = U(t− t0)T†(t0)|ψ(t0)〉 (16.45)

and then

|ψ(t)〉 = Û(t, t0)|ψ(t0)〉, (16.46)

where

Û(t, t0) = T (t)U (t− t0)T†(t0) (16.47)

is the time evolution operator in the Interaction Picture. Remembering that T is given byEq. (16.42) and U by Eq. (16.40), we finally obtain

Û(t, t0) = exp

(

i

~H0t

)

exp

[

− i

~H(t− t0)

]

exp

(

− i

~H0t0

)

. (16.48)

It appears from the right side of Eq. (16.48) that the evolution operator Û(t, t0), as indicatedin the notation depends separately on t and t0.

A first-order differential equation for Û(t, t0) can be obtained from Eq. (16.48). Sinceonly the two first exponential functions depend on t we get

i~d

dtÛ(t, t0) = −H0

Û(t, t0) + exp

(

i

~H0t

)

H exp

[

− i

~H(t− t0)

]

exp

(

− i

~H0t0

)

= −H0Û(t, t0) +

ˆH Û(t, t0), (16.49)

the last member of this equation being obtained inserting the unit operator 1 =

exp(−iH0t/~) exp(iH0t/~) after H [H = H 1]. Since ˆH(t) = ˆH0 + ˆHI(t) = H0 + ˆHI(t),we finally obtain the following dynamical equation for the evolution operator in the Inter-action Picture:

i~d

dtÛ(t, t0) =

ˆHI(t)Û(t, t0). (16.50)


The time evolution of Û(t, t0) thus is determined solely by the interaction HamiltonianˆHI(t). The corresponding dynamical equation for the evolution operator in the SchrodingerPicture is readily obtained from Eq. (16.40) since H is time independent. Thus,

i~d

dtU(t− t0) = HU(t− t0). (16.51)

A formal integration of Eq. (16.50) from t0 to t leads to an integral equation for Û(t, t0),viz.,

Û(t, t0) =Û(t0, t0) +

1

i~

∫ t

t0

ˆHI(t′) Û(t′, t0)dt

′. (16.52)

Since Û(t0, t0) = 1, of course, we reach the following important integral equation for thetime evolution operator in the Interaction Picture:

Û(t, t0) = 1 +1

i~

∫ t

t0

ˆHI(t′) Û(t′, t0)dt

′. (16.53)

By successive iterations, Eq. (16.53) leads to a perturbative expansion of the evolution

operator in orders of the interaction Hamiltonian. To first order in ˆHI , we have

Û(t, t0) = 1 +1

i~

∫ t

t0

ˆHI(t′)dt′, (16.54)

an approximation on which the expression to be given for the first-order field correlationmatrix in Sec. 16.4 is based.

16.4 Atomic and field correlation matrices

In a somewhat simplified manner we now want to address the following question: How cana given single-photon state be observed? The delicate question of how to produce a single-photon state is not discussed in this subsection. In order to obtain information on thequantum state of a free electromagnetic field, this field must interact with massive matterparticles, e.g., electrons. The interaction will bring initially uncoupled field and detectorsystems into an entangled state, and it may therefore in general seem difficult to find outwhat the initial quantum field was, not least in the single-photon case, where the field-matter interaction (observation) destroys the photon. In the optical domain one most oftenuses detectors based on the photoelectric effect, and to observe a photoelectric signal atleast one photon must be annihilated.

Let us assume that the field-matter system, described in the Interaction Picture, at theinitial (i) instant t = 0 is in the quantum state |ψi(0)〉. At a later time t = t this state hasevolved into

|ψi(t)〉 = Û(t)|ψi(0)〉, (16.55)

Û(t) ≡ Û(t, 0) being the time evolution operator in the Interaction Picture. The probability


per unit time, Pi→f (t), that the total system has gone from the initial state |ψi(0)〉 to the

final (f) state |ψf (t)〉 after a time t equals

Pi→f (t) =∣

∣

∣〈ψf (t)| Û(t)|ψi(0)〉∣

∣

∣

2

. (16.56)

If the evolution operator is calculated to first order in the interaction Hamiltonian [Eq.(16.54)], Pi→f (t) is given by

Pi→f (t) =

∣

∣

∣

∣

〈ψf (t)|1 +1

i~

∫ t

0

ˆHI(t′)dt′|ψi(0)〉

∣

∣

∣

∣

2

. (16.57)

It is now assumed that the radiation field interacts with a single atom, with effectivelyonly one mobile electron. From a fundamental point of view a single-electron atom is a usefulphoton detector because it to a certain extent allows local field measurements. However,one must emphasize that a rigorous study of the probing of quantum fields locally by point-like test charges involves deep problems in quantum electrodynamics; see, e.g., the articlesby Bohr and Rosenfeld [35, 36]. At the initial time t = 0, the atom is assumed to be inthe ground state |a(0)〉 of the atomic Hamiltonian HA. In the Schrodinger Picture, whereHA is time independent, the state would evolve according to |a(t)〉 = |a(0)〉 exp(−iωat) inthe absence of coupling to the field. The quantity ~ωa = Ea is the ground-state energy,satisfying HA|a(0)〉 = Ea|a(0)〉. In the Interaction Picture one has

|a(t)〉 = exp

(

i

~HAt

)

|a(t)〉 = |a(0)〉 ≡ |a〉 (16.58)

indicating that |a(t)〉 is time independent and equal to the Schrodinger Picture state att = 0. Let us assume that the initially free field is in the one-photon state |Φ(0)〉. Hence,

|Φ(t)〉 = exp

(

i

~HF (t)

)

|Φ(t)〉 = |Φ(0)〉 ≡ |Φ〉. (16.59)

In Eq. (16.59) the free-field (T-photon) Hamiltonian Hph [Eq. (15.94)] is denoted by HF .The time independent single-photon state |Φ〉 is the one given in Eq. (15.80). In the absenceof field-matter interactions |Φ〉 is the state vector in both the Heisenberg and InteractionPicture [and in the Schrodinger Picture at t = 0]. The initial state of the total system isthe tensor product state

|ψi〉 = |a〉 ⊗ |Φ〉 ≡ |a,Φ〉. (16.60)

The photon-atom interaction now is assumed to be present in the time interval (0|t). Let|b〉 = |b〉 be one of the excited atomic states [|a〉 6= |b〉] and |Γ〉 = |Γ〉 a member of an or-thonormal basis for the global free field (e.g., a global Fock state), and let us then determinethe probability per unit time that the atom-photon system has gone from the uncoupledinitial state in Eq. (16.60) into the uncoupled final state

|ψf 〉 = |b〉 ⊗ |Γ〉 ≡ |b,Γ〉 (16.61)

at time t. Since 〈a|b〉 = 0, Eq. (16.57) tells us that

Pi→f (t) =1

~2

∣

∣

∣

∣

∫ t

0

〈b,Γ| ˆHI(t′)|a,Φ〉dt′

∣

∣

∣

∣

2

. (16.62)


By now the explicit expression for the interaction Hamiltonian is needed. For a singleparticle of mass m and charge q the nonrelativistic Hamiltonian is given by [127, 53]

HI = − q

2m

(

p · AT (r) + AT (r) · p)

+q2

2mAT (r) · AT (r) (16.63)

in the Coulomb gauge [A(r) = AT (r)]. The quantity p is the particle momentum opera-tor. The expression for HI refers to the Schrodinger Picture, where all operators are timeindependent. For the transverse vector potential we thus have

AT (r) =∑

q,s

(

~

2ǫ0V ωq

)1

2[

aqs(0)εqseiq·r + h.c.

]

(16.64)

in the plane-wave expansion. In the Interaction Picture one has

ÂT (r, t) = exp

(

i

~HF t

)

AT (r) exp

(

− i

~HF t

)

, (16.65)

and by means of the expression given in Eq. (15.94) for the field Hamiltonian (HF = HT )one is led to the expression in Eq. (15.30) for the vector potential. [Note that no tilde (∼)was placed on top of the vector potential in Eq. (15.30)]. For free fields Eq. (15.30) alsorepresents the transverse vector potential operator in the Heisenberg Picture. Since we hereare interested only in transitions from the atomic ground state to excited states the termproportional to AT · AT in the interaction Hamiltonian cannot contribute to Pi→f (t) [Eq.(16.62)] due to the fact that it contains field operators only. The remaining part of theinteraction Hamiltonian can be written in the compact form

HI = − q

mp · AT (r) (16.66)

since ∇ · AT (r) = 0. The reader may prove this by letting the operator p · AT (r) act onthe general r-space wave function ψ(r, t) = 〈r|ψ〉(t). Hence,

p ·(

AT (r)ψ)

=

[

~

i∇ · AT (r)

]

ψ + AT (r) · pψ = AT (r) · pψ, (16.67)

a result which shows that AT · p can be replaced by p · AT in Eq. (16.63) in the Coulombgauge. In the interaction Hamiltonian given in Eq. (16.66) one now recalls that r is thedynamical particle position coordinate (position operator in the r-representation). For whatfollows it is sufficient to assume that r can be replaced by the here fixed coordinate r0 ofthe nucleus of our one-particle atom. With this assumption HI is reduced to the form

HI = − q

mp · AT (r0). (16.68)

Before proceeding it must be emphasized that the approximation above certainly notalways is a good one. For instance, it cannot be made if one is interested in studies ofelectric quadrupole, magnetic dipole, and higher order multipole phenomena in atoms, orin various studies of conduction electron coupling to light in condensed matter systems.The interaction Hamiltonian in Eq. (16.68) is not equivalent to the often employed electricdipole Hamiltonian. Equivalence is only obtained if the full HI in Eq. (16.63), with r = r0,is used [127, 53, 244].


With HI given by the approximate expression in Eq. (16.68) in the Schrodinger Picture,we obtain in the Interaction Picture an interaction Hamiltonian

ˆHI(t) = exp

[

i

~

(

HA + HF

)

t

]

HI exp

[

− i

~

(

HA + HF

)

t

]

= − q

m

exp

[

i

~HAt

]

p exp

[

− i

~HAt

]

·

exp

[

i

~HF t

]

AT (r0) exp

[

− i

~HF t

]

= − q

mˆp(t) · ÂT (r0, t). (16.69)

By combining Eqs. (16.60), (16.61), and (16.69) one has

〈b,Γ| ˆHI(t)|a,Φ〉 = − q

m〈b|ˆp(t)|a〉 · 〈Γ| ÂT (r0, t)|Φ〉, (16.70)

and when this result is inserted into Eq. (16.62) one obtains the following formula for thetransition probability per unit time:

Pi→f (t) =( q

m~

)2∫ t

0

∫ t

0

〈a|ˆp(t′)|b〉 ·[

〈Φ| ÂT (r0, t′)|Γ〉〈Γ| ÂT (r0, t

′′)|Φ〉]

· 〈b|ˆp(t′′)|a〉dt′′dt′, (16.71)

remembering the Hermiticity of ˆHI(t). Using the compact vector notation

AB : CD ≡ (A ·C) (B ·D) , (16.72)

Eq. (16.71) can be written in the compact form

Pi→f (t) =( q

m~

)2∫ t

0

∫ t

0

〈a|ˆp(t′)|b〉〈b|ˆp(t′′)|a〉

: 〈Φ| ÂT (r0, t′)|Γ〉〈Γ| ÂT (r0, t

′′)|Φ〉dt′′dt′. (16.73)

The total probability (per unit time) that the system at time t has gone from the initialstate to any arbitrary excited (exc) state, Pi→exc(t), is obtained by summing Pi→f (t) overall excited atomic states and the complete basis set of the global free field. In the summationover atomic states the ground state can be included since 〈a|ˆp(t′)|a〉 = 0 for the odd parityground state. In the excitation probability per unit time

Pi→exc(t) =∑

b

∑

Γ

Pi→f (t), (16.74)

one now uses the completeness theorems∑

b |b〉〈b| = 1 and∑

Γ |Γ〉〈Γ| = 1. It appears viaEq. (16.73) that this leads to the following result

Pi→exc(t) =( q

m~

)2∫ t

0

∫ t

0

〈a|ˆp(t′)ˆp(t′′)|a〉

: 〈Φ| ÂT (r0, t′) ÂT (r0, t

′′)|Φ〉dt′′dt′. (16.75)

The excitation probability per unit time therefore essentially is a two-time averaged specialproduct (:) of the atomic correlation matrix

GA(t′, t′′) ≡

( q

m

)2

〈a|ˆp(t′)ˆp(t′′)|a〉, (16.76)


and the field correlation matrix

GF (r0, t′, t′′) ≡ 2ǫ0c

~〈Φ| ÂT (r0, t

′) ÂT (r0, t′′)|Φ〉 (16.77)

at the atomic position, r0. The factor 2ǫ0c/~ in Eq. (16.76) is included in the definitionof the field correlation matrix for later convenience. It is also clear from the derivationleading up to Eq. (16.77) that the initial free-field radiation state need not necessarily bea single-photon state. In terms of the correlation matrices in Eqs. (16.76) and (16.77) onehas

Pi→exc(t) = (2ǫ0c~)−1∫ t

0

∫ t

0

GA(t′, t′′) : GF (r0, t

′, t′′)dt′′dt′. (16.78)

It appears from Eq. (16.77) that the field correlation matrix at the atomic position (r0)relates solely to the initial quantum state of the electromagnetic field. Although we shallbe interested mainly in one-photon states, |Φ〉 need not be such a state. The only thingrequired for the use of Eq. (16.77) is that |Φ〉 is a pure state.

For a statistical mixture of pure radiation states |R〉, specified by the normalized prob-ability PR [

∑

R PR = 1] the field correlation matrix takes the generalized form

GF =∑

R

PR〈R|O|R〉, (16.79)

with O = (2ǫ0c/~)ÂT (r0, t

′) ÂT (r0, t′′). The correlation matrix in Eq. (16.79) can be ex-

pressed in an elegant manner in terms of the density matrix operator ρ defined to be

ρ =∑

R

PR|R〉〈R|. (16.80)

Let |Γ〉 represent some complete set of states for the radiation field. Inserting the unitoperator 1, resolved according to the closure theorem

1 =∑

Γ

|Γ〉〈Γ|, (16.81)

behind O in Eq. (16.79) one obtains

GF =∑

R,Γ

PR〈R|O|Γ〉〈Γ|R〉

=∑

Γ

〈Γ|[

∑

R

PR|R〉〈R|]

O|Γ〉

=∑

Γ

〈Γ|ρO|Γ〉. (16.82)

The last member of Eq. (16.82) shows that the generalized field correlation matrix may bewritten as

GF (r0, t′, t′′) =

2ǫ0c

~Tr

ρ ÂT (r0, t′) ÂT (r0, t

′′)

, (16.83)

where Tr· · · is the trace of the operator · · · for any complete set of free radiation states.


16.5 Single-photon correlation matrix: The wave function finger-print

The (one-photon) field correlation matrix GF (r0, t′, t′′) relates to the field at the atomic

position, r0, and imagining that one moves the atom around, the correlation matrix displaysproperties of the photon field at different points (r) in space. As we shall soon learn, itis useful to generalize the field correlation matrix concept. Hence, we define a first-order(superscript (1)) (single-photon) field correlation matrix, G(1), related to two space-timepoints (r, t) and (r′, t′), as follows:

G(1)(r, t, r′, t′) =2ǫ0c

~〈Φ|AT (r, t)AT (r

′, t′)|Φ〉. (16.84)

For notational brevity the tilde indicating that the transverse vector potential is given in theInteraction Picture is omitted in Eq. (16.84), and in all subsequent equations. Let us nowdivide the vector potentials entering Eq. (16.84) into their positive and negative-frequencyparts. Using also the relativistic notation x = (ct, r) and x′ = (ct′, r′), the first-ordercorrelation matrix is written

G(1)(x, x′) =2ǫ0c

~〈Φ|A(+)

T (x)A(+)T (x′) + A

(−)T (x)A

(−)T (x′)

+ A(+)T (x)A

(−)T (x′) + A

(−)T (x)A

(+)T (x′)|Φ〉. (16.85)

Limiting ourselves to single-photon states [|Φ〉 given by Eq. (15.80)] one has

〈Φ|A(+)T (x)A

(+)T (x′)|Φ〉 = 〈Φ|A(−)

T (x)A(−)T (x′)|Φ〉 = 0, (16.86)

since the destruction (A(+)T A

(+)T ) or generation (A

(−)T A

(−)T ) of two plane-wave photons from

|Φ〉 cannot give us back the state |Φ〉. In view of this, the single-photon correlation matrixis reduced to

G(1)(x, x′) =2ǫ0c

~〈Φ|A(+)

T (x)A(−)T (x′) + A

(−)T (x)A

(+)T (x′)|Φ〉. (16.87)

In general, it is not possible to reduce the formula for G(1)(x, x′) further. If a certain ap-proximation is adopted, a reduction bringing the correlation matrix in contact with single-photon wave mechanics emerges. In order to reach this point let us analyze the two termsin Eq. (16.87) assuming that x = (ct, r) and x′ = (ct′, r) [same space point]. Let us start

with the one containing the operator A(+)T (x)A

(−)T (x′). Recalling the physics in the expres-

sion for the transition (i → f) probability per unit time given in Eq. (16.73) it appears

that A(−)T (x′) relates to a (polychromatic) photon CREATION at the space-time point x′

with a SIMULTANEOUS EXCITATION of the atom from the ground state to a state ofhigher energy. Such a process which in a sense does not conserve energy does occur inquantum mechanics, but is for instance usually less important in situations where quasi-monochromatic light interacts (almost) resonantly with a given atomic transition. In the

part of the process which relates to A(+)T (x) a photon is DESTRUCTED at the space-

time point x simultaneously with a DEEXCITATION of the atom to the ground state. Aschematic illustration of the two energy nonconserving processes inherent in the first part ofthe correlation function is shown in Fig. 16.1. In the second part of G(1), containing the op-

erator combination A(−)T (x)A

(+)T (x′), a photon is DESTROYED at x′ simultaneously with

an atomic EXCITATION from the ground state, and a photon is CREATED at x while the


atom is DEEXCITATED to the ground state. The two processes inherent in this part ofG(1) are both energy conserving, and they are also illustrated in Fig. 16.1. In the languageof quantum optics studies of (two-states) dynamics with neglect of energy nonconservingterms is done under the so-called rotating-wave approximation [211, 214, 244].

a

x´ = (ct´,r) x = (ct,r)

b

FIGURE 16.1The two fundamental channels contributing to the first-order single-photon (black arrow)correlation matrix, G(1)(x, x′). In the upper part of the figure a T-photon (open arrow)is created at the space-time point x′ = (ct′, r) [resulting in a two-photon state] with asimultaneous excitation of the atom (located at r) from the ground state to the excited state(|a〉 → |b〉). At the space-time point x = (ct, r) the T-photon is destructed simultaneouslywith a deexcitation of the atom (|b〉 → |a〉). In each of the two elementary processes shownin the upper part of the figure energy is not conserved. In the lower part of the figure aT-photon is destroyed at x′ under a simultaneous excitation of the atom. At x, a T-photonis generated in the field vacuum simultaneously with a deexcitation of the atom.

In line with the neglect of the energy nonconserving processes, it is useful to define afirst-order (single-photon) correlation matrix by the simplified expression

G(1)(r, t, r′, t′) =2ǫ0c

~〈Φ|A(−)

T (r, t)A(+)T (r′, t′)|Φ〉. (16.88)

A quite closely related formula, namely,

G(1)(r, t, r′, t′) ≡ Tr

ρE(−)T (r, t)E

(+)T (r′, t′)

, (16.89)

where ET = E(+)T + E

(−)T = −∂(A(+)

T + A(−)T )/∂t is the transverse electric field operator of

the free field, is often used in correlation studies of mixed field states, containing possiblymore than one photon [155, 53, 52, 214].

The single-photon correlationmatrix given in Eq. (16.88) gives one a beautiful fingerprintof the photon wave function Φ(r, t) = 〈R|Φ〉(r, t), as we now shall realize. Between thetwo vector-potential operators in Eq. (16.88) one may insert the unit operator, 1, withoutchanging G(1)(x, x′). Utilizing in turn the closure theorem [Eq. (16.81)], the first-ordercorrelation matrix takes the form

G(1)(x, x′) =2ǫ0c

~

∑

Γ

〈Φ|A(−)T (x)|Γ〉〈Γ|A(+)

T (x′)|Φ〉. (16.90)


Since |Φ〉 is a one-photon state, and A(+)T (x′) in plane-wave expansion is a superposition

of annihilation operators (aqs(0)), the state A(+)T (x′)|Φ〉 is the global vacuum state, |0〉,

multiplied by some function of x′. Combining Eqs. (15.120) and (15.80) one thus obtains inthe helicity basis

A(+)T (x′)|Φ〉 =

[

L−3∑

q,s

(

~

2ǫ0ωq

)1

2

eqsei(q·r′−ωqt

′)φqs

]

|0〉, (16.91)

after having used the commutation relation in Eq. (15.28) and the ground-state conditionaqs|0〉 = 0. The quantity in front of the vacuum state vector, |0〉, in Eq. (16.91) is just thesingle-photon wave function, multiplied by the factor [~/(2ǫ0c)]

1/2. [In the continuum limitthe function in front of |0〉 is identical to the expression in Eq. (15.125) [with Eq. (15.92)inserted] up to the above-mentioned factor]. Hence,

A(+)T (x′)|Φ〉 =

(

~

2ǫ0c

)1

2

Φ(x′)|0〉. (16.92)

Since 〈Γ|0〉 vanishes unless |Γ〉 is the global vacuum state, the expression for the correlationmatrix in Eq. (16.90) is reduced to

G(1)(x, x′) =2ǫ0c

~〈Φ|A(−)

T (x)|0〉〈0|A(+)T (x′)|Φ〉 = Φ∗(x)Φ(x′), (16.93)

where the last member of Eq. (16.93) follows using Eq. (16.92) and its corresponding brarelation (for x). The final result in Eq. (16.93) may also be reached without use of Eq.(16.92), remembering the expression for the photon mean position operator in Eq. (15.121).Thus,

G(1)(x, x′) =〈Φ|R(x)〉〈R(x′)|Φ〉 = Φ∗(x)Φ(x′), (16.94)

in view of the photon wave function definition in Eq. (15.118). By now, we have reached theimportant conclusion that the single-photon first-order correlation matrix equals the dyadicproduct of the photon wave function and its complex conjugate at the space-time points xand x′ [(Φ∗(x)Φ(x′)ij) = Φ∗

i (x)Φj(x′)].

Within the framework of the approximation leading from Eq. (16.87) to Eq. (16.88),measurements of the first-order correlation matrix thus may provide us with informationon the single-photon wave function. If the photon wave function vanishes in one of thespace-time points x and x′ the correlation matrix will be zero. In the special case where xand x′ coincide, the trace of the correlation matrix, viz.,

Tr

G(1)(x, x)

= Φ∗(x) ·Φ(x), (16.95)

equals the probability density for observing (detecting) the transverse polychromatic photonin the space-point r at time t.

In cases where it is necessary to take into account the counter-propagating terms inthe first-order one-photon correlation matrix [Eq. (16.87)] two-photon intermediate statesare involved. In consequence, one obtains an extra term to G(1)(x, x′) involving a certaintensorial product of specific two-photon wave functions [106].

17

Interference phenomena with single-photon states

17.1 Wave-packet mode interference

The wave aspect of light is observed experimentally through interference phenomena, andin a certain sense such observations always require a division of the given (initial) light fieldinto two or more space-time parts. The division can only be done by introducing matter(massive particles) in the field region, and the light-matter interaction inevitably changesthe initial state of the field. Not least when one is dealing with single and few-photoninterference, it is good to remember again a central point of Niels Bohr, namely that noelementary phenomenon is a phenomenon until it is a registered (observed) phenomenon. Itturns out that it is possible to observe interference with a quantized electromagnetic fieldwhich contains only one photon. Since the irreversible final detection destroys the photon,it is in practice necessary to repeat the experiment starting from the same initial (given)conditions. Below we shall discuss the basics of single-photon interference in the simplestterms.

We have seen in Sec. 16.1 that it is possible to expand an arbitrary one-photon state,|Φ〉, after a complete set of orthonormalized single-photon wave-packet states, |Wi〉, viz.[Eq. (16.10)],

|Φ〉 =∑

i

|Wi〉〈Wi|Φ〉. (17.1)

The one-photon wave-packet state |Wi〉 is generated by acting with the creation operator

b†i (0) on the global vacuum state, and different sets of bases can be constructed by properchoice of the unitary coefficient matrix, tij, relating the wave-packet and plane-wave basisstates [Eq. (16.5)]. Since all observations are carried out in space-time the single-photon wavefunction,

Φ(r, t) = 〈R(r, t)|Φ〉 =∑

i

〈R(r, t)|Wi〉〈Wi|Φ〉, (17.2)

necessarily plays a central role in one-photon interference experiments. Since

Wi(r, t) = 〈R(r, t)|Wi〉 (17.3)

is the photon wave function of wave-packet number i, one has

Φ(r, t) = L− 3

2

∑

i

ciWi(r, t), (17.4)

where ci = L3/2〈Wi|Φ〉. The ci’s are the probability amplitudes relating to the chosen basisset.

In one-photon detection processes the first-order correlation matrix G(1)(r, t; r′, t, ) is of

197


central importance. We argued for this in Sec. 16.4 in the case of a single-atom detector,but the conclusion reached holds for more complicated particle-system detectors, of course.Since all (single-photon) interference phenomena are closed by an irreversible photoelectricdetection process, it is plausible that G(1)(r, t; r′, t, ) must play a crucial role. Leaving outthe energy nonconserving processes (rotating-wave approximation) the one-photon first-order correlation matrix factorizes into a dyadic product containing only the photon wavefunction; see Eq. (16.94). By inserting the superposition in Eq. (17.4) into Eq. (16.94) oneobtains

G(1)(x, x′) = L−3∑

i

|ci|2W∗i (x)Wi(x

′) + L−3∑

i,j( 6=i)

c∗i cjW∗i (x)Wj(x

′). (17.5)

Generally speaking, the observation of single-photon interference relates to the presenceof the terms in the double sum (i 6= j) of the correlation matrix. Thus, by placing asuitable configuration of matter in the initial photon field, one seeks to filter-out two ormore wave-packet terms (Wi(r, t)) in the superposition in Eq. (17.4). These terms in turnare brought to interfere at the position of a detector. I use the words “brought to” because itphysically only makes sense to detect photons (quantized fields) in a matter-field interactionprocess. It is clear that for a given basis set |Wi〉 one cannot in general expect to establishan experimental arrangement that selects precisely (some of) the basis one-photon wavefunctions that enter an already chosen set of Wi’s. In practice the procedure is the opposite:One first chooses an experimental arrangement of matter particles that appears simple andbeforehand adequate for the goal to be achieved; then one chooses the basis set whichdescribes the situation in terms of as few photon basis wave functions as possible.

The one-photon wave functions, Wi(r, t), which appear in Eq. (17.4) do not relate tothe mean electric field (or transverse vector potential) in the various wave-packet modes,because all mean values are zero; that is,

〈Wi|ET (r, t)|Wi〉 = −〈Wi|∂

∂tAT (r, t)|Wi〉 = 0, ∀i. (17.6)

The result above follows directly from the fact that AT (r, t) is linear in the plane-mode

creation (a†j(0)) and annihilation (aj(0)) operators. The interference thus is not related tomean-field interference.

17.2 Young-type double-source interference

The basics of the general considerations in Sec. 17.1 may be illustrated assuming that oursingle-photon wave function is composed of only two wave-packet modes (i = 1, 2):

Φ(r, t) = L− 3

2 (c1W1(r, t) + c2W2(r, t)) . (17.7)

In this case the first-order correlation matrix takes the form

G(1)(x, x′) =L−3(

|c1|2W∗1(x)W1(x

′) + |c2|2W∗2(x)W2(x

′))

+ L−3 (c∗1c2W∗1(x)W2(x

′) + c1c∗2W

∗2(x)W1(x

′)) (17.8)

In the spirit of the famous Young double-slit experiment, let us imagine that a single photonwave function, Φ(r, t), is incident on an opaque screen with two holes, A and B. The trans-mitted one-photon wave function then consists of a superposition of two pieces ΦA(r, t) and

Interference phenomena with single-photon states 199

ΦB(r, t). If the holes are sufficiently small (much smaller that all relevant wavelengths en-tering the experiment, loosely speaking), one may approximate the transmitted one-photonstate by a superposition of two spherical one-photon wave-packet basis states originatingfrom the positions rA and rB of the two subwavelength holes. For a subwavelength (atomic)detector placed at r the first-order one-photon correlation matrix [x = (r, t)] of relevancehence is

G(1)(x, x) =L−3(

|KA|2W∗A(x)WA(x) + |KB|2W∗

B(x)WB(x))

+ L−3 (K∗AKBW

∗A(x)WB(x) + c.c.) , (17.9)

where WA(x) and WB(x) are the one-photon wave functions emerging from A and B, andKA and KB are the associated probability amplitudes. In the special case where the twoholes are identical and the point of observation is in the far field, the probability density ofdetection is given by

TrG(1)(x, x) =L−3[

|KA|2|W(r − rA, t− tA)|2 + |KB|2|W(r − rB, t− tB)|2]

+ L−3 [K∗AKBW

∗(r− rA, t− tA) ·W(r− rB, t− tB) + c.c.] . (17.10)

The identity of the two mode functions has allowed us to omit the subscripts A and B[WA = WB ≡ W]. Since the radiative part of the field propagates with the vacuum speedof light one has t− tA = |r− rA|/c and t− tB = |r− rB |/c.

A microscopic Young-type double-source experiment can be carried out replacing thescreen with the holes by two two-level atoms A and B; see Fig. 17.1. The ground statesof the atoms are denoted by |1A〉 and |1B〉, and the excited states by |2A〉 and |2B〉. At acertain time an incident single-photon wave packet may be absorbed in either atom A or B.The related quantum states are |2A〉 ⊗ |1B〉 ⊗ |0〉 and |1A〉 ⊗ |2B〉 ⊗ |0〉, where |0〉 denotesthe vacuum state of the field. There also is probability that the atom does not absorb thephoton in the initial state |Φi〉. If we do not observe the states of any of the atoms, thequantum state of the coupled atom-field system is given by the superposition

|Ξ〉 = [α1|2A〉 ⊗ |1B〉+ α2|1A〉 ⊗ |2B〉]⊗ |0〉+ β|1A〉 ⊗ |1B〉 ⊗ |Φi〉, (17.11)

where α1, α2, and β are the probability amplitudes associated with the various atom-fieldstates. In the spirit of the two-hole experiment we assume that the part β|1A〉⊗ |1B〉⊗ |Φi〉does not reach the detector (or any of several detectors). We therefore take

|ψi〉 = [α1|2A〉 ⊗ |1B〉+ α2|1A〉 ⊗ |2B〉]⊗ |0〉 (17.12)

as the initial (i) state of our Young-type experiment. After a sufficiently long time the twoatoms have with certainty decayed to their ground states, and a photon has been emittedfrom atom A or B, the respective single-photon states being |ΦA〉 and |ΦB〉. The final (f)state of the total system hence has the form

|ψf 〉 = |1A〉 ⊗ |1B〉 ⊗ |Φ〉, (17.13)

where

|Φ〉 = KA|ΦA〉+KB|ΦB〉 (17.14)

is the final one-photon state related to the emission process. We can assume that the single-photon states entering Eq. (17.14) are normalized, i.e., 〈Φ|Φ〉 = 〈ΦA|ΦA〉 = 〈ΦB|ΦB〉 = 1,but the states |ΦA〉 and |ΦB〉 are not in general orthogonal. With 〈ΦA|ΦB〉 6= 0 one has|KA|2 + |KB|2 6= 1. In order to determine the first-order correlation matrix one needs the


>

>

>

>

FIGURE 17.1Schematic illustration of a microscopic Young-type double-source diffraction process. Anincident electromagnetic field (wave packet) interacts with a three-layer system of weaklyinteracting atoms (an ultrathin dielectric screen). The diffraction from the two holes in thescreen originates in the scattering of the field from the atoms in the vicinity of the holes(framed regions). If the linear extensions of the holes are sufficiently small compared to thecharacteristic wavelengths composing the incoming wave-packet field, the scattering fromthe (framed) hole regions can be replaced by the scattering from two atoms. In resonancediffraction the atoms may be conceived as two-level electric-dipole antennas in a first ap-proximation. In the essentially macroscopic theory [see [176] and references therein] it isconcluded that the diffraction of light from an opaque (metallic) screen with a mesoscopichole to a certain extent is equivalent to the scattering from a source with only electric- andmagnetic-dipole response. From a microscopic point of view this conclusion is not correct,in general.

wave function associated to |Φ〉. Denoting the mean position state by |R(r, t)〉, as before,one obtains

Φ(r, t) = 〈R(r, t)|Φ〉= KA〈R(r, t)|ΦA〉+KB〈R(r, t)|ΦB〉= KAΦA(r, t) +KBΦB(r, t), (17.15)

where ΦA(r, t) and ΦB(r, t) are the photon wave functions related to the emissions from


atom A and B. Inserting Eq. (17.15) into Eq. (16.94) the correlation matrix takes the form

G(1)(x, x′) =|KA|2Φ∗A(x)ΦA(x

′) + |KB|2Φ∗B(x)ΦB(x

′)

+K∗AKBΦ

∗A(x)ΦB(x

′) +KAK∗BΦ

∗B(x)ΦA(x

′). (17.16)

For a single detector the relevant probability density at x hence is given by

Tr

G(1)(x, x)

=|KA|2|ΦA(x)|2 + |KB|2|ΦB(x)|2

+ [K∗AKBΦ

∗A(x) ·ΦB(x) + c.c.] . (17.17)

Interference then only occurs if the wave-packet photons from the two atoms overlap (sig-nificantly) in the space-time point x.

17.3 Interference between transition amplitudes

It was mentioned in Sec. 17.1 that the single-photon interference phenomenon cannot berelated to mean-field interference, cf. Eq. (17.6). Rather, the interference originates in inter-ference between transition amplitudes, a general quantum physical property. To elaborateon this interpretation let us return to the first expression given for the first-order correlationmatrix in Eq. (16.93). By inserting here the expansion of the one-photon state vector aftera complete set of wave-packet states [Eq. (17.1)] one obtains

G(1)(x, x′) =2ǫ0c

~

∑

i,j

〈Φ|Wj〉〈Wj |A(−)T (x)|0〉〈0|A(+)

T (x′)|Wi〉〈Wi|Φ〉, (17.18)

and hereafter in terms of the mean position state for transverse photons [Eq. (15.115)]

G(1)(x, x′) =∑

i,j

〈Φ|Wj〉〈Wj |R(x)〉〈R(x′)|Wi〉〈Wi|Φ〉. (17.19)

The expression on the right side of Eq. (17.19) may be interpreted in the following manner.When the “initial” one-photon state, |Φ〉, is considered as a superposition of single-photonwave-packet states, |Wi〉, there is a probability that the photon in state |Wi〉 is absorbed atthe space-time point x′. The destruction of the photon, |Φ〉 ⇒ |0〉, thus can follow every (∀i)path, |Φ〉 ⇒ |Wi〉 ⇒ |0〉, and the amplitudes associated with the individual paths interfere.To regain the original single-photon state |Φ〉 a photon may be pulled out of the vacuumand sent into the wave-packet state Wi at space-time point x. Here it contributes to the“final” photon state. The amplitudes of all (∀j) the emission processes, |0〉 ⇒ |Wj〉 ⇒ |Φ〉,interfere to build up the wave function at x.

17.4 Field correlations in photon mean position state

So far we have studied first-order field correlation effects for the arbitrary single-photonstate

|Φ〉 = L− 3

2

∑

q,s

q−1

2Φqsa†qs(0)|0〉, (17.20)


where Φqs is the scalar T-photon wave function in the wave-vector representation. Theexpression in Eq. (17.20) is the photon state in the Interaction Picture, or equivalently theHeisenberg Picture in the absence of field-matter interaction. From a fundamental point ofview the photon mean position state

|R(x0)〉 =(

2ǫ0c

~

)1

2

A(−)T (x0)|0〉, (17.21)

describing a T-photon localized in the neighborhood of the space-time point x0 = (ct0, r0),plays an important role in photon physics, and it is therefore of substantial interest toinvestigate the first-order field correlations in the state.

17.4.1 Correlation supermatrix

In order to apply the formula for the first-order correlation matrix given in Eq. (16.88) oneneeds to transfer the Schrodinger Picture representation |R(x0)〉 to the Interaction Picture,|R(x0)〉. By means of the expression for the free-field photon Hamiltonian HF ≡ Hph [Eq.(15.94)] one obtains

|R(x0)〉 = exp

(

i

~HF t

)

|R(x0)〉 = |R(r0, 0)〉, (17.22)

so that |R(x0)〉 is the value of |R(x0)〉 at time t0 = 0, as expected. At this point onemust be aware of the circumstance that |Φ〉 is a scalar and |R(x0)〉 a three-vector. Let ei[i = x, y, z] be unit vectors along the axes of a Cartesian (x, y, z)-coordinate system. Theresolution

|R(x0)〉 =∑

i

eiei · |R(x0)〉 (17.23)

gives one three scalar states

|R(x0)〉i = |R(x0)〉 · ei, i = x, y, z, (17.24)

and each of these corresponds to a certain |Φ〉. A comparison of the ith scalar state

|R(r0, 0)〉i = L− 3

2

∑

q,s

q−1

2 ei · e∗qse−iq·r0 a†qs(0)|0〉 (17.25)

to Eq. (17.20) shows that the wave function, Φ(i)qs , of |R(r0, 0)〉i in the wave vector repre-

sentation is given by

Φ(i)qs = ei · e∗qse−iq·r0 . (17.26)

For what follows it is useful to write the state |R(r0, 0)〉i in the form

|R(r0, 0)〉i = ei ·[

(

2ǫ0c

~

)1

2

A(−)T (r0, 0)|0〉

]

. (17.27)

Inserting Eq. (17.27) into the general expression for the first-order single-photon correlationmatrix [Eq. (16.88)] one obtains

G(1)i (x, x′) = ei ·G(x, x′|r0) · ei, (17.28)


where

G(x, x′|r0) =(

2ǫ0c

~

)2

〈0|A(+)T (r0, 0)A

(−)T (x)A

(+)T (x′)A(−)

T (r0, 0)|0〉. (17.29)

The subscript i on G(1)i (x, x′) is meant to indicate that the field correlation matrix is for the

ith component of the photon mean position state. The three correlation matrices associatedto |R(x0)〉 appear as projections of the correlation supermatrixG(x, x′|r0) on the Cartesianaxes; cf. Eq. (17.28).

17.4.2 Relation between the correlation supermatrix and the transversephoton propagator

Let |Γ〉 represent a complete set of states for the radiation field. Insert then the unity

operator, in the form given in Eq. (16.81), between A(−)T (x) and A

(+)T (x′) in Eq. (17.29):

G(x, x′|r0) =(

2ǫ0c

~

)2∑

Γ

〈0|A(+)T (r0, 0)A

(−)T (x)|Γ〉〈Γ|A(+)

T (x′)A(−)T (r0, 0)|0〉. (17.30)

A little reflection may convince the reader that only the global vacuum state |Γ = 0〉contributes to the sum in Eq. (17.30). The expression for the field correlation supermatrixhence may be written in the form

G(x, x′|r0) =(

2ǫ0c

~

)2

〈0|A(+)T (r0, 0)A

(−)T (x)|0〉〈0|A(+)

T (x′)A(−)T (r0, 0)|0〉. (17.31)

Utilizing the relation

〈ψ2|O|ψ1〉 = 〈ψ1|O†|ψ2〉∗, (17.32)

valid for arbitrary state vectors |ψ1〉 and |ψ2〉, and identifying the operator O with

A(+)T (r0, 0)A

(−)T (x), Eq. (17.31) can be written

G(x, x′|r0) =(

2ǫ0c

~

)2

〈0|A(+)T (x)A

(−)T (r0, 0)|0〉∗〈0|A(+)

T (x′)A(−)T (r0, 0)|0〉. (17.33)

An interesting interpretation of the structure given in Eq. (17.33) for the single-photoncorrelation supermatrix follows from the fact that

GT (xB − xA) =iǫ0c

2

~〈0|A(+)

T (xB)A(−)T (xA)|0〉 (17.34)

is the transverse photon propagator, a tensorial quantity which describes how the transverseelectromagnetic field in vacuum propagates forward in time (tA < tB) from a space-timepoint xA = (ctA, rA) to another point xB = (ctB , rB). The reader may find a detailedanalysis of the properties of the transverse photon propagator in my book, Quantum Theoryof Near-field Electrodynamics [127]. The correlation supermatrix thus can be written as acombination of two transverse photon propagators, viz.,

G(x, x′|r0) =(

2

c

)2

G∗T (r− r0, t)GT (r

′ − r0, t′), (17.35)

assuming t, t′ > 0. As illustrated schematically in Fig. 17.2, a T-photon created at the


P

AP

X

X0

X´

FIGURE 17.2Schematic illustration of the first-order field correlation in a single-photon mean positionstate with the photon localized in the vicinity of the space-time point x0 = (0, r). Therelated supermatrix G(x, x′|r0) correlates the T-photon wave packet field in the space-timepoints x′ = (ct′, r′) and x = (ct, r) by combining two [(i) and (ii)] elementary propagationprocesses. In (i) a T-photon (P) created at x0 propagates forward in time until it is an-nihilated at x′. This process is correlated to (ii), where a photon created at x propagatesbackward in time to be annihilated at x0 [or equivalently, an antiphoton (AP) created at xpropagates forward in time to be annihilated at x0].

mean position r0 at time t = 0 propagates to the space point r′, where it is annihilatedlater at time t = t′ [associated propagator: GT (r

′ − r0, t′)]. The supermatrix in Eq. (17.35),

correlates this process to one where a photon created at r at time t = t propagates backwardin time and is absorbed at r0 at time t = 0. [associated propagator:G∗

T (r−r0, t)]. A photonpropagating backward in time corresponds to an antiphoton propagating forward in time.Physically, the photon is its own antiphoton; cf. the discussion in Secs. 10.4 and 13.3.

18

Free-field operators: Time evolution and

commutation relations

18.1 Maxwell operator equations. Quasi-classical states

Since each of the monochromatic (ωq = cq) plane-wave modes in the expansion of the

transverse vector potential operator (AT (r, t)) given in Eq. (15.30) satisfies the free-space

wave equation [ exp[i(q · r− cqt)] = 0], AT (r, t) also obeys this equation; that is,

AT (r, t) = 0. (18.1)

Starting from Eqs. (15.33), (15.34), and (18.1), one can derive a set of Maxwell operatorequations for free space form-identical to the ones given for the classical fields in Eqs. (2.1)and (2.2). Thus, by taking the curl of Eq. (15.33) and using Eq. (15.34), one obtains

∇× ET (r, t) = − ∂

∂tB(r, t). (18.2)

The operator equation associated to Eq. (2.2) is gotten by first acting with the curl operatoron Eq. (15.34). Hence,

∇× B(r, t) = ∇×(

∇× AT (r, t))

= −∇2AT (r, t) = −c−2 ∂2

∂t2AT (r, t). (18.3)

The third member of Eq. (18.3) follows because ∇ · AT (r, t) = 0, and the last member is

a consequence of Eq. (18.1). Eliminating AT in favor of ET by utilizing Eq. (15.33), onefinally obtains

∇× B(r, t) = c−2 ∂

∂tET (r, t), (18.4)

that is, the operator form of Eq. (2.2).Having established the Maxwell operator equations it is of interest to try to find the

quantum state |αi〉, i ≡ q, s, which in the best possible manner reproduces the classicalMaxwell equations. Specifically, we require that the quantum mechanical mean values of theMaxwell operator equations in the state |αi〉 coincide with the classical Maxwell equations.We have seen above that the quantized Maxwell equations can be obtained starting fromEqs. (15.33), (15.34), and (18.1). It is obvious therefore that to reach our goal we just needto require that the mean value of the transverse vector potential operator coincides withthe classical transverse vector potential, i.e.,

〈αi|AT (r, t)|αi〉 = AT (αi; r, t). (18.5)

In the notation on the right side of Eq. (18.5), αi refers to the set of amplitudes, αi(0) =

205


αqs(0), which enter the monochromatic plane-wave expansion of a given AT (r, t); seeEq. (15.11). The canonical quantization procedure discussed in Sec. 15.1 results in thetranscription in Eq. (15.30), and from this it is clear that the condition in Eq. (18.5) isequivalent to

〈αi|ai(0)|αi〉 = αi, ∀i. (18.6)

In order that the mean values of the Hamilton operator for the transverse electromagneticfield [with the vacuum contribution omitted] and the transverse field momentum operatorcoincide with the corresponding classical quantities, it is required that

〈αi|a†i (0)ai(0)|αi〉 = α∗iαi, ∀i, (18.7)

as the reader readily may realize comparing Eqs. (15.94) and (15.44) to Eqs. (15.12) and(15.42). Introduction of the operator

bi(0) = ai(0)− αi1, (18.8)

where 1 is the identity operator, allows one to write Eqs. (18.6) and (18.7) as follows:

〈αi|bi(0)|αi〉 =0, ∀i, (18.9)

〈αi|b†i (0)bi(0)|αi〉 =0, ∀i. (18.10)

To obtain Eq. (18.10) use has been made of Eq. (18.9) and its complex conjugate form. Eq.(18.10) can also be written in the form

∥

∥

∥bi(0)|αi〉∥

∥

∥

2

= 0, (18.11)

where ‖ · · · ‖ denotes the norm. Since the norm of bi(0)|αi〉 is zero, one has

bi(0)|αi〉 = 0, ∀i. (18.12)

A result which in turn implies that Eq. (18.9) is satisfied. Reintroducing ai(0) in the lastequation, we get

ai(0)|αi〉 = αi(0)|αi〉, ∀i. (18.13)

The result in Eq. (18.13) shows that |αi〉 is a tensor product state, namely,

|αi〉 = |α1〉 ⊗ |α2〉 ⊗ · · · ⊗ |αi〉 ⊗ · · · ≡∏

i

|αi〉, (18.14)

with

ai(0)|αi〉 = αi(0)|αi〉, ∀i. (18.15)

Because of the requirement in Eq. (18.5), |αi〉 is called a quasi-classical state, or a globallycoherent state. The state |αi〉 is called a single-mode (i) coherent state [155, 214, 77, 78]. Itappears from Eq. (18.15) that |αi〉 is an eigenstate for the T-photon annihilation operatorwith eigenvalue αi. Since ai(0) is a nonhermitian operator the eigenvalues (αi) are not realin general. It follows from Eq. (18.15) that

〈αi|a†i (0) = α∗i (0)〈αi|, ∀i, (18.16)

Free-field operators: Time evolution and commutation relations 207

as well as

A(+)T (r, t)|αi〉 = A

(+)T (αi; r, t)|αi〉, (18.17)

〈αi|A(−)T (r, t) = A

(−)T (αi; r, t)〈αi|. (18.18)

The quasi-classical ket (|αi〉) and bra (〈αi|) states hence are eigenstates for the positive(A

(+)T ) and negative (A

(−)T )-frequency parts of the transverse vector potential operator,

respectively. The associated eigenvalues are just the positive (A(+)T ) and negative (A

(−)T )-

frequency parts of the corresponding classical potential.States containing a fixed number of photons cannot be quasi-classical states since the

mean value of AT (r, t) vanishes in any number state:

〈ni|AT (r, t)|ni〉 = 0, (18.19)

for all |ni〉 states. If the coherent states are defined by Eq. (18.13), one may include theglobal vacuum state, |0〉, among the coherent states, but the photon vacuum is a conceptoutside the framework of classical electrodynamics. The single-photon states in focus in thisbook are not quasi-classical states, and as such these states have no corresponding classicalfields.

18.2 Generalized Landau–Peierls–Sudarshan equations

It follows from the general analysis of complex analytical signals given in Sec. 2.3 that ifthe transverse vector-potential operator satisfies the free-space wave equation in Eq. (18.1),the positive-frequency part of the potential also satisfies this equation, i.e.,

A(+)T (r, t) = 0. (18.20)

Upon a formal factorization of the d’Alembertian operator (see Sec. 10.2) it is realized thatall solutions to the operator equation of propagation

i∂

∂tA

(+)T (r, t) = c

√

−∇2A(+)T (r, t) (18.21)

also are solutions to Eq. (18.20). The dynamical equation in (18.21) is an example of a gen-eralization of the Landau–Peierls–Sudarshan equation [given in Eq. (2.128)] to the operatorlevel. Other field operators obeying a wave equation of the form in Eq. (18.20), likewisesatisfy a dynamical operator equation of the form in Eq. (18.21).

Let the two sides of Eq. (18.21) now operate on the quasi-classical state |αi〉. Thisgives a dynamical equation for the state vector A

(+)T (r, t)|αi〉, namely,

i∂

∂tA

(+)T (r, t)|αi〉 = c

√

−∇2A(+)T (r, t)|αi〉, (18.22)

and use of the eigenvalue equation in Eq. (18.17) then shows that the positive-frequencypart of the related classical transverse vector potential obeys the evolution equation

i∂

∂tA

(+)T (αi; r, t) = c

√

−∇2A(+)T (αi; r, t). (18.23)

The analytical parts of the transverse electric and magnetic fields belonging to the given


A(+)T (αi; r, t) of course satisfy dynamical equations form-identical to the one in Eq.

(18.23). With the establishment of Eq. (18.23) we have hence recovered Sudarshan’s equa-tion for analytical signals in the classical domain. Sudarshan’s equation is also obeyed bythe free-field second-order space-time correlation function, a result which has a number ofinteresting physical consequences in coherence theory [155].

From the operator Landau–Peierls–Sudarshan equation in Eq. (18.21), the bridge tophoton wave mechanics can be made. Thus, if one makes the sandwich of Eq. (18.21) withthe vacuum state 〈0| and the polychromatic single-photon state |Φ〉 the following dynamical

equation is obtained for the matrix element 〈0|A(+)T (r, t)|Φ〉:

i∂

∂t〈0|A(+)

T (r, t)|Φ〉 = c√

−∇2〈0|A(+)T (r, t)|Φ〉. (18.24)

Expressed in terms of the position state for the transverse photon in Hilbert space [Eq.(15.121)], Eq. (18.24) takes the form

i∂

∂t〈R(r, t)|Φ〉 = c

√

−∇2〈R(r, t)|Φ〉, (18.25)

and since 〈R(r, t)|Φ〉 is the vectorial photon wave function in space-time [see Eq. (15.123)],one finally obtains the spatially nonlocal dynamical equation

i~∂

∂tΦ(r, t) = c~

√

−∇2Φ(r, t) (18.26)

for the one-photon wave function in direct space. The symbolic notation introduced bythe definition in Eq. (2.127) combined with the expansion given for Φ(r, t) in Eq. (15.125)implies that

√

−∇2Φ(r, t) =∑

s=+,−

∫ ∞

−∞Φs(q)es(κ)e

i(q·r−cqt) d3q

(2π)3. (18.27)

Combining Eqs. (18.26) and (18.27) one regains the dynamical equation given for the photonwave function in Eq. (15.127) [or equivalent in Eq. (15.133)].

18.3 Commutation relations

The commutation relations between the field operators AT , ET , and B play a fundamentalrole in quantum electrodynamics. Physically, these commutation relations allow one tofind those observables of the electromagnetic field which can be measured independentlyof one another, at least in principle. The question of limitations on the measurability ofelectromagnetic field quantities goes deeper than to the field commutation relations. Theappearence of the Dirac delta function in the commutation relations (see below) means thata non-ambiguous meaning can be given only to space-time integrals of the field components.The measurable field quantities hence are no longer represented by true point functions butby functions of (small) space-time regions. These “region-functions” correspond to (possiblyweighted) average values of the idealized fields over the region in question. To measure afield interaction with electrically charged matter, particles (test bodies) are needed. Ideallyspeaking, one wants the test body to have a negligible influence on the electromagnetic field.The test body will move nonuniformly under the influence of the field but the accompanying


radiation reaction must be small compared to the action of the external field which onewants to measure. Despite the fact that test bodies in classical electrodynamics usually areconsider to be point charges, “use of” point-like test bodies gives rise to certain difficultiesbecause measurable field quantities relate to region-functions. As mentioned previously,the question of the measurability of electromagnetic field quantities was analyzed in aningenious manner by Bohr and Rosenfeld in a paper from 1933 [35]. A more general andexhaustive treatment by these authors appeared in the literature (with long delay) in 1950[36].

Notwithstanding the fundamental importance of the various field commutation rela-tions appearing in the presence of field-matter interactions, we here limit ourselves to abrief discussion of free-field commutators. The results given below for the various free-fieldcommutators can be derived starting from Eqs. (15.30)-(15.34) and utilizing the equal-timecommutation relations in Eqs. (15.28) and (15.29) taken at time t = 0. The calculations,which are rather lengthy, can be found in several places in the literature, and there is noneed to repeat these calculations here. Below, explicit forms given in [127, 53], e.g., arecited. These forms are particularly useful in near-field electrodynamics, and for the under-standing of the spatial localization problem for transverse photons. The linearity of therelations among AT , ET , and B implies that all of the commutators are c-numbers and notoperators. In free space the field theory must exhibit translation invariance in space andtime. This implies that commutators relating to two different space-time points x = (ct, r)and x′ = (ct′, r′) depend only on x − x′ = (c(t − t′), r − r′) ≡ (cτ,R). With respect torotations the commutators are second-rank tensors.

18.3.1 Commutation relations at different times (τ 6= 0)

Let us first consider the commutator of ET and B, which has the simplest structure:

[

ET (r, t), B(r′, t′)]

=~

4πiǫ0U× R

[

1

R(δ′′(R + cτ) + δ′′(R− cτ))

− 1

R2(δ′(R + cτ) + δ′(R − cτ))

]

, (18.28)

where δ′ and δ′′ are the first and second derivatives of the delta function with respect toits argument, and R = R/R. It appears from Eq. (18.28) that the commutator of thetransverse electric and magnetic fields is zero outside the light cone, given by

R = c|τ |. (18.29)

In the idealized sense in which field quantities are presented by true point functions andmeasurable by means of point charges, the physical variables ET and B hence can be mea-sured independently of one another outside the light cone. When speaking of measurabilitybelow we refer to the idealized sense of the word. Note that the commutator consists of far(R−1)- and mid-field (R−2) separation parts, involving different generalized functions. The

Cartesian coordinate [R = (X,Y, Z)] form of the U× R-tensor, namely,

U× R =1

R

0 −Z YZ 0 −X−Y X 0

(18.30)

shows that the components of ET and B on a given axis always commutate even on thelight cone. The reason for this lack of correlation stems from the fact that the free Maxwellequations [(2.1) and (2.2)] do not couple components of ET and B in the same direction.


The commutators of the transverse electric field and the magnetic field with themselvesalso vanish outside the light cone, as the following relations show:

[

ET (r, t), ET (r′, t′)

]

=c2[

B(r, t), B(r′, t′)]

=~c

4πiǫ0

(

U− RR) 1

R(δ′′(R + cτ)− δ′′(R− cτ))

−(

U− 3RR)

[

1

R2(δ′(R+ cτ) − δ′(R− cτ))

− 1

R3(δ(R+ cτ) − δ(R− cτ))

]

. (18.31)

The commutators above not only have parts with separation dependences R−1 and R−2

but also a near-field (R−3) part. For far-field (R−1) separated points the commutator has

the tensorial form U− RR, a necessity because electromagnetic field propagation betweenpoints r′ and r can correlate only field components perpendicular to the R-direction. It isalso obvious from the wave equations for ET [Eq. (2.5), with E = ET ] and B [Eq. (2.6)] thatonly components of ET (or B) on the same axis are correlated as indicated by the diagonalform of the tensor. The near-field part of the commutators has a tensorial form given byU−3RR, a fact which reflects a deep relation to the near-field part of the Feynman photonpropagator’s projection onto the subspace of the transverse photons [127].

The most interesting commutator is the one of AT and ET because it contains a partwhich is nonvanishing outside the light cone. Explicitly, one has

[

AT (r, t), ET (r′, t′)

]

=i~

4πǫ0

(

U− RR) 1

R(δ′(R+ cτ) + δ′(R− cτ))

−(

U− 3RR)

[

1

R2(δ(R+ cτ) + δ(R− cτ))

+1

R3(θ(cτ −R)− θ(cτ +R))

]

, (18.32)

where θ is the Heaviside unit step function. In addition to far- and near-field contributionswhich are nonzero on the light cone only, the commutator has a near-field part proportionalto θ(cτ−R)−θ(cτ+R). For space-time points (x, x′) which are timelike separated (R < c|τ |)the step function factor vanishes, but for spacelike separated points (R > c|τ |) the factorequals−1. This last circumstance does not imply that light can propagate with superluminalspeed over near-field distances, it rather reflects the fact that a transverse photon interactingwith a charged-particle source/detector cannot be better localized than to the rim zone ofthe particle domain. For an electric point-dipole the rim zone has the linear extension of thenear-field zone [123]. By integrating Eq. (18.28) over time (t) the reader may easily obtain

the explicit form of the commutator [AT (r, t), B(r′, t′)], remembering that ET = −∂AT/∂t.The non-commutativity of the various field quantities is a nonclassical feature, and thecommutators given in Eqs. (18.28), (18.31), and (18.32) therefore vanish when the quantumof action becomes negligible (~ → 0). Since the commutators are c-numbers, their quantummechanical mean values are the same for all quantum states of the field, including thesingle-photon states of particular interest in this book.

18.3.2 Equal-time commutation relations

The results obtained in the previous subsection for the various field commutators hold forsure for τ 6= 0. In the limit τ → 0, the explicit expressions in Eqs. (18.28) and (18.32)


become ambiguous. By returning to the plane-wave expansions for ET , B and AT it is easy,however, to determine the equal-time commutators of the field operators. The importantcommutation relation between AT and ET thus is

[

AT (r, t), ET (r′, t)]

=~

iǫ0δT (R), (18.33)

where δT (R) is the transverse delta function, a dyadic quantity. The plane-wave expansionof this singular quantity is given in Eq. (2.97), and in the text below this equation it wasnoted that δT (R) decays as R−3. The reader may find more information on the transversedelta function in [127, 53]. The right-hand side of Eq. (18.33) is independent of time, and

the fields AT (r, t) and ET (r′, t) hence are correlated at near-field separations in a manner

which does not involve propagation of signals (with the speed of light). The commutator of

B and ET may be derived from Eq. (18.33), taking the curl (∇×) on both sides. Thus,

[

∇× AT (r, t), ET (r′, t)]

=~

iǫ0∇× δT (R), (18.34)

and then

[

B(r, t), ET (r′, t)]

=~

iǫ0U×∇δ(R). (18.35)

Since the differences involving δ, δ′, or δ′′ in Eq. (18.31) all vanish in the limit τ → 0, oneobtains without ambiguity

[

ET (r, t), ET (r′, t)]

=[

B(r, t), B(r′, t)]

= 0, (18.36)

as one would have expected because ET and B must propagate with the speed of light.

Part V

Photon embryo states

19

Attached photons in rim zones

In theoretical studies of (single) photon emission from an atom it is of central importance todescribe the time development of the field-atom interaction. When the interaction has beenbrought to an end a transverse photon has been released from the atom. Traditionally, far-field detection of the T-photon is used to obtain information on the field-atom interactionprocess. In far-field detection schemes it is in the overwhelming majority of cases sufficientto assume that the released photon field spreads out from a single point in space. This pointwe call “the position of the atom.” Although it certainly is necessary to take into accountthe spatial variation of the electromagnetic field across the atomic domain in studies ofelectric quadrupole, magnetic dipole, and higher-order multipole interaction processes, it isnevertheless an extremely good approximation to consider the atom as a point-like entityin optical far-field studies. In the literature one often encounters the point of view that theemitted T-photon comes from (is generated in) a volume coincident with the electronic sizeof the atom. Since atomic radii are of the order 1-10 A the above view makes it intuitivelyeasy to consider the atom effectively as a point entity in far-field optics.

Not least from near-field electrodynamic investigations we have learned that the sourcedomain of a transverse photon cannot in general be considered to be identical to the domainoccupied by the atomic electron distribution. In order to give a quantity account of thiscircumstance it is useful to start from the integral relation

Aµ(x) =µ0

c

∫ ∞

−∞g(x− x′)Jµ(x′)d4x′, (19.1)

which gives us the four-potential, Aµ(x), generated by a general (not necessarily atomic)four-current source of density Jµ(x′). The presence of the Huygens scalar propagator,g(x−x′), indicates that the four-potential wavelet emitted from the space-time source pointx′ reaches the point x with a delay dictated by the speed of light in vacuum. In photonwave mechanics based on the potential description, the transverse photon wave function[i.e., AT (x) properly normalized] emerges from

AT (x) =µ0

c

∫ ∞

−∞g(x− x′)JT (x

′)d4x′, (19.2)

once the source has stopped radiating. It appears from Eq. (19.2) that the source domainof a T-photon is not to be identified with the electronic three-current density distributionJ(x′). Only the transverse part, JT (x

′), of this vector field distribution is responsible forthe T-photon generation. Since

JT (r, t) =

∫ ∞

−∞δT (r− r′) · J(r′, t)d3r′ (19.3)

where δT (r − r′) is the transverse delta function, it is seen that the relation between Jand JT is nonlocal in space but local in time. Usually, it is a very good approximation toassume that the electronic current density, J, has finite support in space (volume V). The

215


form of the transverse delta function (e.g., in spherical contraction) however shows thatJT in general also is nonvanishing in a certain region outside V. As mentioned previouslythis region I have called the rim zone (of matter). The behavior of the photon fields in therim zone exhibits interesting and surprising features, as we shall realize in the followingchapters.

In Chapt. 20 we investigate the rim zone photon fields in a (model) situation where weimagine that our particle system possesses translational invariance against all infinitesimaldisplacement perpendicular to a given direction, below taken as the z-direction. The essentialpart of the physics can be understood assuming that the relevant vector fields have plane-wave character perpendicular to z; i.e., ∼ exp[i(q‖ · r − ωt)], with q‖ = (q‖,x, q‖,y, 0). Theansatz above reduces the relevant integral relation to one-dimensional form:

Aµ(z) = µ0

∫ ∞

−∞g(z − z′)Jµ(z′)dz′, (19.4)

where

g(z − z′) =i

2

[

(ω

c

)2

− q2‖

]− 1

2

exp

i

[

(ω

c

)2

− q2‖

]1

2

|z − z′|

. (19.5)

The form of the scalar propagator shows that the four-potential field propagates away fromthe source plane (z′) without damping for q‖ < ω/c, and decays exponentially for q‖ > ω/c.The expression in Eq. (19.5) relates to the so-called angular spectrum representation of thespatial part of monochromatic fields in vacuum domains, including rim zones [155, 127]. Inthis representation a two-dimensional plane-wave (wave vector: q‖) expansion of the field ismade. Perpendicular to the plane of the q‖-vectors the spatial spectrum consists of modesthat propagate without damping (q‖ < ω/c) and decay exponentially (q‖ > ω/c). The lasttype of modes is known as evanescent modes, and it is these we focus on in Secs. 20.2and 20.3. Experimentally, evanescent fields, and the closely related electromagnetic surfacewaves, can be generated in various manners [193, 200, 232, 179, 136, 135, 4, 54, 176].

Important qualitative insight can be obtained starting from the assumption that theparticle four-current density is confined to a sheet the thickness of which is much smallerthan the decay length of the evanescent field. If one makes an expansion of the sheet four-current density in derivatives of the Dirac delta function one obtains in lowest order athree-current density J(z) = Iδ(z). The longitudinal part, JL(z), of J(z) may be calculatedwith the help of the longitudinal dyadic delta function. It turns out that JL(z) is confinedto the plane spanned by q‖ and ez, and decays exponentially away from the z = 0 plane,with a decay constant |q‖|. The source domain of the longitudinal (and transverse) photonsthus is spread over a finite range of extension ∼ exp(−q‖|z|). The current density (sourcedomain) producing scalar photons, J0(z), is confined to the sheet plane (z = 0). From thegiven JT (z), JL(z), and J0(z) one can determine the evanescent T-, L-, and S-potentialsvia Eqs. (19.4) and (19.5).

Fourier integral decompositions of the z-dependencies of the AT (z), AL(z), and AS(z)potentials make the bridge to the T-, L-, and S-photons associated to the evanescent fieldsfrom the sheet. From the analysis in Sec. 20.4 we reach the conclusion that the scalar(AS(Q; t)) and longitudinal (AL(Q; t)) photon variables in the wave-vector (Q)-time (t)domain satisfy the following dynamical equations:

(

cQ− i∂

∂t

)

AS(Q; t) =1

ǫ0c

Q

q0

eQ · IQ+ q0

e−iωt (19.6)

(

cQ− i∂

∂t

)

AL(Q; t) =1

ǫ0c

eQ · IQ+ q0

e−iωt, (19.7)

Attached photons in rim zones 217

where eQ = Q/Q, and q0 = ω/c. In free space (I = 0), Eqs. (19.6) and (19.7) reduce(after multiplication by ~) to the quantum mechanical wave equations for the scalar andlongitudinal photons.

In Sec. 20.5 the photon wave mechanical description is extended to the field-quantizedlevel by a canonical quantization procedure, where the properly scaled four-potential iselevated to the operator level, i.e.,

(

2ǫ0cQ

~

)1

2

Aµ(Q; t) = αµ(Q; t) ⇒ aµ(Q; t). (19.8)

The Heisenberg equation of motion for the set of annihilation operators, aµ(Q; t), takesthe usual form, viz.,

∂

∂taµ(Q; t)+ iΩQaµ(Q; t) =

i

(2ǫ0~ΩQ)1

2

Jµ(Q; t). (19.9)

The monochromatic sheet four-current density operator, given by

Jµ(Q; t =

(

Q

q0eQ · I, I

)

e−iωt, (19.10)

is an operator in the particle subspace.We finish our analysis of the monochromatic photon emission related to the evanescent

field from a delta-function confined current density sheet with a brief discussion of thenear-field (NF) photon picture in second quantization (Sec. 20.6). Since the NF photon is amatter-attached photon the Heisenberg equation of motion for the NF-photon annihilationoperator, aNF (Q; t), has the unique solution

aNF (Q; t) =i

ω(ǫ0~ΩQ)

− 1

2Q− q0Q+ q0

Q

q0

(

e−iωt − 1)

eQ · I. (19.11)

The solution for aNF (Q; t) is an implicit solution because the sheet current density opera-tor amplitude (I) in general depends on the near-field photon annihilation (and creation)operators belonging to the various Q-modes. The situation is much simpler if the sheetcurrent density is a classical quantity, and I hence a c-number. With the replacement I ⇒ I(c-number) we denote the right side of Eq. (19.11) by αNF (Q; t). In the Heisenberg Picturethe quantum state |Ψ〉 of our system is time independent, so that

aNF (Q; t)|Ψ〉 = αNF (Q; t)|Ψ〉. (19.12)

The state |Ψ〉 hence is an eigenstate for the near-field annihilation operator belonging tothe Q-mode, and therefore a coherent state with the characteristic parameter |αNF (Q; t)|2.

In Chapt. 21 the photon tunneling process is studied. The tunneling photons are T-photons, and they are able to tunnel across a vacuum gap between two media only if theseare (effectively) in rim zone contact. In a broader perspective it turns out that electro-dynamic near-field interaction, spatial photon localization, and photon tunneling have thesame roots. If one seeks to measure the photon field inside the tunneling gap one is facedwith the fundamental measurement problem in quantum physics.

A paradigm of photon tunneling appears in relation to the physics of frustrated totalinternal reflection (FTIR). In the FTIR process evanescent fields play an indispensable role,and what we shall learn about evanescent fields generated by a sheet current density distri-bution in Chapt. 20 will be of invaluable importance for a qualitative physical understandingof the tunneling process. To describe the photon tunneling as a particle-like phenomenon


one cannot work with monochromatic fields, and we therefore begin with an analysis ofthe scattering of a wave-packet band from a single current-density sheet (Sec. 21.2). Allfields still have plane-wave form, ∼ exp(iq‖ · r), parallel to the plane of the sheet. The basicintegral relation for the scattered (scatt) four-potential now has the form

Aµscatt(z, t) = µ0

∫ ∞

−∞g(z − z′, t− t′)Jµ(z′, t′)dz′dt′, (19.13)

where the relevant scalar propagator is given by

g(Z, τ) =c

2θ (cτ − |Z|)J0

(

q‖√

(cτ)2 − Z2)

, (19.14)

J0 being the Bessel function of zeroth order. The step function θ(cτ − |Z|) indicates thatthe field propagation is time-like (cτ > |Z|) in the covariant description. The transversepart of the scattered potential is given by

AT (z, t) = µ0

∫ ∞

−∞g(Z, τ)JT (z

′, t′)dz′dt′, (19.15)

or equivalently by

AT (z, t) = µ0

∫ ∞

−∞

[∫ ∞

−∞g(Z, τ)δT (z

′)dz′]

· I(t′)dt′, (19.16)

where δT (z′) is the transverse delta function. Although all observations made on the basis

of Eqs. (19.15) and (19.16) will be the same, the metaphorical description appears different.In Eq. (19.15), the field propagates outward with the vacuum speed of light from everysource point in the JT -distribution, which extends over the entire rim zone. In Eq. (19.16),the dyadic Green function

GT (z, τ) =

∫ ∞

−∞g(z − z′, τ)δT (z

′)dz′ (19.17)

describes the field emission generated by source distribution which only is nonvanishing onthe sheet. Since δT (z

′) is time independent, it is obvious that the field emission in this picturewill contain a part which apparently does not obey the Einsteinian causality. In fact this partwhich extends over the rim zone is only nonvanishing in front of the light cone. Altogether,Eq. (19.17) gives one the impression that light “propagates” with superluminal speed inthe rim zone. In turn, this leads to the conclusion that the field apparently propagateswith a speed exceeding the vacuum speed of light in a tunneling process between twocurrent-density sheets in rim-zone contact. Quantum statistically, the conclusion obtainedquantizing Eq. (19.15) is that the theoretical tunneling time, so to speak, is zero. Since “aphenomenon is only a phenomenon when it is a registered phenomenon” [29, 30, 31] theoperational tunneling time will be finite because the role of the detector has to be includedin the analysis; cf. the (photon) measurement theory in quantum physics.

In Sec. 21.3 we discuss how an incident field generating evanescent tunneling potentialsfrom a sheet can be generated by total internal reflection (TIR). In order to produce anevanescent potential the incident field must satisfy the condition q‖ > ω/c. In Sec. 21.5 weturn attention toward single-photon incident fields. Specifically, we construct a polychro-matic wave-packet band which contains precisely one photon, and which single wave-vectorcomponent in the plane of the sheet, Q‖, satisfies the criterion Q‖ > ω/c for all frequencycomponents in the polychromatic state. Choosing the x-axis of the Cartesian coordinate sys-tem along the Q‖-direction, and imagining that a homogeneous dielectric medium with a

Attached photons in rim zones 219

frequency independent (and thus real) refractive index n = ε1/2 is used to generate the one-photon incident field in a TIR-geometry, it turns out that the polychromatic single-photonwave function is given by

Φ(r, t) = eiQ‖xey ×

∫ Q‖

√ε−1

0

κΦ(q⊥) exp

i

[

q⊥z −c

n

(

Q2‖ + q2⊥

)1

2

t

]

dq⊥

2π(

Q2‖ + q2⊥

)1

2

,

(19.18)

where Φ(q⊥) is the mode amplitude, and κ = (Q‖ex + q⊥ez)/(Q2‖ + q2⊥)

1/2.Using the inhomogeneous vacuum concept, the photon tunneling process in the FTIR-

configuration can be conceived as a tunneling between two current-density sheets locatedat the two medium-vacuum interfaces, as we shall realize in Sec. 21.3; see also [118]. Thephoton tunneling between coupled sheets is discussed in Sec. 21.6.

In Chapt. 22 we study the near-field photon emission in 3D. Attention is focused on thefour-potential generated by a classical point-particle (of chargeQ) moving along a prescribedtrajectory. In the wake of some general considerations on source fields in 3D, the near-field(ANF

T ) and far-field (AFFT ) parts of the transverse vector potential of the point-particle are

determined.The important result obtained for ANF

T reads

ANFT (r, t) =

Qc

4πǫ0

∫ t

tr

(t′ − t)R−3(t′)(

U− 3R(t′)R(t′))

· β(t′)dt′, (19.19)

where R(t) = r − r0(t) is the vectorial distance between the particle position at time t,

r0(t), and the point of field observation, and R(t) = R(t)/R(t). The quantity β(t) is theparticle velocity v0(t) normalized to the speed of light in vacuum, i.e., β(t) = v0(t)/c,and tr the well-known retarded time. It appears from Eq. (19.19) that the near-field partof the transverse vector potential generated by the particle solely stems from the part ofthe particle trajectory, r0(t

′), which “radiation” is space-like connected to the space-time(r, t) point of observation. This at first sight perhaps surprising result associates to the factthat a transverse photon emitted from the particle quantum statistically is born somewherein the particle’s near-field zone. The sum of the T- and L-parts of the vector potentialis just the well-known Lienard–Wiechert result for the vector potential of a point-charge[225, 101, 127, 131, 248, 223].

In Subsec. 22.2.1 we specialize the analysis to the situation where the point-particle isin uniform motion in vacuum. In a first step the four-potential of the particle, Aµ(r, t),is given as an integral over the domain of wave four-vectors (ω/c,q). The constant velocity(V) of the particle locks the component of the wave vector parallel to V, i.e., q‖ = q‖V/V =q‖eV, to a value q‖ = ω/V . The spectrum of wave vectors (k) entering the integral expressionfor the four-potential hence is confined to k = q⊥ + (ω/V )eV, where q⊥ is the componentof the wave vector perpendicular to V. The resulting three-dimensional integral formula forAµ(r, t) thus is over the (q⊥, ω)-domain. The integrand has a denominator given by

D0(q⊥, ω) = q2⊥ +( ω

V

)2[

1−(

V

c

)2]

. (19.20)

Since V < c one has D0(q⊥, ω) > 0. This means that there is no resonance contribution tothe four-potential. However, if the point-particle moves through a medium (which we forsimplicity assume exhibits translational invariance in space-time, and rotational invariance


in space) it turns out that the (q⊥, ω)-integrand in the integral expression for the transversepart of the vector potential has a characteristic denominator

D(q⊥, ω) = q2⊥ +( ω

V

)2[

1−(

V

c

)2

εT (k, ω)

]

, (19.21)

where εT (k, ω) is the transverse (T) dielectric function (here taken as real quantity). Reso-nance [D(q⊥, ω) = 0] now may occur provided

V >c

√

εT (k, ω). (19.22)

Physically, this means that the particle must move with a speed exceeding the phase velocityof light [at (k, ω)]. The inequality in Eq. (19.22) is the condition needed for obtainingtransverse Cerenkov shock waves [Subsec. 22.2.3]. The expressions for the longitudinal andscalar parts of the four-potential contain in the (q, ω)-domain a characteristic denominator[q2−(ω/c)2]εL(q, ω), where εL(q, ω) is the longitudinal (L) dielectric function. If one assumesthat εL(q, ω) is real, the resonance condition εL(q, ω) = 0 relates to the Landau shock waves.These waves may be generated provided the particle speed exceeds the phase velocity ofrelevant matter waves (“sounds”) in the medium.

Part V is finished by a discussion of the momenta associated to the transverse and lon-gitudinal parts of the Cerenkov field. The so-called longitudinal part of the field momentumrepresents the field part of an effective canonical particle momentum given by

peff = π +

∫ ∞

−∞ρsc(r, t)AT (r, t)d

3r, (19.23)

where π is the particle’s kinetic momentum, and ρsc(r, t) the longitudinally screened (sc)particle charge density [Qδ(r−r0(t))]. In vacuum we regain from Eq. (19.23) the expressionfor the canonical momentum (p) of a point-particle located instantaneously at r0(t), viz.,p = π +QAT (r0(t)) [127, 53].

20

Evanescent photon fields

20.1 Four-potential description in the Lorenz gauge

In Sec. 12.1 it was realized that the content of the microscopic Maxwell–Lorentz equations in(8.49)-(8.52) is contained in the following inhomogeneous wave equations for the components(µ = 0− 3) of the contravariant four-potential Aµ(x):

∂ν∂νAµ(x)− ∂µ∂νA

ν(x) = −µ0Jµ(x), (20.1)

see Eq. (12.6). The form in Eq. (20.1) is gauge invariant, but the wave equations for thedifferent components of the potential are coupled. With the aim of establishing a covarianttheory for evanescent fields, the Lorenz condition

∂νAν(x) = 0 (20.2)

is used as a subsidiary condition. In all Lorenz gauges the inhomogeneous wave equationsfor the components of Aµ(x) decouple; that is,

Aµ(x) = −µ0Jµ(x). (20.3)

The complete solution to Eq. (20.3) can be written in the integral form [206, 101, 127]

Aµ(x) = Aµinc(x) + µ0

∫ ∞

−∞DR(x− x′)Jµ(x′)d4x′, (20.4)

and the physical interpretation of Eq. (20.4) is the following. An incident (inc) four-potentialAµ

inc(x), which one often may consider as a prescribed quantity, excites a system ofcharged particles. The electromagnetic field generated by the nonuniform particle motion to-gether with the incident field results in a selfconsistent current density distribution Jµ(x).The four-potential generated by the infinitesimal current element Jµ(x′)d4x′ spreads out infree space-time in a manner described by the retarded (R) scalar propagator [206, 101]

DR(X) =1

2πθ(

X0)

δ(

X2)

, (20.5)

where X = x−x′. The propagator in Eq. (20.5) is manifest covariant, and the step functionθ(X0) = θ(c(t− t′)) ensures that the field emitted from the source point r′ at time t′ reachthe point of observation, r at a later time t(> t′). The often used Huygens scalar propagatorg(X) is just DR(X) multiplied by the speed of light:

g(X) = cDR(X). (20.6)

It can be shown [127] that g(X) can be rewritten in the form

g(X) ≡ g(R, τ) =1

4πRδ

(

R

c− τ

)

, (20.7)

221


where R = |R| = |r − r′| and τ = t − t′. For our study of evanescent fields it is useful torewrite Eq. (20.4) in the standard form

Aµ(r, t) = Aµinc(r, t) + µ0

∫ ∞

−∞g(R, τ)Jµ(r′, t′)d3r′dt′, (20.8)

and then assume that the components of all vector fields, Fµ(r, t), have the generic form

Fµ(r, t) = Fµ(z;q‖, ω)ei(q‖·r−ωt). (20.9)

Physically, we thus imagine that our particle system possesses infinitesimal translationalinvariance in time, and in space in the direction given by the (real) vector q‖ = (q‖,x, q‖,y, 0).The form in Eq. (20.9) hence relates to a situation where the incident field is monochromatic(angular frequency: ω) and has plane-wave character perpendicular to the z-direction of theCartesian coordinate system. The ansatz in Eq. (20.9) allows one to reduce the integralrelation in Eq. (20.8) to one-dimensional form. Using the abbreviation

Fµ(z;q‖, ω) ≡ F(z) (20.10)

one thus obtains [129]

Aµ(z) = Aµinc(z) + µ0

∫ ∞

−∞g(Z)Jµ(z′)dz′, (20.11)

where Z = z− z′. The scalar Green function (propagator) appearing in Eq. (20.11) is givenby [155, 127]

g(Z) =i

2q0⊥eiq

0

⊥|Z|, (20.12)

where

q0⊥ =

[

(ω

c

)2

− q2‖

]1

2

, (20.13)

in a generalized sense, is the component of the vacuum field wave vector in the z-direction.When q‖ > q0 ≡ ω/c, q0⊥ becomes imaginary, i.e., q0⊥ = iκ0⊥, with

κ0⊥ =

[

q2‖ −(ω

c

)2]

1

2

(> 0). (20.14)

The associated form of the scalar propagator, viz.,

g(Z) =1

2κ0⊥e−κ0

⊥|Z|, q‖ > q0, (20.15)

shows that the four-potential generated by the sheet current Jµ(z′)dz′ (located at z′)decays exponentially away from the sheet plane with a spatial decay constant κ0⊥. Vacuumfields of the type

∼ e−κ0

⊥|Z| exp[

i(

q‖ ·R− ωt)]

(20.16)

are called evanescent, or inhomogeneous, and it is the wave mechanics and the field-quantized description related to these unusual vacuum fields which are in focus in this

Evanescent photon fields 223

chapter. Evanescent fields can be generated in a number of ways, but since the generationprocess is unimportant in the present context, it is sufficient for the following analysis toconsider the four-potential created by the prevailing current density distribution. Hence, westart from the integral relation

Aµ(z) =µ0

2κ0⊥

∫ ∞

−∞e−κ0

⊥|z−z′|Jµ(z′)dz′, (20.17)

divided in a given frame into its transverse (T)

AT (z) =µ0

2κ0⊥

∫ ∞

−∞e−κ0

⊥|z−z′|JT (z′)dz′, (20.18)

longitudinal (L)

AL(z) =µ0

2κ0⊥

∫ ∞

−∞e−κ0

⊥|z−z′|JL(z′)dz′, (20.19)

and scalar (S)

AS(z) ≡ A0(z) =µ0

2κ0⊥

∫ ∞

−∞e−κ0

⊥|z−z′|J0(z′)dz′ (20.20)

parts. In Eq. (20.20) A0 and J0/c ≡ JS denote the contravariant scalar potential and chargedensity, respectively.

20.2 Sheet current density: T-, L-, and S-parts

Important qualitative insight in the structure of the four-potential parts given by Eqs.(20.18)-(20.20) can be obtained assuming that the current-density distribution, J(z), isnonvanishing only in a thin sheet, centered for simplicity on the plane z = 0. As we shallrealize in Chapt. 21, the sheet model plays an important role for our understanding of thephoton tunneling process which is associated to frustrated total reflection.

To introduce the current-density sheet concept we begin with a Fourier-integral decom-position of the current density, i.e.,

J(z) = (2π)−1∫ ∞

−∞J (q⊥) e

iq⊥zdq⊥. (20.21)

Next, we make a Taylor series expansion of J(q⊥) around q⊥ = 0,

J (q⊥) = J(0) + J′(0)q⊥ + · · · . (20.22)

By combining Eqs. (20.21) and (20.22) we get an expansion of the current density in deriva-tives of the Dirac delta function, namely,

J(z) = J(0)δ(z)− iJ′(0)d

dzδ(z) + · · · . (20.23)

The result in Eq. (20.23) follows immediately from the fact that the nth order derivative ofthe delta function, with plane-wave expansion

δ(z) = (2π)−1∫ ∞

−∞eiq⊥zdq⊥, (20.24)


is given by the integral formula

dn

dznδ(z) =

in

2π

∫ ∞

−∞qn⊥e

iq⊥zdq⊥. (20.25)

In the simplest sheet-current density approximation only the lowest order term in the ex-pansion in Eq. (20.23) is kept. Hence, with J(q⊥ = 0) ≡ I = (Ix, Iy, Iz), we take

J(z) = Iδ(z) (20.26)

in the subsequent analysis. Under what circumstances do we expect the sheet approxima-tion in Eq. (20.26) to be valid? Remembering that the current density originates in theexcitation by the selfconsistent electromagnetic field acting on the particles, one would ina qualitative sense demand that the field variation across the z-region where the currentdensity is nonvanishing is negligible. A rigorous study of the sheet approximation along thisline of reasoning may be found in [129, 110, 111]. In the context of Eq. (20.17), the approx-imation in Eq. (20.26) may be applied if the current density distribution has an extensionin the z-direction much smaller than the decay length (1/κ0⊥) of the evanescent field. Fromthe three-vector in Eq. (20.26) one can determine the last component (J0) of the sheetfour-current density using the equation of continuity [Eq. (5.5)]. With no loss of generalitywe orient the Cartesian coordinate system so that its x-axis points along the q‖-direction.Thus,

q‖ = q‖ex (20.27)

in what follows. Charge conservation hence leads to

J0(z) =1

iq0

(

iq‖Ixδ(z) + Izd

dzδ(z)

)

. (20.28)

Note that J0(z) contains a term proportional to the first derivative of the delta function,unless the direction of the sheet current density is confined to the plane z = 0, that is Iz = 0.In cases where the incident field has a component in the z-direction it is usually important toinclude a nonvanishing Iz in the calculation. To determine I = (Ix, Iy, Iz) it is necessary tocombine the microscopic Maxwell–Lorentz equations with a suitable constitutive equationrelating the (correct) microscopic current density J(z) to the selfconsistent electromagneticfield inside the charge distribution [111], and then make the sheet approximation in thegeneral result [110].

Let us turn the attention now toward a discussion of the transverse and longitudinalparts of the entire sheet current density

J(x, z) = Iδ(z)eiq‖x. (20.29)

A calculation not to be repeated here, leads to the general result [116]

JI(z) =

∫ ∞

−∞δI(z − z′) · J(z′)dz′, I = T, L, (20.30)

where the sheet-contracted longitudinal (δL) and transverse (δT ) dyadic delta functions aregiven by

δL(Z) = U− δT (Z)= ezezδ(Z) +

q‖2e−q‖|Z| (ex + iezsgn(Z)) (ex + iezsgn(Z)) , (20.31)


where sgn(Z) = +1for Z > 0 and −1 for Z < 0. The presence of the term containing thefactor exp(−q‖|Z|) implies that JT and JL are connected in a spatially nonlocal manner toJ. By combining Eqs. (20.26), (20.30), and (20.31) one obtains

JT (z) = (U− ezez) · Iδ(z)−q‖2e−q‖|z| (ex + iezsgn(z)) (ex + iezsgn(z)) · I, (20.32)

JL(z) = ezez · Iδ(z) +q‖2e−q‖|z| (ex + iezsgn(z)) (ex + iezsgn(z)) · I. (20.33)

It appears from Eqs. (20.32) and (20.33) that even though the current density itself is

>d-q d q z0

1 1

FIGURE 20.1Two ED current density sheets located in the planes z = 0 and z = d. The source domainsof scalar (S) photons emitted from the sheets are confined to the sheet planes. The sourcedomains of longitudinal (L) and transverse (T) photons are exponentially confined witha characteristic spatial decay constant q‖, provided the sheets are excited by an incidentfield having simple plane-wave form along the sheet planes [∼ exp(iq‖z)]. Qualitatively, thesource domains of the T (L)-photons overlap in space when the distance between the sheetsis smaller than the reciprocal decay constant (d < q−1

‖ ).

confined to the plane z = 0, its transverse and longitudinal parts are nonvanishing outsidethis plane. The T- and L-parts decay exponentially away from the z = 0 plane, with adecay constant q‖. In a broader context this implies that the source domain of transverseand longitudinal photons may be considered as spread over a finite range of extension∼ exp(−q‖|z|) in the z-direction. The source domain of the scalar photons is confined tothe sheet plane (z=0), cf. Eq. (20.28). As illustrated schematically in Fig. 20.1, the sourcedomains of T(or L)-photons from two sheets overlap appreciably in space if the sheets arewithin a distance smaller than ∼ q−1

‖ from each other. As we shall understand in Chapt. 21,

this offers us a new perspective in the understanding of the physics behind photon tunneling[118].

20.3 Evanescent T-, L-, and S-potentials

The various parts of the potential associated with the sheet current density in Eq. (20.26)may now be calculated from Eqs. (20.18)-(20.20) inserting Eqs. (20.28), (20.32), and (20.33)


in the relevant places. Let us consider first the determination of the transverse vector poten-tial. It appears by inspection of Eqs. (20.18) and (20.32) that the integral along the z′-axismust be divided into three different pieces depending on whether z > 0 or z < 0:

∫ ∞

−∞(· · · )dz′ =

∫ z

−∞(· · · )dz′ +

∫ 0

z

(· · · )dz′ +∫ ∞

0

(· · · )dz′, z < 0, (20.34)

and

∫ ∞

−∞(· · · )dz′ =

∫ 0

−∞(· · · )dz′ +

∫ z

0

(· · · )dz′ +∫ ∞

z

(· · · )dz′, z > 0. (20.35)

Once these divisions are made, only integrals of simple exponential functions need to becalculated. We leave it to the reader to verify that the final result for the gauge invarianttransverse part of the vector potential is

AT (z) = A(1)T (z) +A

(2)T (z), (20.36)

where

A(1)T (z) =

µ0

2κ0⊥(U− ezez) · Ie−κ0

⊥|z|, (20.37)

and

A(2)T (z) =

q‖2ǫ0κ0⊥ω

2

κ0⊥ (ex + iezsgn(z)) (ex + iezsgn(z)) e−q‖|z|

−[

q‖ (exex − ezez) + iκ0⊥ (exez + ezex) sgn(z)]

e−κ0

⊥|z|

· I. (20.38)

With a knowledge of I one thus can determine AT (z). Let us take a brief look at the

structure of AT (z). The part A(1)T (z) stems from the term in JT (z) containing the delta

function; see Eq. (20.32). In a sense one may say that A(1)T (z) has it source in the sheet

plane (z = 0). The term only depends on component, Ixex + Iyey, of the sheet current inthe plane of the sheet. If I is perpendicular to the plane spanned by the unit vectors q‖/q‖(here ex) and ez only A

(1)T contributes to AT (z). The spatial decay of A

(1)T (z) away from

z = 0 is exponential with a decay constant κ0⊥. For q‖ → ω/c [from above], κ0⊥ → 0, so

that the decay length becomes infinite, as expected. The term A(2)T contains a superposition

of two exponential decays exp(−q‖|z|) and exp(−κ0⊥|z|). If we denote the parts of AT (z)and JT (z) which contain the exponential factor exp(−q‖|z|) by AT (z|q‖) and JT (z|q‖) itappears that

1

iǫ0ωJT (z|q‖) = iωAT (z|q‖) ≡ ET (z|q‖). (20.39)

Hence, the part of AT (z) which decays according to exp(−q‖|z|) can be characterized as atransverse self-field term [127, 129]. Note also that the y-component of I does not contribute

to A(2)T (z). Since κ0⊥ < q‖, the term proportional to exp(−κ0⊥|z|) gives the dominating con-

tribution to A(2)T (z) far from the sheet, i.e., qualitatively in the region where the transverse

current density essentially is zero. In the quasi-static (electrostatic) limit, it is for c → ∞and thus κ0⊥ → q‖, one obtains

AT (z|c→ ∞) = 0. (20.40)


With the usual replacement µ0 = (ǫ0c2)−1 it readily follows that A

(1)T (z|c → ∞) = 0 and

the reader may show by a simple calculation that also A(2)T (z|c → ∞) = 0. The result in

Eq. (20.40) is in agreement with the fact that a transverse vector potential vanishes in theelectrostatic limit in general.

To determine the longitudinal part of the vector potential let us add Eqs. (20.18) and(20.19). In view of Eq. (20.30), and the first member of Eq. (20.31), one obtains

AT (z) +AL(z) =µ0

2κ0⊥

∫ ∞

−∞e−κ0

⊥|z−z′|J(z′)dz′

=µ0

2κ0⊥e−κ0

⊥|z|I (20.41)

for the sheet current density in Eq. (20.26). Using the division for AT (z) given in Eq.

(20.36), and the explicit formula for A(1)T (z) [Eq. (20.37)], one gets

AL(z) =µ0

2κ0⊥e−κ0

⊥|z|ezez · I−A(2)T (z). (20.42)

Inserting the expression given in Eq. (20.38) for A(2)T (z), straightforward calculations lead

to the following gauge dependent result for the longitudinal vector potential:

AL(z) =1

2ǫ0κ0⊥ω2

[

(

q‖ex + iκ0⊥ezsgn(z)) (

q‖ex + iκ0⊥ezsgn(z))

e−κ0

⊥|z|

−q‖κ0⊥ (ex + iezsgn(z)) (ex + iezsgn(z)) e−q‖|z|

]

· I. (20.43)

The final formula for AL(z) contains parts decaying as exp(−q‖|z|) and exp(−κ0⊥|z|). Thepart with the exp(−q‖|z|)-factor, which we denote by AL(z|q‖), is characterized as a longi-tudinal self-field term [127] for obvious reasons. Thus, by comparing the self-field terms inthe transverse and longitudinal vector potentials, one sees that

AL(z|q‖) = −AT (z|q‖). (20.44)

Beforehand, one would have expected this result since JL(z) = −JT (z) outside the sheet.Inspection of the term proportional to exp(−κ0⊥|z|) in Eq. (20.43) [denoted by AL(z|κ0⊥)]reveals that the “direction” of AL(z|κ0⊥) is that of the inhomogeneous complex wave vector

q = q‖ex + iκ0⊥ezsgn(z). (20.45)

This is in agreement with the fact that AL(z) is a longitudinal (rotational-free) vector field.Returning for a moment to the sum of the parts in the formula for AT (z) which contain thefactor exp(−κ0⊥|z|), this sum, denoted by AT (z|κ0⊥), necessarily must display the transverse(divergence-free) nature of AT (z). In fact, one can rewrite AT (z|κ0⊥) in the form

AT (z|κ0⊥) =e−κ0

⊥|z|

2ǫ0κ0⊥ω2

[

q20eyey + (ey × q) (ey × q)]

· I. (20.46)

The AT (z|κ0⊥)-part of the transverse vector potential hence is seen to contain two mutuallyorthogonal terms [∼ ey and ∼ ey × q] polarized orthogonal to the complex wave vector q.The structural similarity to AL(z|κ0⊥) becomes manifest by rewriting this in the form

AL(z|κ0⊥) =e−κ0

⊥|z|

2ǫ0κ0⊥ω2qq · I. (20.47)


If one adds the transverse and longitudinal vector potentials the self-field terms cancel,and we are left with the expected division into a sum of two transverse modes and onelongitudinal mode, namely,

A(z) = AT (z) +AL(z)

=e−κ0

⊥|z|

2ǫ0κ0⊥ω2

[

q20eyey + (ey × q) (ey × q) + qq]

· I. (20.48)

Although the sum of the self-field terms vanishes, these play a significant role for under-standing the physics related to photon localization and near-field electrodynamics, as weshall realize later on. It appears from Eq. (20.45) that the complex wave vector q has op-positely directed z-components in the two half-spaces (z ≷ 0). This must necessarily be sobecause the x-component q‖ex is forced to be the same in both half-spaces by the ansatzin Eq. (20.9).

The scalar potential is determined by inserting the expression for J0(z), given in Eq.(20.28), into Eq. (20.20):

AS(z) =µ0

2iκ0⊥q0

∫ ∞

−∞e−κ0

⊥|z−z′|(

iq‖Ixδ(z′) + Iz

d

dz′δ(z′)

)

dz′. (20.49)

Upon a partial integration of the second part of the integral the reader may verify to herselfthat the gauge dependent scalar potential is given by

AS(z) =e−κ0

⊥|z|

2ǫ0κ0⊥cωq · I. (20.50)

Let us finally consider the longitudinal part of the electric field, EL(z). From the generalconnection in Eq. (10.27), one obtains

EL(z) = iωAL(z)− icqAS(z). (20.51)

A glance at the expressions in Eqs. (20.47) and (20.50) shows that

ωAL(z|κ0⊥) = cqAS(z), (20.52)

so that

EL(z) = iωAL(z|q‖). (20.53)

This result is not surprising since only the longitudinal part of the electric field survivesin the quasi-static limit, i.e., E(z|c → ∞) = EL(z) [127, 129]. The relation in Eq. (20.52)ensures that terms containing the factor exp(−κ0⊥|z|) in AL(z) and AS(z) cancel. Thetransverse part of the electric field, ET (z) = iωAT (z), clearly vanishes in the quasi-staticlimit because of the result obtained in Eq. (20.40). A schematic illustration relating to thespatial decays of the various parts of the four-potential and the electric field is shown inFig. 20.2.

Let us finally note that the results obtained for the T-, L-, and S-parts of the vectorpotential can be transferred to results for propagating modes for which q‖ < ω/c. One justneeds to make the replacement κ0⊥ → −iq0⊥ in the relevant equations, cf. Eqs. (20.16) and(20.45).


a

q||

κ⊥0

TA, TE TA + LA

LA LE

SA TE + LE

b

c

d

e

f

FIGURE 20.2Schematic illustration of various (normalized) decay profiles relating to the evanescentelectric field (four-potential) generated by an ED sheet current density J(x, z;ω) =Iδ(z) exp[i(q‖x − ωt)]. The transverse parts of the vector potential and the electric field(TA, TE) both consist of a superposition of profiles with decay constants q‖ and κ0⊥ =

[q2‖ − (ω/c)2]1/2; see Fig. a. The longitudinal part of the vector potential (LA) contains the

same profiles (Fig. b). The grey toned regions indicate the extension of the source domainsof the T- and L-fields. The scalar potential (SA) has only one decay constant, κ0⊥ (Fig. c).The profiles of the total vector potential (TA + LA) and the total electric field (TE + LE)both are characterized solely by κ0⊥ (Figs. d and f), and the longitudinal part of the electricfield (LE) has the decay constant q‖ (Fig. e).

20.4 Four-potential photon wave mechanics

Starting from the expressions obtained for the z-dependencies of the various parts of thefour-potential in Sec. 20.3, Fourier-integral decomposition allows one to establish contactto the wave mechanical description of the transverse, longitudinal, and scalar photons inthe wave-vector representation (Chapts. 10 and 11). With the here convenient replacementq‖ → Q‖, the components of the four-potential Aµ(r, t) are given by the integral form

Aµ(r, t) =

[

(2π)−1∫ ∞

−∞Aµ(Q⊥)e

iQ⊥zdQ⊥

]

ei(Q‖x−ωt), (20.54)

where

Aµ(Q⊥) =∫ ∞

−∞Aµ(z)e−iQ⊥zdz (20.55)

is the Fourier transform of Aµ(z). With the relabelling Aµ(Q⊥) ≡ Aµ(Q), the potentialfour-vector

Aµ(Q; t) = Aµ(Q)e−iωt (20.56)

is the one which relates to the dynamical variables of the T-, L-, and S-photons in thewave-vector representation.

Let us consider first the evanescent [κ0⊥ = (Q2‖ − q20)

1/2] scalar potential. The Fourier


transform of Eq. (20.50), viz.,

AS(Q⊥) =1

2ǫ0κ0⊥cω

[∫ ∞

−∞

(

Q‖ex + iκ0⊥ezsgn(z))

e−iκ0

⊥|z|e−iQ⊥zdz

]

· I (20.57)

is easily obtained dividing the integral into z < 0 and z > 0 parts. Thus,

AS(Q⊥) =1

2ǫ0κ0⊥cω

[

Q‖ex − iκ0⊥ezκ0⊥ − iQ⊥

+Q‖ex + iκ0⊥ezκ0⊥ + iQ⊥

]

· I (20.58)

in the first step. Since

(

κ0⊥)2

+Q2⊥ = Q2 − q20 , (20.59)

where Q2 = Q2‖+Q

2⊥, the expression for AS(Q⊥) ≡ AS(Q) can be contracted to the simple

form

AS(Q) =µ0

q0

Q · IQ2 − q20

, (20.60)

where

Q = Q‖ex +Q⊥ez (20.61)

is the (real) wave vector of the monochromatic (ω) mode with fixed Q‖ > q0. For a latercomparison to results for the T- and L-potentials it is useful to write Eq. (20.60) as follows:

AS(Q) =µ0

Q2 − q20

Q

q0eQ · I, (20.62)

where eQ = Q/Q is a unit vector in the direction of Q. We expect that the right sideof Eq. (20.62) can be expressed in terms of the scalar part of the four-current density,J0(Q⊥) ≡ J0(Q), given, in view of Eq. (20.28), by

J0(Q⊥) =1

iq0

∫ ∞

−∞

[

iQ‖Ixδ(z) + Izd

dzδ(z)

]

e−iQ⊥zdz. (20.63)

Upon a partial integration of the term with Iz , one immediately obtains

JS(Q) ≡ J0(Q) =Q · Iq0

, (20.64)

and hereafter

AS(Q) =µ0

Q2 − q20JS(Q). (20.65)

A knowledge of the scalar potential [Eq. (20.62)] readily allows one to determine thelongitudinal potential, AL(Q) = AL(Q)eQ, via the Lorenz gauge condition [Eq. (11.13)],which in the monochromatic case amounts to

QAL(Q) = q0AS(Q). (20.66)

By combining Eqs, (20.62) and (20.66) one obtains

AL(Q) =µ0

Q2 − q20eQeQ · I. (20.67)


To determine the transverse potential, AT (Q) = AT (Q⊥), we just take the Fouriertransform of Eq. (20.41). Hence,

AT (Q⊥) +AL(Q⊥) =µ0

2κ0⊥I

∫ ∞

−∞e−κ0

⊥|z|e−iQ⊥zdz =µ0

Q2 − q20I. (20.68)

Inserting Eq. (20.67) in the equation above, we obtain

AT (Q) =µ0

Q2 − q20(U− eQeQ) · I. (20.69)

The longitudinal [Eq. (20.67)] and transversal [Eq. (20.69)] parts of the vector potentialhence equal the projections of µ0I/(Q

2 − q20) parallel and perpendicular to the Q-direction,respectively.

From the expressions in Eqs. (20.65) and (20.67) the reader may show that the time-dependent quantities AS(Q; t) = AS(Q) exp(−iωt) and AL(Q; t) = eQ ·AL(Q) exp(−iωt)satisfy the first-order differential equations

(

cQ− i∂

∂t

)

AS(Q; t) =1

ǫ0c

Q

q0

eQ · IQ+ q0

e−iωt, (20.70)

(

cQ− i∂

∂t

)

AL(Q; t) =1

ǫ0c

eQ · IQ+ q0

e−iωt. (20.71)

We extend our perspective by considering Eqs. (20.70) and (20.71) as dynamical equationsfor the scalar [AS(Q; t)] and longitudinal [AL(Q; t)] photon variables associated with themonochromatic sheet four-current density

Jµ(Q; t) =

(

Q

q0eQ · I, I

)

e−iωt. (20.72)

The expression given for AS(Q; t) and AL(Q; t) in Eqs. (20.62) and (20.67) are the particularsolutions to Eqs. (20.70) and (20.71) associated to the S- and L-potentials generated by theparticle dynamics in the sheet. For I = 0, Eqs. (20.70) and (20.71) [upon multiplicationby ~, possibly] become identical to the (unnormalized) wave equations for the scalar andlongitudinal photons. The solutions to the homogeneous parts of Eqs. (20.70) and (20.71)are monochromatic with angular frequency Ω = cQ(> 0). The transverse vector potential,AT (Q; t) = AT (Q) exp(−iωt), generated by the sheet current density, and given in Eq.(20.69), satisfies the dynamical equation

(

cQ− i∂

∂t

)

AT (Q; t) =1

ǫ0c

(U− eQeQ) · IQ+ q0

e−iωt, (20.73)

as the reader may prove to herself. If wished, one may divide Eq. (20.73) into dynamicalequations for each of the two helicity species, cf. the analysis in Sec. 10.2.

20.5 Field-quantized approach

Let us now discuss the extension of the four-photon wave mechanical description (Sec. 20.4)to the field-quantized level, starting with the L and S-photons. It appears from Eq. (20.72)that the scalar part of the longitudinal sheet four-current density, JL(Q), is given by

JL(Q) ≡ eQ · JL(Q) = eQ · I, (20.74)


and this shows that the wave mechanical equations in (20.70) and (20.71) have the samestructure, viz.,

(

cQ− i∂

∂t

)

AI(Q; t) =1

ǫ0c

JI(Q)

Q+ q0e−iωt, I = L, S. (20.75)

Analogous to what was done in connection to the canonical quantization of the transversevector potential in Sec. 15.1 scaled potentials αI(Q; t) are introduced via the definitions

AI(Q; t) =

(

~

2ǫ0cQ

)1

2

αI(Q; t), I = L, S, (20.76)

cf. Eq. (15.7). The scaled potentials satisfy the dynamical equations

∂

∂tαI(Q; t) + iΩQαI(Q; t) =

i

(2ǫ0~ΩQ)1

2

2Q

Q + q0JI(Q)e−iωt, I = L, S, (20.77)

where

ΩQ = cQ (20.78)

is the mode frequency.With the replacements

αI(Q; t) ⇒ aI(Q; t), (20.79)

and

JI(Q) ⇒ JI(Q), (20.80)

the dynamical equations in (20.77) become the Heisenberg equations of motion for theannihilation operator belonging to the longitudinal (aL(Q; t)) and scalar (aS(Q; t)) fieldoperators [127, 53]:

∂

∂taI(Q; t) + iΩQaI(Q; t) =

i

(2ǫ0~ΩQ)1

2

2Q

Q+ q0JI(Q)e−iωt, I = L, S. (20.81)

Note that JL(Q) and JS(Q) are operators in particle space, relating to the amplitudes ofthe longitudinal and scalar parts of the sheet current density.

Two linearly independent polarization states exist for a given transverse electromagneticplane-wave mode, and if one chooses these as the positive and negative-helicity states, Eq.(20.73) can be split into two scalar equations, which afterward may be extended to thequantum electrodynamic level in a manner analogous to the one employed for the L andS-modes. Denoting as usual the positive and negative-helicity unit vectors by e+ and e−,respectively, we insert the relation

U− eQeQ = e∗+(κ)e+(κ) + e∗−(κ)e−(κ), (20.82)

where κ = Q/Q, into Eq. (20.73). From the resulting equation,

(

cQ− i∂

∂t

)

AT (Q; t) =1

ǫ0c

[

e∗+(κ)e+(κ) + e∗−(κ)e−(κ)]

· IQ+ q0

e−iωt, (20.83)


we obtain dynamical equations for the scalar quantities

AT,s(Q; t) ≡ es(κ) ·AT (Q; t), s = +,−, (20.84)

by utilizing Eqs. (10.52) and (10.53). Thus,(

cQ− i∂

∂t

)

AT,s(Q; t) =1

ǫ0c

Js(Q)

Q+ q0e−iωt, s = +,−, (20.85)

where

Js(Q) = es(κ) · I, s = +,−, (20.86)

are the sheet current densities belonging to the two helicity eigenstates. As the reader mighthave expected, the structures of Eqs. (20.85) are the same as that for the L and S-modes[Eqs. (20.75)]. From here on the extension to the second-quantized level follows the sameline of reasoning as for the L- and S-potentials. Hence, the scaled potentials αs(Q; t), givenvia

AT,s(Q; t) =

(

~

2ǫ0cQ

)1

2

αT,s(Q; t), s = +,−, (20.87)

satisfy the dynamical equations

∂

∂tαT,s(Q; t) + iΩQαT,s(Q; t) =

i

(2ǫ0~ΩQ)1

2

2Q

Q+ q0JT,s(Q)e−iωt, s = +,−, (20.88)

and the replacements

αT,s(Q; t) ⇒ aT,s(Q; t), (20.89)

JT,s(Q) ⇒ JT,s(Q), (20.90)

give us the Heisenberg equations of motion for the annihilation operators aT,+(Q; t) andaT,−(Q; t) belonging to the two helicities, viz.,

∂

∂taT,s(Q; t) + iΩQaT,s(Q; t) =

i

(2ǫ0~ΩQ)1

2

2Q

Q+ q0JT,s(Q)e−iωt, s = +,−, (20.91)

JT,s(Q) being an operator in particle space.The results obtained in Eqs. (20.81) and (20.91) for monochromatic evanescent fields

are special cases of the following most general Heisenberg equations of motion for thefour [I = T+, T−, L, S] annihilation operators [127, 53]. One may write these equations incompact form using the covariant notation in Sec. 20.1. Hence

∂

∂taµ(Q; t)+ iΩQ aµ(Q; t) =

i

(2ǫ0~ΩQ)1

2

Jµ(Q; t)

, (20.92)

where

Jµ(Q; t)

=

∫ ∞

−∞

Jµ(r, t)

e−iQ·rd3r (20.93)

is the spatial Fourier transform of the four-current density operator in direct space,Jµ(r, t). If the standard 3D-Fourier integral representation in Eq. (20.93) is replacedby the angular spectrum representation [155, 127], one obtains for the individual modes ofthe inhomogeneous (evanescent) part of the spatial spectrum precisely the results in Eqs.(20.81) and (20.91). In a sense this is clear from our earlier considerations. Returning thefirst-quantized point of view the sheet four-current density is given by Eq. (20.72). Thisequation, in combination with the dynamical equations in (20.70), (20.71), and (20.73),leads to the result mentioned above.


20.6 Near-field photon: Heisenberg equation of motion and coher-ent state

We finish our journey into the physics of evanescent fields with a brief look at the dynamics ofthe second-quantized near-field (NF) photon. The annihilation operator for the NF-photonbelonging to the mode Q is at time t given by

aNF (Q; t) =i√2(aL(Q; t)− aS(Q; t)) , (20.94)

i.e., and extension of Eq. (11.31) to the field-quantized level. From the dynamical equationsfor the longitudinal and scalar photon annihilation operators [Eq. (20.81)] we obtain theHeisenberg equation of motion

∂

∂taNF (Q; t) + iΩQaNF (Q; t) =

1

(ǫ0~ΩQ)1

2

Q

Q+ q0

(

JS(Q)− JL(Q))

e−iωt. (20.95)

From Eq. (20.72) the difference between JS(Q) and JL(Q) can be related to eQ · I, and Eq.(20.95) thereafter simplified to

∂

∂taNF (Q; t) + iΩQaNF (Q; t) =

1

(ǫ0~ΩQ)1

2

Q− q0Q+ q0

Q

q0eQ · Ie−iωt. (20.96)

With the abbreviation

β(Q) ≡ 1

(ǫ0~ΩQ)1

2

Q− q0Q+ q0

Q

q0eQ · I (20.97)

the general solution to Eq. (20.96) takes the form

aNF (Q; t) = aNF (Q; 0)e−iΩQt +i

ωβ(Q)

(

e−iωt − 1)

. (20.98)

The first term on the right side of Eq. (20.98) describes the free evolution of the annihilationoperator

afreeNF (Q; t) = aNF (Q; 0)e−iΩQt, (20.99)

but, since we know from Eq. (11.40) [extended to the second-quantized level] that afreeNF (Q; t)also must satisfy the time evolution

afreeNF (Q; t) = aNF (Q; 0)eiΩQt, (20.100)

we must conclude that

aNF (Q; 0) = 0, (20.101)

in agreement with the conclusion in Eq. (11.41): There are no states containing near-fieldphotons in free space. For this reason the general solution to the Heisenberg equation ofmotion for the NF-photon annihilation operator must be

aNF (Q; t) =i

ωβ(Q)

(

e−iωt − 1)

. (20.102)


Although this operator equation at first sight looks simple, one must not forget that theamplitude of the sheet current density operator I in general depends on the NF-photon fieldoperators for the various Q’s. The solution to Eq. (20.102) hence is implicit and in generalcomplicated to calculate.

There is a simple case, however. Thus, if the sheet current density source is classical,so that I, and consequently β(Q), are c-numbers, β(Q) = β(Q), the quantum state of thequantized near-field is a coherent state [127], here denoted |ψcoh〉. In the Heisenberg Picturethe quantum states are time independent. The identification as a coherent state follows fromthe definition that |ψcoh〉 is an eigenstate for the annihilation operator, that is

aNF (Q; t)|ψcoh〉 = αNF (Q; t)|ψcoh〉. (20.103)

The eigenvalue αNF (Q; t) has the time evolution

αNF (Q; t) =i

ωβ(Q)

(

e−iωt − 1)

. (20.104)

The coherent near-field photon state cannot be a single-photon state. Elementary consider-ations [155, 53, 77] show that the probability, p(n), for having n near-field photons in thestate |ψcoh〉 is given by the Poisson distribution

p(n) =|αNF (Q; t)|2n

n!exp

(

−|αNF (Q; t)|2)

(20.105)

with parameter

|αNF (Q; t)|2 =2|β(Q)|2

ω2(1− cosωt). (20.106)

The mean number of NF-photons is given by

∞∑

n=0

np(n) = |αNF (Q; t)|2 = 〈ψcoh|a†NF (Q; t)aNF (Q; t)|ψcoh〉, (20.107)

a quantity which oscillates as sin2(ωt/2).

21

Photon tunneling

21.1 Near-field interaction. The photon measurement problem

A paradigm of optical (photon) tunneling appears in relation to the physics of frustratedtotal internal reflection (FTIR); see e.g., [123, 118, 50, 224, 51]. In the FTIR process evanes-cent fields play an indispensable role, and the fact that the source domains of transverseand longitudinal photons generated by a current density sheet (located at z = 0) extendoutside the sheet, with a current density profile given by exp(−q‖|z|), indicates that thephysics in the rim zone is crucial for an understanding of the photon tunneling process.We shall reach the FTIR tunneling process starting from a broader framework. Thus, itturns out that optical near-field interaction, spatial photon localization, and photon tun-neling have the same roots. The considerations to follow also lead to connections to thephoton measurement problem which we shall address in Part VIII. In a broader context theproblem brings us in contact with the so-called measurement theory in quantum physics,which in itself raises deep questions about the physical/philosophical structure of humanobservations [168, 157, 247].

Let us now return to the integral relation in Eq. (20.8), and here assume that the four-current density Jµ(r, t) can be divided into two parts (A and B) which do not overlapelectronically in space at any time. Quantum mechanically, the overlap of the probabilitycurrent densities of A and B must be insignificant. The difference between the total (prevail-ing) four-potential Aµ(r, t) ≡ Aµ

tot(r, t) and the incident potential Aµinc(r, t) defines

the scattered potential, i.e.,

Aµscatt(r, t) ≡ Aµ

tot(r, t) − Aµinc(r, t). (21.1)

Formally, the scattered four-potential is the sum of the potentials scattered from the currentdensity distributions Jµ

A(r, t) and JµB(r, t), namely,

Aµscatt(r, t) =µ0

∫

V µAg(R, τ)Jµ

A(r′, t′)d3r′dt′

+ µ0

∫

V µBg(R, τ)Jµ

B(r′, t′)d3r′dt′, (21.2)

where the integrals are over the non-overlapping four-volumes V µA and V µ

B of the self-consistently determined current density distributions belonging to A and B. To relate thescattering problem in Eqs. (21.1) and (21.2) to photon wave mechanics (in a given inertialframe) one splits the four-potential and four-current densities into their T-, L-, and S-parts.For the transverse part the vectorial scattering problem takes the form

AT (r, t) =AincT (r, t)

+ µ0

∫

V AT

g(R, τ)JAT (r

′, t′)d3r′dt′ + µ0

∫

V BT

g(R, τ)JBT (r

′, t′)d3r′dt′, (21.3)

237


using a slightly different (but obvious) notation than in Eqs. (21.1) and (21.2). Note that theintegrations in Eq. (21.3) are over the transverse space-time volumes V A

T and V BT , i.e., the

volumes occupied by the transverse current density distributions JAT (r

′, t′) and JBT (r

′, t′).One may look at the scattering problem in a slightly different way: An incident field,

AincT , excites (polarizes) an object (system) A. The induced polarization gives rise to an

emitted (a scattered) field which, together with the incident field, polarize the object (sys-tem) B. In turn, object B emits a field. The relation between the incident field, Ainc

T , andthe selfconsistently determined scattered field, Ascatt

T = AT − AincT , carries information

about the mutual interaction between the objects A and B.If one considers A as the object (sample) under study, and B as a detector, then the

scattering problem links up with the photon measurement theory; see Part VIII. When thetwo transverse space-time domains overlap, i.e.,

V AT ∩ V B

T 6= ⊘, (21.4)

the scattering process defines the canonical near-field optics problem: The scattered field,Ascatt

T , is a measure of the induced (by AincT ) electrodynamic near-field interaction between

sample (A) and probe (B). Objects which satisfy the dynamic criterion in (21.4) but do notoverlap electronically, that is

V A ∩ V B = ⊘, (21.5)

are said to be in rim-zone contact. It has been argued that the optical tunneling betweentwo (mesoscopic) objects has its roots in the near-field interaction between the objects [121].Experimentally, it is true certainly that all observations up to now have been carried outon objects (systems) in rim-zone contact [51]. With main emphasis on optical (photon)tunneling in the FTIR-geometry, we shall in the following sections step by step develop ourphysical understanding of the tunneling process.

21.2 Scattering of a wave-packet band from a single current-density sheet

In our study of evanescent fields in Chapt. 20 it was assumed that all the vector fields weremonochromatic and had plane-wave character parallel to the sheet plane (correspondingto a driving current-density distribution with translational invariance in two directions);see Eq. (20.9). In order to understand the optical tunneling process we can no longer workwith monochromatic fields. Substantial insight in the physics can however still be obtainedeven if one assumes that the fields have simple plane-wave form, exp(iq‖ · r), parallel tothe sheet plane. Although (at least) two sheets are needed for a quantitative discussionof the tunneling process useful information is gained studying the scattering from a singlesheet (say A, in the notation of Sec. 21.1). The subsequent analysis thus is based on theassumption that all vector (and scalar) fields, Fµ(r, t), have the form

Fµ(r, t) = Fµ(z, t;q‖)eiq‖·r. (21.6)

Ultimately, the expression in Eq. (21.6) has its roots in a specific choice for the incidentfield, Aµ

inc(r, t). In Sec. 21.3 we shall see how the incident field may be designed so thatthe various vector fields obey Eq. (21.6) and the field scattered by the sheet is evanescent.

Photon tunneling 239

We shall call the so designed incident field a wave-packet band (or strip). For notationalsimplicity we use the abbreviation

Fµ(z, t;q‖) ≡ Fµ(z, t) (21.7)

in the following. The ansatz in Eq. (21.6) allows us to reduce the basic integral relation inEq. (20.8) to the form

Aµscatt(z, t) ≡ Aµ(z, t)−Aµ

inc(z, t) = µ0

∫ ∞

−∞g(Z, τ)Jµ(z′, t′)dz′dt′, (21.8)

where Z = z − z′, and τ = t− t′.The dynamics of the scattering process is contained in the electromagnetic scalar propa-

gator, g(Z, τ), which describes how the field generated by the four-current density prevailingat time t′ in a sheet located at the z′-plane contributes to the scattered four-potential in thez-plane at (a later) time t. To obtain the explicit expression for g(Z, τ) it is convenient firstto transform the retarded Huygens scalar propagator, given in Eq. (20.7), to the wave-vectordomain. The Fourier integral

g(q, τ) =

∫ ∞

−∞(4πR)−1 δ

(

R

c− τ

)

e−iq·Rd3R (21.9)

is easily carried out in spherical coordinates with the polar axis directed along q. Theintegration over the azimuth angle gives 2π, and with the substitution µ = cos θ, θ beingthe polar angle, one obtains the intermediate result

g(q, τ) =1

2

∫ ∞

0

∫ 1

−1

Rδ

(

R

c− τ

)

e−iqRµdµdR

=1

q

∫ ∞

0

δ

(

R

c− τ

)

sin(qR)dR. (21.10)

The last integral is nonvanishing only for τ > 0, as it must be because our starting pointwas the retarded scalar propagator, and we finally have

g(q, τ) =c

qθ(τ) sin(qcτ). (21.11)

As indicated, g(q, τ) only depends on the magnitude of the wave vector. This is obviouslycorrect because the Huygens propagator is isotropic in R-space. With q = (q2‖ + q2⊥)

1/2, the

g(Z, τ)-propagator is given by the Fourier integral

g(Z, τ) =1

2π

∫ ∞

−∞g(q, τ)eiq⊥Zdq⊥

=cθ(τ)

2π

∫ ∞

−∞sin(

cτ√

q2‖ + q2⊥

) eiq⊥Z

√

q2‖ + q2⊥dq⊥

=cθ(τ)

π

∫ ∞

0

sin(

cτ√

q2‖ + q2⊥

) cos (q⊥|Z|)√

q2‖ + q2⊥dq⊥. (21.12)

The last integral can be expressed in terms of the zeroth-order Bessel function, J0. Thus


[118],

∫ ∞

0

sin(

cτ√

q2‖ + q2⊥

) cos (q⊥|Z|)√

q2‖ + q2⊥dq⊥

=

π2 J0

(

q‖√

(cτ)2 − Z2)

, 0 < |Z| < cτ,

0, |Z| > cτ,(21.13)

remembering that τ > 0. The result in Eq. (21.13) allows one to write the sought-forpropagator as follows:

g(Z, τ ; q‖) =c

2θ (cτ − |Z|)J0

(

q‖√

(cτ)2 − Z2)

, (21.14)

in a notation where the parameter q‖(> 0) has been reinserted in g(Z, τ) ≡ g(Z, τ ; q‖) for thesake of clarity. As expected, the expression for g(Z, τ) shows that the connection betweenthe contribution to the four-potential at (z, t) originating in the four-current density at(z′, t′) is time-like (cτ > |Z|). Near the light “cone,” i.e., for cτ → |Z|, g → c/2. Behind the“cone” [cτ > |Z|], g(Z, τ) oscillates and decays in space-time, (Z, τ), in a manner dictatedby the Bessel function J0(q‖

√

(cτ)2 − Z2).To make further progress toward an understanding of the photon tunneling process (in a

given inertial frame), we extract the T and L dynamics from Eq. (21.8). For the transversepart of the problem one obtains

AT (z, t) = AincT (z, t) + µ0

∫ ∞

−∞g(Z, τ)JT (z

′, t′)dz′dt′. (21.15)

If one assumes that the incident vector potential is purely transverse,Ainc(z, t) = AincT (z, t),

the longitudinal dynamics is governed by

AL(z, t) = µ0

∫ ∞

−∞g(Z, τ)JL(z

′, t′)dz′dt′. (21.16)

In optical tunneling experiments [51] the source of the incident field usually is so far awayfrom the tunneling barrier (region) that the incident field is a transverse vector field. In thephoton language the incident photon starts as a free particle, usually.

We may reduce Eqs. (21.15) and (21.16) further without losing the main points of photontunneling by assuming that the current density distribution is confined to a sheet, locatedat z = 0 (cf. the description in Sec. 20.2). The relevant three-current density thus is givenby

J(z, t) = I(t)δ(z), (21.17)

with T- and L-parts

JI(z, t) = δI(z) · I(t), I = T, L. (21.18)

Although the current density is confined to a sheet, the T and L-dynamics still contain adouble integration (over z′ and t′) because the transverse and longitudinal delta functionsare nonvanishing outside the plane z = 0, see Eq. (20.31). In the sheet case the basicequations take the forms

AT (z, t) = AincT (z, t) + µ0

∫ ∞

−∞g(Z, τ)δT (z

′) · I(t′)dz′dt′, (21.19)


and

AL(z, t) = µ0

∫ ∞

−∞g(Z, τ)δL(z

′) · I(t′)dz′dt′. (21.20)

In optical tunneling experiments so-called apparent superluminality occurs [123, 51, 121,48, 49, 42, 43]. In FTIR-studies it has been shown that a photon “crosses” a tunnel barrier(vacuum gap) of width d in a time t which is smaller than the time d/c it would take thephoton to propagate the same distance in free space, t < d/c [51]. If one incorrectly assumesthat the photon in the vacuum gap is free the observations above would lead one to theconclusion that the photon in a tunneling experiment crosses the barrier with superluminalspeed. In turn this would imply that the Einstein causality would be broken and quantumelectrodynamics be wrong, as it sometimes has been claimed; see [51] and references therein.On the basis of Eqs. (21.19) and (21.20) it is possible to conclude, however, that the apparentsuperluminality observed in FTIR-tunneling experiments can be explained without breakingEinstein causality. To reach this very satisfactory conclusion it must be remembered thatall tunneling experiments are optical near-field experiments. In turn this implies, as wehave realized in Chapt. 20 (and elsewhere), that the photon in the tunneling region is notfree, but in an embryo state, as I have called it [123, 127]. In the framework of the four-potential description both Eq. (21.19) and Eq. (21.20) must be kept in the analysis. Itappears from the explicit form of the scalar propagator g(Z, τ ; q‖) [Eq. (21.14)] that theEinstein causality is obeyed in light propagation between a given space-time source plane,(z′, t′), and the space-time plane of observation, (z, t). The door is still open for apparentsuperluminal propagation, however. To understand this, let us focus on Eq. (21.20). Byinserting the expression given for the longitudinal delta function in Eq. (20.31), AL(z, t)splits into two parts,

AL(z, t) = A(1)L (z, t) +A

(2)L (z, t). (21.21)

The first part, associated with δ(z′), is given by

A(1)L (z, t) = µ0ezez ·

[∫ ∞

−∞g(z, t− t′)I(t′)dt′

]

=µ0c

2ezez ·

[

∫ t− |z|c

−∞J0

(

q‖√

c2(t− t′)2 − z2)

I(t′)dt′]

. (21.22)

If the source domain of the longitudinal vector potential (and thus also that of the trans-verse vector potential) was confined to the sheet plane (z′ = 0), Eq. (21.22) would be thefinal result for AL(z, t), and no apparent superluminality would appear, in conflict withthe experimental observations. We know however that the source domain of the photonextends over a finite interval along the z′-axis [with a source current density proportional toexp(−q‖|z′)]. In a slightly different (but physically completely equivalent) picture, one maysay that a photon emitted with a given q‖ from the sheet located at z′ = 0 in the initialinstant of emission is exponentially confined in the z′-direction, the spatial decay constantbeing q‖. The Einstein causal contribution to AL(z, t), which originates in source planes inthe exponential tail, exp(−q‖|z′|), is given by

A(2)L (z, t) =

µ0q‖2

∫ ∞

−∞g(z − z′, t− t′)e−q‖|z′| (ex + iezsgn(z

′)) (ex + iezsgn(z′)) · I(t′)dz′dt′.

(21.23)

Let us consider the contribution to Eq. (21.23) from a source plane (z′) located in the same


halfspace as the plane of observation (z), i.e., z, z′ > 0 or z, z′ < 0. The explicit expressionfor g(Z, τ) [Eq. (21.14)] then shows that the field generated in the z′ source plane at t′,starts to arrive in the plane of observation (z) at the time

t = t′ +|z − z′|

c< t′ +

|z|c; (21.24)

see also Fig. 21.1. The inequality in (21.24) indicates that if one wrongly had assumed that

the source plane of the field was z′ = 0, superluminality would be associated with A(2)L (z),

and thus of course also with the correspondent term in the transverse part of the vectorpotential [Eq. (21.19)]. The lack of spatial photon localization hence is the root to apparentsuperluminality in optical tunneling processes. Although one may say that the theoreticaltunneling time is zero because the response time from the source plane z′ = z is zero, theoperational (experimentally determined) tunneling time is finite, with a value determinedby the signal level required to excite the detector. In a quantum statistical sense this relatesto the probability for generating the photon in a given distance (|z′|) from the sheet plane(z′ = 0).

The presence of apparent superluminality may be further illuminated by determining theexplicit expression for the longitudinal part of the electric field in the space-time domain.In the frequency domain EL(z;ω) is given by Eq. (20.53) where

AL(z;ω|q‖) = − q‖2ǫ0ω2

(ex + iezsgn(z)) (ex + iezsgn(z)) · I(ω)e−q‖|z|, (21.25)

cf. Eq. (20.43), and the remarks below this equation. In the time domain the longitudinalelectric field hence is proportional to the folding integral of the Fourier transforms I(t) [ofI(ω)] and

1

2π

∫ ∞

−∞ω−1e−iωτdω =

1

πi

∫ ∞

0

ω−1 sinωτdω

=1

2isgn(τ). (21.26)

Gathering the details one obtains

EL(z, t; q‖) =− q‖4ǫ0

e−q‖|z| (ex + iezsgn(z)) (ex + iezsgn(z)) ·[∫ ∞

−∞sgn(t− t′)I(t′)dt′

]

=− q‖2ǫ0

e−q‖|z| (ex + iezsgn(z)) (ex + iezsgn(z))

·[

1

2

(∫ t

−∞I(t′)dt′ −

∫ ∞

t

I(t′)dt′)]

. (21.27)

If one assumes that the longitudinal electric field vanishes in the remote past, i.e., fort→ −∞, one must have

∫ ∞

−∞I(t′)dt′ = 0, (21.28)

so that

EL(z, t; q‖) = − q‖2ǫ0

e−q‖|z| (ex + iezsgn(z)) (ex + iezsgn(z)) ·[∫ t

−∞I(t′)dt′

]

. (21.29)

In the evanescent tails, exp(−q‖|z|), the longitudinal electric field, originating in the sheetcurrent density, does not relate to Einstein causal propagation between z = 0 and z =z. This result again underlines the close connection between apparent superluminality inphoton tunneling and our inability to localize the photon completely along the z-directionin the sheet case.


–q–1||

q–1||0

∫gδTdz

z´ z

gU

z

–q–1||

q–1||0 z´ z

z

FIGURE 21.1Apparent superluminality and subluminality related to the transverse electromagnetic fieldemitted by an ED current density sheet (located in the plane z = 0) and received in theplane of observation (located at z = z > 0). The field coming from a given plane (z = z′)within the exponentially confined transverse current density domain (spatial decay constantq‖) propagates toward the plane of observation with the vacuum speed of light (c). Therelated photon propagator is gU (top figure). If one incorrectly assumes that the sourceof the field is confined to the plane of the sheet (bottom figure), the related propagator is∫

gδT dz′. In consequence the phase velocity of light propagating between the sheets z = z′

and z = z(> 0) must be renormalized to vSL = [1− z′/z]−1c. For 0 < z′ < z, the apparentspeed of light is superluminal (vSL > c), whereas for z′ < 0 < z it is subluminal (vSL < c).

21.3 Incident fields generating evanescent tunneling potentials

In the FTIR-configuration optical tunneling experiments often are carried out using quasi-monochromatic light from a laser beam. In themonochromatic plane-wave approximation wehave seen in Chapt. 20 that the electromagnetic field associated with the sheet dynamicsis evanescent provided q‖ > ω/c. How do we connect this result to the space-time (z, t)description of the transverse and longitudinal vector potentials [Eqs. (21.19) and (21.20)]in view of the fact that the scalar propagator g(Z, τ ; q‖) [Eq. (21.14)] is time-like? Thisquestion may be answered making a Fourier-integral transformation of g(Z, τ ; q‖) to the


frequency domain as a first step. Thus,

g(Z;ω, q‖) =c

2

∫ ∞

−∞θ(cτ − |Z|)J0

(

q‖√

(cτ)2 − Z2)

eiωτdτ

=c

2

∫ ∞

|Z|/cJ0

(

q‖√

(cτ)2 − Z2)

(cosωτ + i sinωτ) dτ. (21.30)

The two parts of Eq. (21.30), containing cosωτ and sinωτ , respectively, can be integratedanalytically. The explicit result one obtains depends on whether cq‖ < ω or cq‖ > ω. Forcq‖ < ω we have [118]

g(Z;ω, q‖) =c

2

[

ω2 − (cq‖)2]− 1

2

[

− sin

( |Z|c

√

ω2 − (cq‖)2)

+ i cos

( |Z|c

√

ω2 − (cq‖)2)]

=i

2q0⊥exp

(

iq0⊥|Z|)

, cq‖ < ω, (21.31)

where q0⊥ = [(ω/c)2 − q2‖ ]1/2, i.e., precisely the z-component of the vacuum wave vector

[Eq.(20.13)]. For cq‖ > ω, the integral containing the sinωτ -factor in Eq. (21.30) is zero,and thus [118]

g(Z;ω, q‖) =c

2

[

(

cq‖)2 − ω2

]− 1

2

exp

(

−|Z|c

√

(

cq‖)2 − ω2

)

=1

2κ0⊥exp

(

−κ0⊥|Z|)

, cq‖ > ω, (21.32)

where κ0⊥ = [q2‖ − (ω/c)2]1/2 is the spatial decay constant belonging to the z-direction

[Eq. (20.14)]. The Fourier analysis hence has given us back the scalar Green functions(propagators) in Eqs. (20.12) and (20.15), as the reader perhaps might have expected.Returning to the time domain, the scalar propagator now may be divided into evanescent(Ev) and propagating (Pr) parts,

g(Z, τ ; q‖) = gEv(Z, τ ; q‖) + gPr(Z, τ ; q‖). (21.33)

The evanescent part, which is of particular importance for FTIR-tunneling, is given by theintegral expression

gEv(Z, τ ; q‖) =1

2π

∫ cq‖

−cq‖

1

2κ0⊥exp

(

−κ0⊥|Z|)

e−iωτdω

=1

π

∫ cq‖

0

1

2κ0⊥exp

(

−κ0⊥|Z|)

cosωτdω. (21.34)

The propagating part of g(Z, τ ; q‖) is given by the sum of integrals over the ω-intervals(−∞| − cq‖) and (cq‖|∞). The two integrals can by lumped together with the result that

gPr(Z, τ ; q‖) =1

π

∫ ∞

cq‖

i

2q0⊥exp

(

iq0⊥|Z|)

cosωτdω. (21.35)

Although optical tunneling certainly exists, when the incident field causes both emissionof propagating and evanescent modes from the particle motion induced in the sheet, it isexperimentally particularly convenient to excite the sheet in such a manner that only (orpredominantly) evanescent field modes are created. For a given q‖ this requires as we have


seen cq‖ > ω, cf. Eq. (21.34). So, how can one achieve that a propagating homogeneous in-cident field can create modes satisfying cq‖ > ω? If the incident field propagates in vacuum,

where the dispersion relation is ω = cq = c[q2‖ + (q0⊥)2]1/2, we certainly must have ω ≥ cq‖.

However, if the halfspace to the left of the sheet (z < 0) is filled with a medium (dielectrics),and the halfspace to the right (z > 0) is vacuum, one can for a propagating wave hittingthe sheet from the left satisfy the condition cq‖ > ω. Without losing the central point itis sufficient for us to assume that the medium is homogeneous with a positive (relative)dielectric constant ε(ω)(> 1). Although the inevitable presence of frequency dispersion inthe dielectric constant leads to losses in the field propagation, as one may conclude fromthe Kramers–Kroenig relations [155, 127, 137, 132], it is assumed that the dispersion inε(ω) is so weak in the frequency range of interest that the losses are unimportant. Theabsence of spatial dispersion in the dielectric response implies that the incident electromag-netic field must be divergence-free (transverse). The dispersion relation in the dielectrics,(ω/c)ε1/2(ω) = q, shows that the condition needed to be satisfied in order to obtain anevanescent mode emitted from the sheet is

q‖ =

[

(ω

c

)2

ε(ω)− q2⊥

]1

2

>ω

c. (21.36)

For a given q‖, an incident field superimposed of modes with components of the z-componentof the wave vector (q⊥) in the interval

0 < q⊥ =ω

c(ε(ω)− 1)

1

2 ≡ qmax⊥ , (21.37)

therefore will satisfy the requirement wanted. Instead of composing the incident field ofmodes with a single q‖, and thus varying q⊥, it is from an experimental point of view usuallymost convenient to form the incident pulse from modes propagating in given directionq = q/q and having q‖ > ω/c, and hence the related q⊥ < (ω/c)(ε(ω) − 1)1/2. From atheoretical point of view the two cases (fixed q‖, fixed q) are not so different in respect toa qualitative discussion of the optical tunneling phenomenon.

Let us now take a closer look at the optical tunneling process associated with thedielectric-vacuum system. If a monochromatic and plane p-polarized transverse electric fieldEinc

T is incident from the dielectric side on the interface there will be a bulk (B) currentdensity

JB(x, z) = iǫ0ω (1− ε(ω))(

Ueiq⊥z + re−iq⊥z)

· EincT (0)eiq‖x (21.38)

in the medium. In Eq. (21.38) r is the amplitude reflection matrix, and EincT (0) ≡

EincT (q‖ex, ω) the amplitude of the incident field [118]. In our heuristic sharp-boundary

model, which of course is an idealization, yet sufficiently good for our purpose, the inducedcurrent density is given by

J(x, z) = JB(x, z)θ(−z) ≡ JB(z)eiq‖xθ(−z). (21.39)

The bulk current density, JB(x, z), is of course divergence-free since the dielectric mediumis (assumed to be) homogeneous, i.e.,

∇ · JB(x, z) = 0. (21.40)

In view of Eq. (21.40) one has

∇ · J(x, z) = −δ(z)ez · JB(z = 0)eiq‖x, (21.41)


a result which implies that the scalar part of the four-current density becomes

JS(x, z) =i

q0δ(z)JB,z(0)e

iq‖x. (21.42)

It appears from Eq. (21.42) that the induced charge density is located solely at the inter-face, and it is clear therefore that the dielectric-vacuum system in the framework of thepresent model behaves like a sheet-current source for field (T-photon) emission. Althoughthe transverse bulk current density is nonvanishing, the only effect of this is an adjustmentof the speed of light from c to c/ε1/2(ω). In a sense we put together “just” two halfspaceswith different “vacuum” speeds of light. In relation to the general sheet model discussed inSec. 20.2, a comparison of Eqs. (20.28) and (21.42) shows that in the model above only atangential (parallel to the surface plane) component of the sheet current exists, that is

I =i

q‖JB,z(0)ex. (21.43)

The longitudinal part of the sheet current density is readily obtained from Eq. (20.33),setting I = Ixex. Thus, including also the exp(iq‖x)-factor, one obtains

JL(x, z) =i

2(ex + iezsgn(z)) JB,z(0)e

q‖(ix−|z|). (21.44)

The result in Eq. (21.43) makes the bridge to the analysis in Sec. 21.2.

21.4 Interlude: Scalar propagator in various domains

Considering the fundamental role of the Lorentz-invariant Huygens propagator in photonphysics and electrodynamics in general, it might be useful to make a pause in the develop-ment of the description of the photon tunneling process, and briefly discuss/compare theexplicit forms of this retarded scalar propagator in four domains, viz., (R, τ), (q; τ), (R;ω)and (q, ω). Starting from the explicit expression for the propagator in the space(R)-time(τ)domain [Eq. (20.7)],

g(R, τ) =1

4πRδ

(

R

c− τ

)

, (21.45)

a spatial Fourier-integral transformation led in Sec. (21.2) to the following expression forthe propagator in the wave-vector(q)-time(τ) domain [Eq. (21.11)]:

g(q, τ) =c

qθ(τ) sin(qcτ). (21.46)

With g(q, τ) at hand a two-dimensional Fourier transformation (in q‖) gave us an explicitexpression for g(Z, τ ; q‖) [Eq. (21.14)], a representation of utmost importance for the un-derstanding of basic elements of the optical tunneling process. The form of the propagatorin the space(R)-frequency(ω) domain is immediately obtained by Fourier transforming Eq.(21.45) in time. Thus,

g(R;ω) =

∫ ∞

−∞(4πR)

−1δ

(

R

c− τ

)

eiωτdτ

=1

4πRexp

(

iω

cR)

, (21.47)


a well-known and often used result.The δ-function singularity in δ((R/c)− τ) impede the transformation to the (q; τ)- and

(R;ω)-domains, because only the space (essentially the radial component) or time coordi-nate was transformed. The singularity needs more attention when one wishes to transformboth R and τ . One possible road is to start from a formal Fourier transformation of q(q; τ),viz.,

g(q;ω) =

∫ ∞

−∞

c

qθ(τ) sin(qcτ)eiωτdτ

=c

2iq

[∫ ∞

0

ei(ω+cq)τdτ −∫ ∞

0

ei(ω−cq)τdτ

]

. (21.48)

The last expression is obtained using the Euler relation sinx = [exp(ix) − exp(−ix)]/(2i),and the fact that the step function cuts off negative τ ’s. The two integrals in Eq. (21.48)[notation α = ω±cq] are just positive-time parts [δ(+)(α)] of the Dirac delta function [δ(α)],viz. [155]

δ(+)(α) =1

2π

∫ ∞

0

eiατdτ =1

2

[

δ(α) +i

πP

(

1

α

)]

, (21.49)

where P stands for the Cauchy Principal Value. By utilizing Eq. (21.49) one obtains

g(q, ω) =cπ

2iq

[

δ(ω + cq)− δ(ω − cq) +i

πP

(

1

ω + cq

)

− i

πP

(

1

ω − cq

)]

. (21.50)

The result, which finally can be contracted to

g(q, ω) =π

2iq

[

δ(ω

c+ q)− δ(

ω

c− q)

]

+ P

(

1

q2 −(

ωc

)2

)

, (21.51)

represents the Huygens propagator in the wave vector-frequency domain. Away from thetwo singularities, located at ω = ±cq, the propagator is simply given by

g(q, ω) =1

q2 −(

ωc

)2 ,ω

c6= ±q. (21.52)

One may regain the formula given for g(q, τ) in Eq. (21.46) by multiplying Eq. (21.52) byexp(−iωτ)/(2π) and carrying out suitable contour integrations. The singularities of g(q, ω)lie on the real ω-axis at ω = ±cq. The integrations run along different contours for τ > 0and for τ < 0, respectively. With Principal integrals at ω = ±cq the reader may prove toherself that it is possible to obtain

g(q; τ) =

∫ ∞

−∞

e−iωτ

q2 −(

ωc

)2

dω

2π=c

qθ(τ) sin(qcτ), (21.53)

in agreement with the calculation in Sec. 21.2.

21.5 Incident polychromatic single-photon state

Theoretical studies of space-time resolved single-photon tunneling require that the incidentelectromagnetic field is in a polychromatic single-photon state. In line with the analyses in


Chapts. 20 and 21, we now construct and discuss a p-polarized single-photon state havingonly one wave-vector component, q‖ = Q‖, parallel to the plane of the current density sheet(or the dielectric-vacuum interface).

In Secs. 15.4-15.6, we showed how a single-photon wave packet in Hilbert space [Eq.(15.80)] by means of a T-photon “mean” position state (Eq. (15.125)) leads to a photonwave function in space-time [Eq. (15.125)]; such a wave function is very appealing in stud-ies of photon-matter interactions. The photon wave function Φ(r, t) in Eq. (15.125) wassynthesized from helicity species (s = +,−) in q-space. For the present purpose it is con-venient to switch from the helicity basis [complex basis vectors: e+(κ), e−(κ)] to a basiswith real basis vectors [εR1 (κ), ε

R2 (κ)]. If we orientate our Cartesian coordinate system such

that the scattering plane coincides with the xz-plane, the unit polarization vectors relatedto p- and s-polarized fields are εR1 (κ) = ey × κ and εR2 (κ) = ey, respectively. Using thetransformation given by Eqs. (10.48) and (10.49), one obtains

∑

s=+,−Φs(q)es(κ) =

1√2

[(

Φ+ + ieiδΦ−)

εR1 +(

iΦ+ + eiδΦ−)

εR2]

, (21.54)

and in the special case where the scalar wave functions Φ+ and Φ− are related by

iΦ+ + eiδΦ− = 0, (21.55)

the vectorial photon wave function in q-space is p-polarized and given by

√2Φ+(κ)ε

R1 = Φp(κ)ey × κ. (21.56)

Our first step hence has reduced the photon wave function in space-time to the form

Φ(r, t) = ey ×[∫ ∞

−∞κΦp(κ)e

i(q·r−ωqt)d3q

(2π)3q

]

. (21.57)

Since all wave vectors lie in the scattering plane (xz-plane), and we assume that they havethe same wave-vector component qx = Q‖ parallel to the sheet (interface) plane, Φp(κ) isgiven by

Φp(κ) = (2π)2δ(

qx −Q‖)

δ(qy)Φ(q⊥) (21.58)

in our usual notation qz ≡ q⊥. By inserting Eq. (21.58) into (21.57) our second step gives


∫ ∞

−∞κΦ(q⊥)e

i(q⊥z−ωqt)dq⊥

2π(

Q2‖ + q2⊥

)1

2

, (21.59)

where κ = (Q‖ex+q⊥ez)/(Q2‖+q

2⊥)

1/2. By now we are left with a single integration, over q⊥.To represent (i) an incident single-photon wave function, and (ii) a state for which Q‖ > ω/cfor all frequency components in the polychromatic state, Φ(q⊥) cannot be nonvanishing forall q′⊥s (−∞ < q⊥ < ∞). For a field incident from the left halfspace (z < 0) on the sheetone must have q⊥ > 0 for all wave-vector components. To obtain an evanescent field in thevacuum halfspace (z > 0), the angle of incidence must be larger than the critical angle forq-modes. For a given (fixed) q‖ = Q‖ this criterion corresponds to

nQ‖ > q =(

Q2‖ + q2⊥

)1

2

, (21.60)


where n = ε1/2 is the (assumed) real refractive index of the homogeneous dielectric medium.I urge the reader to prove the inequality in (21.60) to herself. It follows from the inequalitythat one must have

q⊥ < Q‖√ε− 1. (21.61)

Without losing the main point in the description one may assume that the refractive indexis independent of ω. In this case the dispersion relation is linear, and given by

ω = v(

Q2‖ + q2⊥

)1

2

, (21.62)

where v = c/n is the speed of light in the medium. On the basis of the considerations above,we can conclude that a Φ(q⊥)-distribution which only is nonvanishing in the interval

0 < q⊥ < Q‖√ε− 1 (21.63)

must be used in Eq. (21.59). The frequency interval (divided by c) which corresponds to(21.63) is

Q‖n

<ω

c< Q‖. (21.64)

The considerations above give us the final expression for the incident polychromatic single-photon wave function, viz.,


∫ Q‖

√ε−1

0

κΦ(q⊥) exp

i

[

q⊥z −c

n

(

Q2‖ + q2⊥

)1

2

t

]

dq⊥

2π(

Q2‖ + q2⊥

)1

2

.

(21.65)

The fact that the photon wave function in Eq. (21.65) is independent of y and has theoscillatory form exp(iQ‖x) in x, implies that Φ(r, t) is only normalizable in the Dirac sensein the x and y coordinates. Since we have only one polarization state (p-polarization) forthe various q-modes, it follows from the general normalization condition in Eq. (15.93) thatthe mode amplitudes Φ(q⊥) must satisfy the normalization condition

1

2π

∫ Q‖

√ε−1

0

|Φ(q⊥)|2dq⊥

(

Q2‖ + q2⊥

)1

2

= 1. (21.66)

To obtain the form of the single-photon wave function in Hilbert space, |Φ〉, which inwave-vector space is represented by Φ(r, t) [Eq. (21.65)], we make use of the connection inEq. (15.92). In the present case, this gives the amplitude weight factors

φ(q⊥) =(

Q2‖ + q2⊥

)− 1

4

Φ(q⊥), (21.67)

and, according to the continuum version of Eq. (15.80), a Hilbert space polychromaticsingle-photon state

|Φ〉 = (2π)−3

2

∫ Q‖

√ε−1

0

φ(q⊥)a†(q⊥; 0)dq⊥|0〉, (21.68)

where a†(q⊥; 0) is the creation operator (at t = 0) belonging to the q⊥-mode.


21.6 Photon tunneling-coupled sheets

In Sec. 21.2 we discussed the scattering of a wave-packet band from a single current-densitysheet, and in Sec. 21.3 it was concluded that a purely evanescent sheet potential might begenerated in a vacuum halfspace (z > 0) if the halfspace to the left of the sheet (z < 0) isfilled with a homogeneous and isotropic dielectric medium with refractive index n = ε1/2.Without loss of qualitative insight, it was possible to assume that the relative dielectricfunction (constant) is independent of the light four-vector qµ = (ω/c,q). Using the IN-HOMOGENEOUS VACUUM concept introduced in Sec. 5.1, our sheet thus is placed in asimple inhomogeneous vacuum with n(r) = n for z < 0, and n(r) = 1 for z > 0. It followsfrom the analysis in Sec. 5.1 that if an extra current-density distribution, J(r, t), is added tothe polarization current density [Eq. (5.3)] of the inhomogeneous vacuum, Eq. (5.11) mustbe replaced by

∇×B(r, t) =n2(r)

c2∂

∂tE(r, t) + µ0J(r, t). (21.69)

For

n(r) = nθ(−z) + θ(z), (21.70)

it is clear that the transverse part of Eq. (21.69) is given by

∇×B(r, t) =n2(r)

c2∂

∂tET (r, t) + µ0JT (r, t) (21.71)

in the two halfspaces. Inserting Eqs. (10.3) [with ∇ × A = ∇ × AT ] and (10.26) in theequation above, it appears that the transverse vector potential in the frequency domain(∂/∂t→ −iω), AT (r;ω), satisfies the inhomogeneous differential equation

[

∇2 +

(

ω

v(r)

)2]

AT (r;ω) = −µ0JT (r;ω), (21.72)

where

v(r) =c

n(r)(21.73)

is the speed of light in the inhomogeneous vacuum [given here by Eq. (21.70)].In the usual manner [118], the solution to Eq. (21.72) is expressed in integral form by

means of the scalar propagator g(r, r′;ω), which is a solution to the singular differentialequation

[

∇2 +

(

ω

v(r)

)2]

g(r, r′;ω) = −δ(r− r′). (21.74)

Thus,

AT (r;ω) = AincT (r;ω) + µ0

∫ ∞

−∞g(r, r′;ω)JT (r

′;ω)d3r′, (21.75)

where the incident transverse vector potential satisfies the homogeneous part of Eq. (21.72),i.e.,

[

∇2 +

(

ω

v(r)

)2]

AincT (r;ω) = 0. (21.76)


Under the assumption that all vector fields have the generic form given in Eq. (20.9), thecentral integral relation between the transverse vector potential and the current density is

AT (z) = AincT (z) + µ0

∫ ∞

−∞g(Z)JT (z

′)dz′, (21.77)

cf. Eq. (20.11). As before [Eq. (20.10)], we have left out the reference to q‖ and ω in thenotation above. For the sheet current density in Eq. (20.26), the related transverse currentdensity is given by

JT (z) = δT (z) · I, (21.78)

cf. Eq. (20.30). By combining Eqs. (21.77) and (21.78) one obtains

AT (z) = AincT (z) + µ0

[∫ ∞

−∞g(z − z′)δT (z

′)dz′]

· I. (21.79)

If one places the sheet at z = z0, the term in the square bracket will be replaced by

DT (z − z0) =

∫ ∞

−∞g(z − z′)δT (z

′ − z0)dz′. (21.80)

The quantity DT (z − z0) may be called the dyadic transverse photon propagator [for theinhomogeneous vacuum in Eq. (21.70)]. We shall not here enter a general analysis of thestructure of this propagator. The reader may get an idea of the structure and the relatedphysical interpretation by consulting Ref. [118]. Herein, the somewhat simpler dyadic prop-agator

DT (z − z0) ≡ DT (z − z0|n = 1), (21.81)

associated to homogeneous vacuum (n = 1), is analyzed and related to the near-field elec-trodynamics of a current density sheet. For n = 1 in the entire space, g(Z) becomes identicalto the scalar propagator given in Eq. (20.12), g(Z) = g(Z). If n = n everywhere in spaceone would have

g(Z) =i

2q⊥eiq⊥|Z|, (21.82)

where

q⊥ =

[

(ω

v

)2

− q2‖

]1

2

, (21.83)

corresponding to the fact that we are dealing with a “vacuum” where the speed of light isreduced to v = c/n.

Let us turn our attention now toward the situation where we have two sheets, locatedat z = 0 and z = d. For simplicity, it is assumed that the sheets have identical physicalproperties. To make contact to the paradigm of optical tunneling, viz., the FTIR-process,we further assume that the regions z < 0 and z > d are filled with a dielectrics withrefractive index n, and that we inbetween (0 < z < d) have vacuum (see Fig. 21.2). Thedyadic propagator describing transverse field propagation in this three-layer inhomogeneousvacuum we denote by DFTIR

T (Z). Although it is possible to establish an explicit expressionfor DFTIR

T (Z), this rather lengthy expression is not needed in the following. For the two-sheet system, the transverse vector potential is given by

AT (z) = AincT (z) + µ0D

FTIRT (z) · I(0) + µ0D

FTIRT (z − d) · I(d), (21.84)


0 d z

t

r

i

q–1||

FIGURE 21.2Schematic illustration of the FTIR-process between two homogeneous and isotropic dielec-tric hemispherical prisms placed in vacuum a distance d apart. The FTIR-process can beconceived as an optical tunneling phenomenon between current density sheets located at theprism/vacuum interfaces, cf. Fig. 21.1. When the vacuum gap is less than the confinementlength of the T-photon source domain, qualitatively, i.e., d < q−1

‖ , an incident transverse

photon has a significant probability of tunneling across the vacuum gap.

where I(0) and I(d) are the prevailing vectorial strengths of the sheet current densities.These strengths, which certainly cannot be considered as prescribed quantities in the FTIR-process, are induced by the local transverse vector potential acting on the sheets. To deter-mine the I’s one must go beyond the sheet approximation, as already indicated in Sec. 20.2.In the framework of linear response theory for media exhibiting translation invariance intime the current density (J(r;ω)) and the transverse vector potential (AT (r;ω)) are relatednonlocally in space [118], i.e.,

J(r;ω) =

∫ ∞

−∞S(r, r′;ω) ·AT (r

′;ω)d3r′, (21.85)

where S(r, r′;ω) is a sort of microscopic response tensor [118]. In the present case, Eq.(21.85) is reduced to the still spatially nonlocal form

J(z) =

∫ ∞

−∞S(z, z′) ·AT (z

′)dz′, (21.86)

where again the references to q‖ and ω are omitted. If one neglects by brute force spatialvariations in AT (z

′) on a length scale comparable to (and shorter than) the effective rangeof the response tensor S(z, z′) around the point z0, where it is centered, one obtains

J(z) =

[∫ ∞

−∞S(z, z′)dz′

]

·AT (z0). (21.87)

With the approximation in Eq. (21.87), the sheet is considered as an electric-dipole (ED)absorber. We reach the sheet approximation given in Eq. (20.26) by a subsequent integrationover z. Thus,

J(z) = I(z0)δ(z − z0) (21.88)


where

I(z0) =

[∫ ∞

−∞S(z, z′)dz′dz

]

·AT (z0). (21.89)

The double integration of S(z, z′) means that the sheet is considered not only as an EDabsorber but also as an ED radiator. Since we have assumed that the two sheets haveidentical physical properties, the quantity

∫

S(z, z′)dz′dz is the same for the two. With theabbreviation

Γ(z) = µ0DFTIRT (z) ·

[∫ ∞

−∞S(z, z′)dz′dz

]

, (21.90)

Eq. (21.84) takes the form

AT (z) = AincT (z) + Γ(z) ·AT (0) + Γ(z − d) ·AT (d). (21.91)

It appears from Eq. (21.91) that the transverse vector potential can be determined for all zif it is known for z = 0 and z = d, provided of course that the dyadic propagator DFTIR

T (z)and the double integral of S(z, z′) have been calculated. The vector potentials AT (0) andAT (d) are obtained setting z = 0 and z = d in Eq. (21.91). In the FTIR-configuration, theincident field, Ainc

T (z), is zero for z > 0. Therefore,

AT (0) =AincT (0) + Γ(0) ·AT (0) + Γ(−d) ·AT (d), (21.92)

AT (d) =Γ(d) ·AT (0) + Γ(0) ·AT (d). (21.93)

Together, the two vectorial equations above allow one to determine the prevailing fields onthe sheets. Once these are known the transverse vector potential van be obtained everywherealong the z-axis from Eq. (21.91). The two contributions Γ(0) ·AT (0) and Γ(0) ·AT (d) maybe called self-field terms because they represent the contributions to the transverse vectorpotential on the given sheet stemming from the sheet itself (back-action).

22

Near-field photon emission in 3D

22.1 T-, L-, and S-potentials of a classical point-particle

In this section we shall study the field emission from a classical point-particle moving along aprescribed trajectory. Since we have seen that transverse photons are conveniently describedin terms of the transverse part of the vector potential in photon wave mechanics, we shallfocus our efforts on the four-potential generated by a point-particle.

22.1.1 General considerations on source fields

We already know that the transverse part of the vector potential, AT (r, t), satisfies theinhomogeneous wave equation

AT (r, t) = −µ0JT (r, t), (22.1)

where, as usual, JT (r, t) denotes the transverse part of the current density [see Eq. (12.27)].The transverse part of the vector potential is gauge invariant, and the inhomogeneoussolution to Eq. (22.1) is given by

AT (r, t) = µ0

∫ ∞

−∞g(R, τ)JT (r

′, t′)d3r′dt′, (22.2)

where g(R, τ) is the Huygens scalar propagator, and R = r− r′, and τ = t− t′, as before.One may consider the transverse current density distribution as the source domain of thetransverse vector potential, and from each space-time point in this distribution the field AT

spreads out as dictated by the scalar propagator [Eq. (20.7)]. If one prefers to consider thecurrent density itself, J(r, t), as the source domain of AT , one must replace JT by J in Eq.(22.2). The relation between the two current densities is given by

JT (r, t) =

∫ ∞

−∞δT (r − r′) · J(r′, t)d3r′, (22.3)

where δT (r−r′) is the transverse delta function, a dyadic quantity. For the present purposeit is useful below to apply for δT (R) its explicit form in spherical contraction, namely[53, 118],

δT (R) =2

3δ(R)U− U− 3RR

4πR3, (22.4)

where R = R/R. By inserting Eq. (22.3) into Eq. (22.2) one obtains

AT (r, t) = µ0

∫ ∞

−∞GT (r− r′, t− t′) · J(r′, t′)d3r′dt′, (22.5)

255


where

GT (r− r′, τ) =∫ ∞

−∞g(r− r′′, τ)δT (r

′′ − r′)d3r′′. (22.6)

The physics contained in Eqs. (22.2) and (22.5) is the same, but the pictures offered by thetwo equations appear different. In Eq. (22.2), the field generated from a given space-timepoint (r′, t′) in the JT -distribution (i) propagates with the speed of light (c), (ii) spreads outisotropically, and (iii) has the typical far-field (R−1) dependence at all source-observationdistances. In Eq. (22.5), the explicit form of the transverse dyadic propagator, to be givenbelow, shows that the field from the point (r′, t′) in the J-distribution (i) is anisotropic,(ii) has both far (R−1) and near-field (R−3) components, and (iii) is in the near-field zonedifferent from zero in front of the light cone (space-like). The last point should not comeas a surprise to the reader in view of Eq. (22.3). Thus, because the relation between J andJT is nonlocal in space but local in time, a truncation in space from the JT -distribution tothe J-distribution must result in a picture where space-like effects appear. Such a result isin no way in conflict with Einstein causality, of course.

The longitudinal and scalar parts of the four-potential are gauge dependent. In theLorenz gauge the longitudinal part of the vector potential, AL(r, t), satisfies the inhomoge-neous wave equation

AL(r, t) = −µ0JL(r, t), (22.7)

as one immediately sees by subtracting Eq. (22.1) from Eq. (12.4). The inhomogeneoussolution, describing the longitudinal vector potential generated by the longitudinal part ofthe current density, JL(r, t), is

AL(r, t) = µ0

∫ ∞

−∞g(R, τ)JL(r

′, t′)d3r′dt′. (22.8)

It is a straightforward matter to establish an integral relation in which the current densityitself is taken as the source for the longitudinal vector potential. Since JL = J − JT , acombination of Eqs. (22.2), (22.5), and (22.8) readily gives

AL(r, t) = µ0

∫ ∞

−∞[g(R, τ)U−GT (R, τ)] · J(r′, t′)d3r′dt′. (22.9)

The scalar part of the four-potential, AS(r, t)[≡ A0(r, t)] has JS(r, t)[≡ J0(r, t)] as its sourcedistribution, and from Eq. (12.5), it appears that the emitted scalar potential is given by

AS(r, t) = µ0

∫ ∞

−∞g(R, τ)JS(r

′, t′)d3r′dt′. (22.10)

It is possible to obtain an explicit expression for the dyadic transverse propagatorGT (R, τ), given an integral form in Eq. (22.6). We shall not derive this expression here,but the reader may consult [119] and my book on the quantum theory of near-field electro-dynamics [118] if she is interested in the derivation. Thus, it turns out that [119]

GT (R, τ) =1

4πRδ

(

R

c− τ

)

(

U− RR)

− c2τ

4πR3θ (τ) θ

(

R

c− τ

)

(

U− 3RR)

. (22.11)

The far-field (R−1) part of the propagator is nonvanishing only on the light cone, andthe near-field (R−3) part is space-like [manifest from the presence of the step functionθ(R/c− τ)], but causal [manifest because of the θ(τ)-factor].

Near-field photon emission in 3D 257

22.1.2 Point-particle potentials

Let us now discuss the various parts of the four-potential associated with a charged point-particle moving along a prescribed trajectory, r0(t), with velocity v0(t) = r0(t). The com-plete (perfect) spatial localization of our classical particle (with charge Q) implies that thescalar part of the four-current density is given by

JS(r, t) = cQδ (r− r0(t)) , (22.12)

and the vectorial part by

J(r, t) = Qv0(t)δ (r− r0(t)) . (22.13)

I urge the reader to prove to herself that the expressions above do satisfy the law of chargeconservation.

For what follows it is convenient to refer to the picture in which the current densityitself, J(r, t), is the source of the transverse and longitudinal scalar potentials. By insertingEq. (22.13) into Eq. (22.5) one obtains

AT (r, t) = µ0Q

∫ ∞

−∞GT (r− r0(t

′), t− t′) · v0(t′)dt′, (22.14)

because the delta function δ(r′ − r0(t′)) immediately allows one to carry out the integra-

tion over r′-space. The expression for GT in Eq. (22.11) shows that the transverse vectorpotential consists of a sum of far-field (FF) and near-field (NF) parts, i.e.,

AT (r, t) = AFFT (r, t) +ANF

T (r, t). (22.15)

With the abbreviations

β(t) =v0(t)

c, (22.16)

R(t) = r− r0(t), (22.17)

and R(t) = R(t)/R(t), the integral expression for the far-field contribution to AT (r, t) reads

AFFT (r, t) =

µ0cQ

4π

∫ ∞

−∞

1

R(t′)

(

U− R(t′)R(t′))

· β(t′)δ(

R(t′)c

− t+ t′)

dt′. (22.18)

The presence of the Dirac delta function enables one to carry out the integration over time.For simplicity, we assume that the delta function has only one singular point, t′ = tr. Thetime tr, which is the solution to the equation

|r− r0(tr)|c

− t+ tr = 0, (22.19)

is for obvious reasons called the retarded (r) time. Let

f(t′) = t′ − t+|r− r0(t

′)|c

, (22.20)

and thus

df(t′)dt′

= 1− β(t′) · R(t′), (22.21)


a positive quantity since |β(t′)| < 1. With only one zero point t′ = tr for f(t′), i.e., f(tr) = 0,

the reader may show, using a Taylor expansion of f(t′) around t′ = tr, that

∫ ∞

−∞F(t′)δ(f(t′))dt′ =

F(tr)

|f ′(tr)|. (22.22)

This relation, when applied to Eq. (22.18), gives the following result for the far-field partof the transverse vector potential:

AFFT (r, t) =

µ0Qc

4π

(

U− RR)

· β(

1− β · R)

R

ret

, (22.23)

using the notation

F(tr) ≡ [F ]ret . (22.24)

It appears from Eq. (22.23) that AFFT at the space-time point (r, t) only depends on the

position (r0(tr)) and velocity (r0(tr)) of the point-particle at the retarded (ret) time.The near-field contribution to AT (r, t), viz.,

ANFT (r, t) =

µ0c2Q

4π

×∫ ∞

−∞θ(t− t′)θ

(

R(t′)c

− t+ t′)

(t′ − t)R−3(t′)(

U− 3R(t′)R(t′))

· v0(t′)dt′,

(22.25)

which the step functions allow one to write in the form

ANFT (r, t) =

Qc

4πǫ0

∫ t

tr

(t′ − t)1

R3(t′)

(

U− 3R(t′)R(t′))

· β(t′)dt′, (22.26)

relates ANFT at the space-time point (r, t) to those positions of the particle, r0(t

′), whichare space-like (t′ > tr) connected to r; see Fig. 22.1 This important result associates to thefact that a transverse photon emitted from the moving point-particle is born in a quantumstatistical sense in the particle’s near-field zone. Since the upper limit on the integral inEq. (22.26) is t, no photon can be detected at the observation point r before it is born, asatisfactory conclusion.

It immediately appears from Eq. (22.9) [with J given by Eq. (22.13)] that the longitudinalvector potential of the point-particle also consists of far and near-field parts,

AL(r, t) = AFFL (r, t) +ANF

L (r, t), (22.27)

and that the near-field contribution is just

ANFL (r, t) = −ANF

T (r, t). (22.28)

The sum of the near-field parts of AT and AL hence is zero at all times. With the help ofEq. (22.22), it is not difficult to show that

µ0Q

∫ ∞

−∞g (r− r0(t

′), t− t′)v0(t′)dt′ =

µ0Qc

4π

β(

1− β · R)

R

ret

. (22.29)


>

>

t rt'

t

(t')0

c(t-t )r[ ,t]q

r

r

FIGURE 22.1A charged point-particle moves along a prescribed trajectory, r0(t

′). An observer (O) placedat the space point rmay at time t possibly register a near-field contribution to the transversevector potential (ANF

T ) generated by the particle. Only the fields from those portions of thetrajectory which are space-like (t′ > tr) connected to r contribute to ANF

T (r, t).

The general form in Eq. (22.9) then implies that the far-field part of the longitudinal vectorpotential is obtained subtracting the right-hand sides of Eqs. (22.29) and (22.23) from eachother, that is

AFFL (r, t) =

µ0Qc

4π

RR · β(

1− β · R)

R

ret

. (22.30)

The sum of the T- and L-parts of the vector potential, namely,

A(r, t) = AT (r, t) +AL(r, t)

= AFFT (r, t) +AFF

L (r, t)

=µ0Qc

4π

β(

1− β · R)

R

ret

, (22.31)

is the famous Lienard–Wiechert result for the vector potential of a point-charge [225, 101,127, 6, 248, 223]. The Lienard–Wiechert formula is of course the one given in Eq. (22.29),remembering Eq. (20.8).

A comparison of the expressions in Eqs. (22.12) and (22.13) makes it clear that thescalar potential can be obtained from Eq. (22.31) making the replacement v0 → c in thenominator. Hence,

AS(r, t) =Q

4πǫ0c

1(

1− β · R)

R

ret

, (22.32)

the Lienard–Wiechert result.


22.2 Cerenkov shock wave

22.2.1 Four-potential of point-particle in uniform motion in vacuum

It is instructive to specialize the description in Sec. 22.1 to the case where the point-particleis moving with constant velocity, v0(t) = V. The result obtained for uniform particlemotion in vacuum provides us with a good starting point for an analysis of the Cerenkovphenomenon [231], and its relation to photon wave mechanics (Subsecs. 22.2.2–22.2.4).

Let us go back to Eq. (20.8), and focus our attention on the part of the four-potentialoriginating in the current density distribution Jµ(r, t), viz.,

Aµ(r, t) = µ0

∫ ∞

−∞g(R, τ)Jµ(r′, t′)d3r′dt′. (22.33)

The translation invariance of the vacuum in space and time, reflected in that the Huygensscalar propagator is a function of the differences r−r′ and t− t′, makes the right side of Eq.(22.33) a folding integral in space-time. In the frequency-wave vector domain one thereforehas the algebraic connection

Aµ(q, ω) = µ0g(q, ω)Jµ(q, ω). (22.34)

The rotational invariance of the vacuum implies that the scalar propagator only dependson the numerical difference R = |r − r′|, so that g(q, ω) = g(q, ω) in the (q, ω)-domain. Inview of Eq. (21.52), one thus has

Aµ(r, t) =µ0

(2π)4

∫ ∞

−∞

Jµ(q, ω)q2 −

(

ωc

)2 ei(q·r−ωt)d3qdω, (22.35)

at least formally. Dealing with an integral of this form, the reader must remember thesingular behavior of the scalar propagator at ω = ±cq must be treated with care, cf. theanalysis in Sec. 21.4.

It appears from Eqs. (22.12) and (22.13) that the four-current density of a point-particlemoving with the constant velocity V is

Jµ(r, t) = Q(c,V)δ(r −Vt), (22.36)

assuming that the particle is at the origin of our coordinate system at time t = 0. In the(q, ω)-domain we then obtain

Jµ(q, ω) = Q(c,V)

∫ ∞

−∞δ(r−Vt)e−i(q·r−ωt)d3rdt

= Q(c,V)

∫ ∞

−∞exp [−i (q ·V − ω) t] dt

= 2πQ(c,V)δ (q ·V − ω) . (22.37)

By combining Eqs. (22.35) and (22.37), the following integral representation is obtained forthe four-potential:

Aµ(r, t) =µ0Q

(2π)3(c,V)

∫ ∞

−∞

δ(q ·V − ω)

q2 −(

ωc

)2 ei(q·r−ωt)d3qdω. (22.38)

Without loss of generality we may assume that the particle propagates along the z-axis,


V = V ez (with V > 0, here), and for convenience, we resolve the given q-vector into itscomponents parallel (q‖ = q‖ez) and perpendicular (q⊥) to V, q = q‖+q⊥. This resolutiongives the delta function the form

δ(q ·V − ω) = δ(q‖V − ω) =1

Vδ(

q‖ −ω

V

)

, (22.39)

and then

Aµ(r, t) =µ0Q

(2π)3V(c,V)

∫ ∞

−∞

δ(

q‖ − ωV

)

q2‖ + q2⊥ −(

ωc

)2 ei(q⊥·r⊥+q‖z−ωt)dq‖d

2q⊥dω. (22.40)

where r⊥ = (x, y, 0). The presence of the delta function δ[q‖ − (ω/V )] allows one to carryout the integration over q‖ or ω immediately. For a later comparison to a certain integral

representation of the Cerenkov four-potential is useful to make the integration over q‖ (first).Hence,

Aµ(r, t) =µ0Q

(2π)3V(c,V)

∫ ∞

−∞

eiq⊥·r⊥ exp[

i(

zV − t

)

ω]

q2⊥ + ω2 (V −2 − c−2)d2q⊥dω. (22.41)

As we shall see shortly, it is possible to carry out the remaining three integrations analyt-ically, but before we do this two remarks to Eq. (22.41) are of physical interest. (i) Thefour-potential depends on z and t only through the combination z − V t. For a particlemoving with constant speed along the z-axis, this was of course to be expected. Since theparticle is at the origo at t = 0, one has r0(t) = V tez = z0(t)ez , and thus z−V t = z−z0(t).Inserting this relation into Eq. (22.41) it is manifest that the z-dependence of the four-potential only depends on the difference z − z0(t), the correct result. (ii) Since V < c, thedenominator in Eq. (22.41) cannot become zero, and the integrand therefore has no singu-larities in the real (q⊥, ω)-domain. The integration over the q⊥-plane is readily carried outin polar coordinates (with the polar axis placed along r⊥) since the factor to exp (iq⊥ · r⊥)in the integrand is a function of q2⊥ only. By means of the result

∫ ∞

−∞eiq⊥·r⊥F

(

q2⊥)

d2q⊥ =

∫ ∞

0

q⊥F(

q2⊥)

∫ 2π

0

eiq⊥r⊥ cos θdθdq⊥

= 2π

∫ ∞

0

q⊥F (q2⊥)J0(q⊥r⊥)dq⊥, (22.42)

where J0(q⊥r⊥) denotes the Bessel function of zero order, and r⊥ = (x2 + y2)1/2 in itsargument, Eq. (22.41) is reduced to a double integral, viz.,

Aµ(r, t) =µ0Q

(2π)2V(c,V)

∫ ∞

0

q⊥J0(q⊥r⊥)∫ ∞

−∞

ei(zV−t)ωdω

q2⊥ + ω2 (V −2 − c−2)dq⊥. (22.43)

The integral over frequency is obtained using the formula [83]

∫ ∞

−∞

ei(zV−t)ω

ω2 + a2dω =

π

aexp

(

−a∣

∣

∣

z

V− t∣

∣

∣

)

, (22.44)

here with

a =q⊥V

√

1−(

Vc

)2= γq⊥V (> 0). (22.45)


As the reader may verify, the expression for the four-potential now is reduced to the singleintegral

Aµ(r, t) =µ0Q

4πγ(c,V)

∫ ∞

0

J0(q⊥r⊥) exp (−γq⊥|z − V t|) dq⊥. (22.46)

The integral over q⊥ is of the type [83]∫ ∞

0

e−αq⊥J0(βq⊥)dq⊥ =(

α2 + β2)− 1

2 , (22.47)

so that we finally reach the following result for the four-potential:

Aµ(r, t) =µ0Q

4πvµ

[

γ2 (z − V t)2+ r2⊥

]− 1

2

, (22.48)

where

vµ = γ(c,V) (22.49)

is the particle’s four-velocity.Looked at in isolation the manner in which we have derived the expression in Eq. (22.48)

for the four-potential of a charged point-particle in uniform motion is not the easiest one.However, the route followed starting from Eq. (22.34) is fruitful for comparison to thederivation of the four-potential in the Cerenkov case (see Subsecs. 22.2.3 and 22.2.4). Withthe help of the Lorentz transformation the result in Eq. (22.48) can be obtained quicklyand in an elegant fashion, as I now shall demonstrate.

We begin in the rest frame (O′) of the point-particle. For a uniformly moving particle thisframe is an inertial frame in Special Relativity. An observer fixed in O′ only sees a Coulombfield from the particle. With the particle placed at the origo of O′, the associated time-independent scalar (A′

S) and vector (A′) potentials are for an observer at r′ = (x′, y′, z′)given by

A′S =

µ0Q

4π

c

r′, (22.50)

A′ =0, (22.51)

where r′ = |r′|. In another inertial frame O, where the particle is seen to move along thecommon z(z′)-axis with velocity V = V ez (V > 0), the particle four-potential (AS ,A) canbe obtained from the Lorentz transformation

AS =γ (A′S + β ·A′) , (22.52)

A⊥ =A′⊥, (22.53)

A‖ =γ(

A′‖ + βA

′S

)

, (22.54)

where the subscripts ‖ and ⊥ relate to the components of the vector potential parallel andperpendicular to V, respectively. In O, the particle’s four-potential hence is given by

Aµ(r′, t′) =µ0Q

4π

cγ

r′(1,β), (22.55)

with coordinates still referring to those in O′. With the origo’s of O and O′ coinciding att = 0, the Lorentz coordinate transformations x′ = x, y′ = y [(x′)2 + (y′)2 = x2 + y2 = r2⊥],and z′ = γ(z − βct) finally lead to

Aµ(r, t) =µ0Q

4πcγ(1,β)

[

γ2 (z − V t)2+ r2⊥

]− 1

2

, (22.56)

i.e., precisely the result given in Eq. (22.48).


22.2.2 Transverse and longitudinal response theory in matter

Let us for a moment return to Eqs. (22.43)-(22.45). It appears from these equations thatthe contribution to the four-potential from a given q⊥(> 0) is an exponentially decayingfunction of |(z/V ) − t| = V −1|z − z0(t)|, the decay constant a being given by Eq. (22.45).The potential belonging to the chosen q⊥ hence is concentrated in a narrow band parallelto the xy-plane and centered on the actual position (z0(t)) of the particle. As a functionof r⊥, the four-potential decays in an oscillatory manner as dictated by the Bessel functionJ0(q⊥r⊥). Imagine that it was possible for our particle to move with a speed exceeding thespeed of light in vacuum (V > c). The integral in Eq. (22.44) would then be replaced byone of type

∫ ∞

−∞

ei(zV−t)ω

b2 − ω2dω = 2

∫ ∞

0

cos[(

zV − t

)

ω]

b2 − ω2dω

=π

bsin[

b∣

∣

∣

z

V− t∣

∣

∣

]

, (22.57)

where

b =q⊥V

√

(

Vc

)2 − 1(> 0) . (22.58)

The integral in Eq. (22.57) can be determined by proper residue calculation, or taken from[83]. The qualitative behavior of the four-potential now would be completely different. Fora given q⊥, the potential would exhibit oscillatory (radiative) character as a function of(z/V )− t. With the insight we possess today, it is not possible for a particle to move withsuperluminal speed in vacuum, but in a medium with refractive index n(> 1), it is possiblefor a particle to propagate with a speed larger than the (phase) velocity of light in themedium, c/n < V < c. The radiation emitted by the particle in this situation is calledCerenkov radiation [101, 91, 231, 172, 130].

Our study of the propagation of a charged point-particle through matter starts fromthe microscopic Maxwell–Lorentz equations that is Eqs. (2.1), (2.4), (5.1), and (5.2). In thepresent case the total current density, J(r, t) ≡ Jtot(r, t) in Eq. (5.1), is divided into thesum of the current density induced by the selfconsistent field in the charged particles ofthe medium [this current density we denote by J(r, t) in the following], and the so-calledexternal current density, Jext(r, t), associated with the extra particle we inject into themedium. Thus,

Jtot(r, t) = J(r, t) + Jext(r, t). (22.59)

In the wave vector-frequency domain the relevant set of Maxwell–Lorentz equations has theform

q×E(q, ω) =ωB(q, ω), (22.60)

iq×B(q, ω) =µ0

(

J(q, ω) + Jext(q, ω))

− iω

c2E(q, ω), (22.61)

iq ·E(q, ω) =ǫ−10

(

ρ(q, ω) + ρext(q, ω))

, (22.62)

q ·B(q, ω) =0, (22.63)

where ρ and ρext are the charge densities associated to J and Jext.For simplicity, it is now assumed that the medium exhibits translational invariance

in space and time, and rotational invariance in space. Furthermore, we assume that the


response to the prevailing electromagnetic field is linear. With these assumptions one obtainsa constitutive equation of the form

J(q, ω) = [σT (q, ω) (U− κκ) + σL(q, ω)κκ] · E(q, ω), (22.64)

where as before κ = q/q. It is the assumed translational invariance in space-time whichhas led to an algebraic constitutive relation in (ω,q)-space [3, 105, 191]. By means ofthe two different conductivity response functions σL and σT we account for the factthat the response to electric fields along [EL(q, ω) = κκ · E(q, ω)] and perpendicular[ET (q, ω) = (U− κκ) · E(q, ω)] to the κ-direction will be different for finite q-values. Inthe long-wavelength limit (q → 0) these two so-called longitudinal (σL) and transverse(σT ) conductivity response functions become identical, σL(q → 0, ω) = σT (q → 0, ω). Therotational invariance of the medium implies that the response functions can depend onlyon the length of the q-vector, as already indicated in Eq. (22.64).

With Eq. (22.64) written as

J(q, ω) = σT (q, ω)ET (q, ω) + σL(q, ω)EL(q, ω), (22.65)

and a division of the external current density into its transverse and longitudinal parts, i.e.,

Jext(q, ω) = JextT (q, ω) + Jext

L (q, ω)

= (U− κκ) · Jext(q, ω) + κκ · Jext(q, ω), (22.66)

the transverse set of Maxwell–Lorentz equations take the following form in the wave vector-frequency representation:

q×ET (q, ω) = ωB(q, ω), (22.67)

and

iq×B(q, ω) =

(

µ0σT (q, ω)−iω

c2

)

·ET (q, ω) + µ0JextT (q, ω), (22.68)

remembering that q · ET (q, ω) = q · B(q, ω) = 0. In free space, Eqs. (22.65) and (22.68)reduce to Eqs. (2.47) and (2.48). If one introduces a (relative) transverse dielectric function(εT ) by the definition [127, 115]

εT (q, ω) = 1 +i

ǫ0ωσT (q, ω), (22.69)

Eq. (22.68) can be written in the useful form

iq×B(q, ω) = µ0JextT (q, ω)− iω

c2εT (q, ω)ET (q, ω). (22.70)

By taking the vectorial product of Eq. (22.65) with q from the left, inserting thereafter Eq.(22.70) in the resulting equation, and remembering finally that q ·ET (q, ω) = 0 one obtainsthe following inhomogeneous wave equation for the transverse electric field in (ω,q)-space:

[

q2 −(ω

c

)2

εT (q, ω)

]

ET (q, ω) = iµ0ωJextT (q, ω). (22.71)

Since ET (q, ω) = iωAT (q, ω), it appears that the transverse vector potential is given by

AT (q, ω) =µ0 (U− κκ) · Jext(q, ω)

q2 −(

ωc

)2εT (q, ω)

. (22.72)


The result in Eq. (22.72) is the starting point for our analysis of the Cerenkov effect. Beforewe proceed to this analysis, we shall derive expressions for the longitudinal and scalar partsof the four-potential in the Lorenz gauge.

In view of Eq. (22.65), the longitudinal part of the Maxwell–Lorentz equation in (22.61)can be given the form

(iǫ0ω − σL(q, ω))EL(q, ω) = JextL (q, ω), (22.73)

and in terms of the (relative) longitudinal dielectric function (εL) defined by

εL(q, ω) = 1 +i

ǫ0ωσL(q, ω), (22.74)

one thus obtains the following connection between EL(q, ω) and JextL (q, ω):

EL(q, ω) =JextL (q, ω)

iǫ0εL(q, ω)ω. (22.75)

Utilizing the (ω,q)-versions of the general relation in Eq. (10.27), viz.,

EL(q, ω) = iωAL(q, ω)− icqAS(q, ω), (22.76)

and the Lorenz gauge condition in Eq. (11.4), namely,

q ·AL(q, ω) =ω

cAS(q, ω), (22.77)

one can obtain relations between AL(q, ω) and Jext(q, ω), and between AS(q, ω) andJextS (q, ω) = cρext(q, ω), as we now shall show. Eliminating AS(q, ω) between Eqs. (22.76)

and (22.77), one gets

ωEL(q, ω) = i(

ω2U− c2qq)

·AL(q, ω)

= i[

ω2 − (cq)2]

AL(q, ω), (22.78)

and if one hereafter combines Eqs. (22.75) and (22.78) the result

AL(q, ω) =µ0κκ · Jext(q, ω)[

q2 −(

ωc

)2]

εL(q, ω)(22.79)

follows. The factor q2 − (ω/c)2 appearing in the denominator cannot be zero, because thiswould require that the vacuum dispersion relation for light, ω = cq, was satisfied in themedium, which it is not. By combining Eq. (22.77), and the equation obtained by takingthe scalar product of Eq. (22.79) with q, one obtains

AS(q, ω) =cµ0qκ · Jext(q, ω)

ω[

q2 −(

ωc

)2]

εL(q, ω). (22.80)

With the help of the equation of continuity for the external charge, viz., q · Jext(q, ω) =(ω/c)Jext

S (q, ω), our final relation between the scalar potential and the external chargedensity (multiplied by c) is reached:

AS(q, ω) =µ0J

extS (q, ω)

[

q2 −(

ωc

)2]

εL(q, ω). (22.81)

At this point the reader should notice the structural similarity between Eqs. (22.72), (22.79)and (22.81).


22.2.3 The transverse Cerenkov phenomenon

If we again assume that the particle is moving along the z-axis, V = V ez (V > 0), theexternal three-current density is given by

Jext(q, ω) = 2πQVδ(

q‖V − ω)

(22.82)

in the wave vector-frequency domain; see Eq. (22.37). The spatial dependence of the trans-verse vector potential belonging to a given angular frequency is obtained by inserting Eq.(22.82) into Eq. (22.72) and making thereafter a Fourier transformation of the resultingequation over q-space.

Hence,

AT (r;ω) =µ0Q

(2π)2V ·

[

∫ ∞

−∞

(U− κκ) δ(

q‖V − ω)

q2 −(

ωc

)2εT (q, ω)

eiq·rd3q

]

=µ0Q

(2π)2ez ·

[

∫ ∞

−∞

(U− κκ) δ(

q‖ − ωV

)

q2 −(

ωc

)2εT (q, ω)

eiq·rd3q

]

. (22.83)

The presence of the delta function δ[q‖ − (ω/V )2] reduces the expression for AT (r;ω) to anintegral over the q⊥-plane, viz.,

AT (r;ω) =µ0Q

(2π)2ez ·

[

∫ ∞

−∞

(U− ekek) eiq⊥·r⊥d2q⊥

q2⊥ +(

ωV

)2 −(

ωc

)2εT (k, ω)

]

exp(

iω

Vz)

, (22.84)

where

k = q⊥ +( ω

V

)

ez, (22.85)

and

ek =k

k. (22.86)

The rotational symmetry of the vector potential around the z-axis, allows one to expressAT (r;ω) in the form of single integral over the magnitude of the q⊥-vector. By introductionof polar coordinates in the q⊥-plane, and placing the polar axis along the given r⊥-direction,the angular integral (I) we need to carry out is

I =1

2π

∫ 2π

0

ez · (U− ekek) eiq⊥r⊥ cos θdθ

=1

2π

∫ 2π

0

ez ·(

U− kk

q2⊥ +(

ωV

)2

)

eiq⊥r⊥ cos θdθ, (22.87)

with

k = q⊥ (r⊥ cos θ + ez × r⊥ sin θ) +ω

Vez, (22.88)

where r⊥ = r⊥/r⊥. Since ez · k = ω/V , it appears that

I =1

2π

∫ 2π

0

[

1−(

ωV

)2

q2⊥ +(

ωV

)2

]

ez

−ωV q⊥

q2⊥ +(

ωV

)2 (r⊥ cos θ + ez × r⊥ sin θ)

eiq⊥r⊥ cos θdθ. (22.89)


The integral part containing the factor sin θ is zero, and the remaining parts can be expressedin terms of the first kind Bessel functions of zero (J0) and first (J1) order since these havethe integral representations

J0(q⊥r⊥) =1

2π

∫ 2π

0

eiq⊥r⊥ cos θdθ, (22.90)

J1(q⊥r⊥) =1

2πi

∫ 2π

0

cos θeiq⊥r⊥ cos θdθ. (22.91)

Remembering that d2q⊥ = q⊥dθdq⊥, we finally obtain the following one-dimensional inte-gral expression for the transverse vector potential at the angular frequency ω:

AT (r;ω) =µ0Q

2π

∫ ∞

0

q2⊥[

q⊥J0(q⊥r⊥)ez − i ωV J1(q⊥r⊥)r⊥]

dq⊥[

q2⊥ +(

ωV

)2] [

q2⊥ +(

ωV

)2 −(

ωc

)2εT (k, ω)

] exp(

iω

Vz)

. (22.92)

It is not possible to evaluate the integral over q⊥ unless a specific model for the transversedielectric function εT (k, ω) is adopted. It is possible, however, to give a qualitative discussionof the general result for AT (r;ω) on the basis of Eq. (22.92).

In a given point of observation, r = r⊥+zez, the transverse vector potential is polarizedin the plane spanned by the unit vectors ez and r⊥, and the state will be elliptically polar-ized, in general. Since J1(0) = 0, AT is linearly polarized along the z-axis for observationpoints located on this axis. This must of course be so because of the rotational symmetryaround the particle trajectory. The asymptotic behaviors of J0(q⊥r⊥) and J1(q⊥r⊥) show

that the magnitude of AT is proportional to r−1/2⊥ far from the z-axis. In most cases it is

possible to neglect the spatial dispersion in the transverse dielectric function, and thus take

εT (k, ω) ≃ εT (0, ω) ≡ εT (ω). (22.93)

The matter-particle interaction implies that a particle travelling with large speed will beslowed down. A quantitative analysis of this process requires that the imaginary part ofεT (ω) is retained. For the present purpose it is sufficient to assume that εT (ω) is real, atleast in a certain frequency range (or in certain ω-bands), and that εT (ω) > 1 here. In thisapproximation the factor

D(q⊥, ω) = q2⊥ + ω2

(

1

V 2− εT (ω)

c2

)

, (22.94)

appearing in the denominator of the integrand in Eq. (22.92) can be zero, provided thespeed of the particle exceeds the phase velocity of light, i.e.,

V >c

ε1

2

T (ω). (22.95)

The resonance value (qRes⊥ ) of q⊥, given by the condition

D(qRes⊥ , ω) = 0, (22.96)

hence is

qRes⊥ (ω;V ) =

ω

V

[

(

V

c

)2

εT (ω)− 1

]1

2

. (22.97)


> >

z (t)0

shockq

qC

z (t )11

FIGURE 22.2Transverse Cerenkov radiation. A charged point-particle moves through a homogeneousdielectric medium with a constant velocity numerically exceeding the speed of light in themedium. At times t1(< t) and t, the particle is at the positions z1 and z0, respectively. Thedirections of the resonant wave vectors of the transverse vector potential of the particle forma circular cone with semiangle (Cerenkov angle) θC . The phase fronts of the field waveletscoming from the various particle positions (z < z0) also form a circular (shock) cone withsemiangle θshock = π/2− θC .

The main contribution to the transverse vector potential in Eq. (22.92) comes from q⊥-valuesin the vicinity of qRes

⊥ [also if an assumed small imaginary part of εT (ω) is included]. Theresonance condition in Eq. (22.97) makes the bridge to the remarks made in the beginningof Subsec. 22.2.2; see in particular Eq. (22.58). The directions of all resonant wave vectorsform a circular cone about the rectilinear trajectory of the particle (the z-axis, here) [Fig.22.2]. The cone’s semiangle θC , called the Cerenkov angle [101, 91, 172], is given by

θC =arccos

ωV

[

(

ωV

)2+(

qRes⊥)2]

1

2

=arccos( c

Vε− 1

2

T (ω))

. (22.98)

For V → c, θC approaches its maximum (max) value

θmaxC = arccos

(

ε− 1

2

T (ω))

. (22.99)

Since the phase speed of the resonant modes is

ω

qRes(ω)=

ω

(

ωV

)2[

(

Vc

)2εT (ω)− 1

]

+(

ωV

)2

1

2

=c

ε1

2(ω)

T

, (22.100)

a phase front emitted when the particle was at the position z1 [at t = t1] has reached

a distance c(t − t1)/ε1/2T (ω) away when the particle reaches the position z0(t) = V t. As


illustrated in Fig. 22.2, the circular shock cone therefore has a semiangle θshock given by

θshock = arcsin

[

c(t− t1)

ε1

2

T (ω) (z0(t)− z1(t1))

]

= arcsin( c

Vε− 1

2

T (ω))

=π

2− θC , (22.101)

as expected.

22.2.4 Momenta associated to the transverse and longitudinal parts ofthe Cerenkov field

It is known that the momentum (P(t)) carried by the electromagnetic field is given by

P(t) = ǫ0

∫ ∞

−∞E(r, t)×B(r, t)d3r (22.102)

Using the Parseval–Plancherel identity [53], P(t) can be expressed as an integral over wave-vector space, i.e.,

P(t) = ǫ0

∫ ∞

−∞E(q; t)×B∗(q; t)

d3q

(2π)3. (22.103)

If one divides the electric field into its transverse and longitudinal parts, E = ET +EL, themomentum of the field appears as a sum, viz.,

P(t) = Ptrans(t) +Plong(t), (22.104)

where

Ptrans(t) [≡ PTT (t)] = ǫ0

∫ ∞

−∞ET (q; t)×B∗(q; t)

d3q

(2π)3(22.105)

is the contribution associated to the transverse electric field, and

Plong(t) [≡ PLT (t)] = ǫ0

∫ ∞

−∞EL(q; t)×B∗(q; t)

d3q

(2π)3(22.106)

is the part related to the longitudinal electric field. As indicated by the notation in thesquare brackets, the integrand in Ptrans is the vectorial product of two transverse vectorfields (ET and B), and that in Plong contains the product of a longitudinal (EL) and atransverse (B) vector field.

In order to determine the two parts of the field momentum associated with the Cerenkoveffect, one must calculate ET (q; t), EL(q; t), and B(q; t) for the uniformly moving point-particle. Let us consider first the transverse electric field. In the (ω,q)-domain it is givenby

ET (q, ω) =iωAT (q, ω)

=2πiµ0Qω(U− κκ) ·Vδ

(

q‖V − ω)

q2 −(

ωc

)2εT (q, ω)

. (22.107)

The last member of Eq. (22.107) is obtained by combining Eqs. (22.72) and (22.82). The


presence of the delta function δ(q‖V −ω) in Eq. (22.107) immediately allows one to calculatethe Fourier integral

ET (q; t) =1

2π

∫ ∞

−∞ET (q, ω)e

−iωtdω. (22.108)

Thus,

ET (q; t) =iµ0Qq‖V

N(

q, q‖V) (U− κκ) ·Ve−iq‖V t, (22.109)


N(q, q‖V ) = q2 −(

q‖V

c

)2

εT(

q, q‖V)

. (22.110)

The longitudinal part of the electric field is given by

EL(q, ω) =2πQ

iǫ0ω

κκ ·Vδ(

q‖V − ω)

εL(q, ω)(22.111)

in the wave-vector-frequency domain, as the reader may verify combining Eq. (22.75) andthe longitudinal part of Eq. (22.82). In the wave-vector-time domain one then obtains

EL(q; t) =Qκκ ·Viǫ0q‖V

e−iq‖V t

εL(

q, q‖V) . (22.112)

From

B(q, ω) = iq×AT (q, ω)

= 2πiµ0Qq× (U− κκ) ·Vδ

(

q‖V − ω)

q2 −(

ωc

)2εT (q, ω)

, (22.113)

we immediately find the following expression for the magnetic field in the (q; t)-domain:

B(q; t) =iµ0Qq

N(

q, q‖V)κ× (U− κκ) ·Ve−iq‖V t, (22.114)

or simpler

B(q; t) =iµ0Qq

N(

q, q‖V)κ×Ve−iq‖V t. (22.115)

Having obtained the expressions for ET (q; t), EL(q; t) and B(q; t), we now calculatethe so-called transverse (TT) and longitudinal (TL) field momentum densities in the (q; t)-domain, viz.,

gTT (q) = ǫ0ET (q; t)×B∗(q; t), (22.116)

and

gLT (q) = ǫ0EL(q; t)×B∗(q; t). (22.117)

As indicated by the notation, these vectors are manifest time independent, a fact we shall


comment on soon. The vectorial product entering the TT-momentum density can be rewrit-ten as follows:

V · (U− κκ)× (κ×V) = V × (κ×V)− (κ ·V)κ× (κ×V)

= κV 2[

1− (κ · eV)2]

= κV 2 sin2 χ, (22.118)

where eV = V/V (= ez here), and χ is the angle between κ and eV. We then get

gTT (q) =µ0

(

Q

c

)2 q‖qV3 sin2 χ

∣

∣N(

q, q‖V)∣

∣

2κ

=µ0

(

Qq

c

)2

V 3 cosχ sin2 χ∣

∣N(

q, q‖V)∣

∣

2κ. (22.119)

It appears from Eq. (22.119) that gTT (q) is in the κ-direction, as one perhaps might haveguessed because the TT-momentum density relates to the transverse part of the electro-magnetic field. Parallel (χ = 0) and perpendicular (χ = π/2) to the particle trajectory gTT

is zero. The transverse part of the field momentum,

PTT =

∫ ∞

−∞gTT (q)

d3q

(2π)3, (22.120)

has a major contribution from the vicinity of the minimum in |N(q, q‖V )|. Neglecting thespatial dispersion in εT (q, q‖V ), and assuming εT to be real, the possible resonance angle,χRes, corresponding to a zero in N and a fixed q, is determined by the condition

cosχRes =c

Vε− 1

2

T

(

0, qV cosχRes)

. (22.121)

If the frequency dispersion in the transverse dielectric function is negligible, there is only

one angle of resonance χRes = θC = arccos[c/(V ε1/2T )]. Physically, it is obvious that PTT

must be time independent for a particle moving with uniform speed, and therefore also thevarious q-components of gTT (q) do not depend on t. Since

(κ ·V)κ× (κ×V) = V 2 cosχ (κ cosχ− eV) , (22.122)

the reader may readily show that the LT-momentum density is given by

qLT (q) =µ0Q

2V

εL(q, q‖V )N∗(q, q‖V )(eV − κ cosχ) . (22.123)

We see from this equation that gTL lies in the plane spanned by κ and eV. In the particlepropagation direction (κ ‖ eV), gTL = 0. A new feature appears in the longitudinal fieldmomentum,

PLT =

∫ ∞

−∞gLT (q)

d3q

(2π)3. (22.124)

Hence, a resonance contribution to PLT appears from the vicinity of a minimum in|εL(q, q‖V )|. If one assumes that the longitudinal dielectric function is real, the condition

εL(q, q‖V ) = 0 (22.125)


relates to Landau shock waves [92], which may be generated when the particle speed exceedsa relevant phase velocity of “sound” in the medium [92]. When a charged particle moves(slightly) above and parallel to a flat metal (jellium) surface with a speed exceeding one ofthe characteristic phase velocities of the surface modes resonant surface dressing may leadto combined Cerenkov–Landau surface shock waves [113, 109]. For εT and εL real, the ratiobetween the numerical magnitudes of the two parts of the field momentum density is givenby

∣

∣

∣

∣

gTT (q)

gLT (q)

∣

∣

∣

∣

=

(

V

c

)2 εL(

q, q‖V)

sinχ cosχ

1−(

Vc

)2εT(

q, q‖V)

cos2 χ, (22.126)

a result I encourage the reader to show.

22.2.5 Screened canonical particle momentum

An interesting reinterpretation of the physical meaning of the longitudinal part of the fieldmomentum appears if one rewrites Eq. (22.106) in a certain manner, as we now shall realize.By combining the equation of continuity for the external current density, viz.,

qκ · JextL (q, ω) = ωρext(q, ω), (22.127)

in the wave-vector-frequency domain, and Eq. (22.75) [with EL(q, ω) = κκ ·EL(q, ω)] oneobtains

EL(q, ω) =κρext(q, ω)

iǫ0qεL(q, ω). (22.128)

The quantity

ρsc(q, ω) =ρext(q, ω)

εL(q, ω)(22.129)

may be recognized by the reader as the longitudinally screened (sc) external charge density[in the (q, ω)-domain]. In the (q; t)-domain

ρsc(q; t) =

∫ ∞

−∞

ρext(q, ω)

εL(q, ω)e−iωt dω

2π

=Qe−iq‖V t

εL(

q, q‖V) , (22.130)

and from this result it appears that the longitudinal electric field EL(q; t) is given by

EL(q; t) =κ

iǫ0qρsc(q; t). (22.131)

If additionally, one expresses the magnetic field in terms of the transverse vector potential,that is

B(q; t) = iqκ×AT (q; t), (22.132)

a combination of Eqs. (22.106), (22.131), and (22.132) enables one to obtain the followingexpression for the longitudinal part of the field momentum

PLT =

∫ ∞

−∞ρsc(q; t)A∗

T (q; t)d3q

(2π)3

=

∫ ∞


3r, (22.133)


where the last member follows from the Parseval–Plancherel identity.In a certain sense PLT represents the field part of an effective (eff) canonical particle

momentum, peff , given by

peff = π +

∫ ∞


3r, (22.134)

where π is the particle’s kinetic momentum. At the quantum physical level peff turnsinto an effective particle momentum operator (peff ⇒ peff ). In vacuum, where ρsc(r, t) =ρext(r, t), we obtain for a point-particle located instantaneously at r0(t) a canonical mo-mentum [peff = p]

p = π +QAT (r0(t)). (22.135)

The connection in Eq. (22.135) is the fundamental one appearing in classical and quantumelectrodynamics in the minimal coupling scheme [53].

Part VI

Photon source domain andpropagators

23

Super-confined T-photon sources

In electromagnetics (optics, e.g.) the concept spatial resolution plays a key role in manyproblems, as it is well-known [155, 38, 54, 176, 13, 58]. Despite the concept’s importance,it does not enter any of the fundamental equations of electrodynamics [Maxwell-Lorentzequations + Schrodinger (Dirac) equation] in a direct manner, so to speak. In the literaturemany different definitions of the limit for the spatial resolution are (and have been) used, andintense discussions concerning “the best definition” have taken place among researchers overthe years. Not least from an experimental point of view it is of importance to understandwhether there exists a fundamental limit for how high a spatial resolution one can obtain.For an experimentalist seeking to resolve finer and finer details in a given structure (of,e.g., a solid) it is obvious that all probing techniques run into the spatial resolution limitproblem in the end.

It is clear that one cannot come up with a precise number (length) for the linear mea-sure of the highest achievable resolution. Thus, the existence of such a number would atleast require that sharp non-differentiable structural transitions existed in matter (systemof massive particles). Sharp transitions are forbidden by quantum mechanics. The non-existence of a specific number does not mean that infinitely high resolution can be obtainedtheoretically, as it sometimes has been claimed.

It is not a purpose of this book to give a detailed account of the spatial resolutionproblem in electrodynamics (physical optics), but, in view of our fundamental interest inthe spatial photon localization problem, it is useful to make the bridge between the twoproblems. It is possible to address the fundamental aspects of the spatial resolution problemby studying the so-called two-point resolution problem (for identical sources). Roughlyspeaking, the limit for the two-point resolution in classical far-field optics is given by thefamous Rayleigh criterion [1, 201], but we have learned, not least from studies in near-fieldoptics [54, 176, 227, 10, 192, 143, 138, 175], that it is possible to obtain spatial resolutions(substantially) beyond the Rayleigh limit. The history of near-field optics is described in[174]. Before I enter a brief discussion of the resolution problem from the perspective ofspatial photon localization let me make the following general comments: (i) A fundamentalunderstanding of the spatial resolution problem in optics cannot be achieved by a studyof free electromagnetic fields. The field-matter interaction always plays a decisive role. (ii)Quantum physics is necessary for the understanding. (iii) Measurement theory in quantumelectrodynamics is needed in (the last stage of) the analysis.

In view of point (iii), it is in a sense obvious that resolution experiments carried out withdifferent, necessarily always classical (macroscopic) equipment (Bohr: The unambiguousinterpretation of any measurement must be essentially framed in terms of the classicalphysics theories [32, 33, 34]), when taken to their limit of ability lead to different results(Bohr: The quantum mechanical formalism permits well-defined applications referring onlyto closed phenomena [29, 30, 31]).

We have realized in Part V that a transverse photon emitted from a given current densitydistribution, J(r, t), has the related transverse current density distribution, JT (r, t), as itssource. Hence, the rim zone of the given electronic source constitutes an indispensable partof the T-photon’s spatial source domain. In a quantum statistical sense, the T-photon is

277


“born” for certainty when the source has stopped its activity (J = 0). During the “birth pro-cess,” JT (r, t) is given as weighted sum of transverse transition current densities, JT

I→J (r, t)belonging to the various many-body transitions (I, J). The complex weight factor belongingto a given I → J transition is given by the product cI(t)c

∗J (t), where cI(t) and cJ(t) are the

amplitude coefficients relating to an expansion of the prevailing many-body wave functionin a given many-body stationary state wave function basis set (I, J, . . . ). The photon wavefunction that (with a certain probability) exists at time t consists of a superposition ofwavelets emerging from the various spatial points in the JT (r, t)-distribution. Each waveletpropagates outward from the given space-point in a manner given by the scalar propagator,i.e., isotropically and with the vacuum speed of light.

To simplify the discussion of the spatial resolution problem, yet without losing the qual-itative conclusion, let us assume that the source is a single two-level atom, and that thetransition between the levels is electric-dipole (ED) allowed. A transverse photon emittedspontaneously in a downward transition turns out to be generated in the near-field zone,which here has algebraic R−3-confinement. If one now places two identical two-level atomsin a certain distance from each other, it is qualitatively clear that one in a detection processcannot tell from which atom the photon came if the atoms are in near-field contact. Twoneutral atoms in near-field contact interact electrodynamically (here via ED-ED interaction,essentially). This interaction results in a dressing of the states of the pair, and in generalthe atom pair must be considered as a two-atom system. For our discussion of the spatialresolution problem in optics this dressing is of no concern when the interatomic coupling isweak [123, 194].

It appears from the previous considerations that the spatial two-point resolution inprinciple may be improved if one is able somehow to reduce the size of the rim zone. If thetwo-level transition is an electric quadrupole (or magnetic dipole) transition, the profile ofthe rim zone would be of the R−4-type. Although this is an improvement in relation to theED-case, the confinement of the rim zone still is of the algebraic type, as it will be alsofor higher-order multipole transitions. For transitions of the nth order (ED↔ n = 1) theconfinement is of the R−n−2-type.

For atoms and mesoscopic objects a tremendous improvement might be obtained if therim zone did not exist. In that case, the absence of a rim zone would imply that

JT (r, t) = J(r, t), (23.1)

or equivalently JL(r, t) = 0. In consequence one would have ∇ · J = ∇ · JL = 0, indicatingthat one would need that the microscopic source current density distribution flow in anincompressible manner during the T-photon emission process. From the equation of conti-nuity ∇ · J + ∂ρ/∂t = 0, it then appears that one must search for processes in which themicroscopic charge density is globally time independent during the photon birth process,i.e.,

∂

∂tρ(r, t) = 0, ∀r. (23.2)

When Eq. (23.1) is (at least approximately) satisfied we say that T-photon source is super-confined [123, 117, 124]. In Chapts. 24 and 25 we shall by two qualitatively different examplesfrom atomic optics see that the source confinement condition in Eq. (23.1) indeed can besatisfied, approximately in the first example, and exactly in the second. With a photonsource domain identical to the electronic current density distribution the rim-zone confine-ment will be exponential, essentially. The possibility of exponential photon confinement hasalso been pointed out from a somewhat different perspective by Bialynicki–Birula [17, 18].

For a qualitative discussion it is sufficient to consider a single-electron atom, and treat

Super-confined T-photon sources 279

the electromagnetic field as a classical quantity. In the nonrelativistic regime a calculationshows that the gauge independent current density can be divided into two parts,

J(r, t) = Jpar(r, t) + Jdia(r, t), (23.3)

which separately are gauge independent [127]. The diamagnetic (dia) part is given by

Jdia(r, t) = −q2

mAT (r, t)|ψ(r, t)|2 (23.4)

for a particle (mass m and charge q) with wave function ψ(r, t). In the Coulomb gauge,the so-called paramagnetic (par) part, Jpar(r, t), has no explicit dependence on the vectorpotential, A = AT . In many situations the diamagnetic contribution to J(r, t) is negligible,but when the gauge invariance is crucial for an understanding Jdia(r, t) must be kept evenif it is small.

It is illustrative to study the photon source domain related to the hydrogen 1s ↔ 2pztransition, in the case where the vector potential is so weak that the diamagnetic effectcan be neglected [112]. The spatial form of the photon source domain is identical to thatof the transverse part [Jpar

T (trans)] of the paramagnetic transition (trans) current density[Jpar(trans)]. In Sec. 24.2 it is shown that the radial (r) dependence of Jpar

T (trans) has theform

F (br) = f(br)e−br + (br)−3, (23.5)

where b = 3/(2a0), a0 being the Bohr radius. For the radial component [of JparT (trans)],

f(br) ≡ fr(br) = (br)−1 + 2(br)−2 + 2(br)−3, and for the tangential component f(br) ≡fθ(br) = 1 + fr(br). It appears from Eq. (23.5) that F (br) → (br)−3 for br → ∞. Thisasymptotic behavior is in agreement with our expectations since 1s↔ 2pz is an ED-allowedtransition.

In Sec. 24.3, we turn our attention toward the 1s↔ 2s transition in hydrogen. A calcula-tion shows that the transverse part of the paramagnetic transition current density vanishesfor this transition, which might be called a breathing mode transition. In consequence noT-photons can be absorbed/emitted in the 1s↔ 2s transition via the paramagnetic mecha-nism. Our calculation confirms the well-known result that paramagnetic transitions betweens-states are forbidden in all multipole orders. Since all s-orbitals have rotational invariance,it is obvious from the outset that a T-photon cannot be coupled to a breathing mode. Thetransverse vector potential present in the diamagnetic transition current density breaks therotational invariance, however, and this allows a coupling to T-photons. Under the assump-tion that the transverse vector potential does not vary across the atomic domain, a lengthycalculation ends up with the following quite simple result for the transverse part of thediamagnetic 1s→ 2s transition current density:

JdiaT (r, t) = Ke−br [br (U− erer)− 2U] ·AT (t), (23.6)

where K is a constant. It appears from Eq. (23.6) that the photon source domain is expo-nentially confined. Although Jdia

L (r, t) is not exactly zero, the main point is that the photonsource domain (Jdia

T ) and the diamagnetic electronic current density (Jdia) both have arange given by ∼ exp(−br), qualitatively.

In Chapt. 25 the current density of a spin-1/2 particle is calculated in the frameworkof relativistic quantum mechanics, with the aim of searching for a super-confined T-photonsource related to the spin dynamics.

The coupling of a relativistic charged spin-1/2 particle to a classical electromagnetic


field, described by the four-potential Aµ, is obtained by making the minimal couplingsubstitution,

pµ ⇒ pµ − qAµ, (23.7)

in the covariant form of the free Dirac equation. From the coupled wave equation, writtenin slash (/) notation as

(p− qA+mc)ψ = 0, (23.8)

one obtains the following well-known result for the charge (q) probability four-current den-sity:

Jµ = cqψγµψ, (23.9)

where ψ is the Dirac conjugate wave function, and γµ the usual γµ-matrices. Note thatthe four-potential does not appear explicitly in Eq. (23.9). In Sec. 25.3 we derive the Gordondecomposition of the Dirac four-current density. Although this decomposition, derived byGordon long ago [82], can be found in many textbooks, details of its derivation are seldomgiven. In [86] the Gordon decomposition is derived for the free Dirac equation. In the Gordondecomposition a part, with component µ given by

Jµspin =

iq

2mpν[

ψSµνψ]

, (23.10)

relates directly to the spin via Sµν = (i/2)[γµ, γν ]. In the weakly relativistic (WR) regime,one obtains from Eq. (23.10) a spin current density three-vector

Jspin,WR =q~

2m∇×

(

φ†σφ)

, (23.11)

where σ is the Pauli (spin) vector, and φ the Pauli two-component spinor. In the WR-regime the spin current density hence is a transverse vector field, Jspin,WR = JT

spin,WR, andconsequently ∂ρspin/∂t = 0. Hence a T-photon emitted in a pure WR spin transition willcome from a super-confined source domain. In a 1s state all spatial dynamics disappears sothat ∂ρ/∂t = ∂ρspace∂t = 0. We are thus left with a possible spin dynamics of the electronand proton in hydrogen. We finish Chapt. 25 with a calculation of the electron spin currentdensity associated to transition between hyperfine-split 1s levels. In our calculation theelectron spin is coupled not only to the spin of the nucleus but also to a time-independentexternal magnetic field.

In Chapt. 26 we analyze the outward propagation of the four-potential Aµ(x) from agiven electronic source (Jµ(x)), starting from the massless Feynman photon propagator.Being interested in the emission of T-photons, special emphasis is devoted to the transversepart of the Feynman propagator. We carry out a study of T-photon time-ordered correlationevents as these appear in a description relating directly to the mean position state |R(r, t)〉for the transverse photon. It turns out from the analysis that the time-ordered (T · · · )T-photon correlation matrix is given by the important formula [127]

〈0|T

A(+)T (x)A

(−)T (x′)

|0〉

=~

iǫ0c2(θ(t− t′)GT (R, τ) + θ(t′ − t)GT (−R,−τ)) , (23.12)

where GT (R, τ) is the transverse photon propagator, and θ(t − t′) the unit step function.The result in Eq. (23.12) is easily extended to the covariant level (Sec. 26.3).

Super-confined T-photon sources 281

We finish Chapt. 26, with a brief review of (i) the covariant quantization procedure forthe electromagnetic field, (ii) the relation between the covariant correlation matrix and theFeynman photon propagator, and (iii) a bit of the physics related to longitudinal and scalarphoton propagators.

24

Transverse current density in nonrelativistic quantum

mechanics

24.1 Single-particle transition current density

We have seen that the source domain of a transverse photon, a quantum physical objectdescribed in this book via the transverse part of the vector potential (AT (r, t)), may beidentified with the transverse part of the charged-particle current density (JT (r, t)). Ingeneral, many-body quantum physics is needed to determine JT (r, t). However, it is possibleto illustrate a number of important principles related to the characterization of the photonsource domain by limiting oneself to a study of the single-particle (electron) current densityconcept in nonrelativistic quantum mechanics.

The starting point for our analysis thus is the Schrodinger equation for the particle wavefunction in the r-representation, ψ(r, t) = 〈r|ψ〉(t). Obviously, we need the dynamical equa-tion for the particle wave function in the presence of an electromagnetic field. In quantumelectrodynamics the field enters the description via the (selfconsistent) potential Aµ(r, t),and since the particle canonical momentum operator

p = π + qA, (24.1)

and the vector potential operator, A, both are gauge dependent, a gauge must be chosen.In the framework of nonrelativistic quantum mechanics it is useful for us to work in theCoulomb gauge. We shall simplify the following considerations further by assuming that theelectromagnetic field can be treated as a classical quantity, A(r, t) ⇒ A(r, t). In view of theremarks above, the Schrodinger equation for our spinless particle of mass m and charge qtakes the form

[

1

2m(p− qAT )

2 + qφ

]

ψ(r, t) = i~∂

∂tψ(r, t), (24.2)

where p = −i~∇ is the particle momentum operator in the r-representation, and φ = φ(r, t)is the Coulomb potential of the charge. The coupling to the dynamical degrees of freedom ofthe electromagnetic field occurs via the purely transverse vector potentialA(r, t) = AT (r, t),a gauge invariant quantity.

In the nonrelativistic domain particles cannot be created nor destroyed, and particleconservation here is expressed in terms of the equation of continuity,

∇ · J(r, t) + ∂

∂tρ(r, t) = 0, (24.3)

for the charge probability density

ρ(r, t) = q |ψ(r, t)|2 . (24.4)

283


On the basis of the Schrodinger equation in (24.2) one is led to the following expression forthe charge probability current density [127, 53]:

J(r, t) = Jpar(r, t) + Jdia(r, t), (24.5)

where

Jpar(r, t) =q~

2mi(ψ∗(r, t)∇ψ(r, t)− ψ(r, t)∇ψ∗(r, t)) (24.6)

is the paramagnetic (par) part of the current density, and

Jdia(r, t) = −q2

mAT (r, t)|ψ(r, t)|2 (24.7)

its diamagnetic (dia) part. The requirement that the Schrodinger equation must be form-invariant against field gauge transformations of the form given by Eqs. (10.5) and (10.6)necessitates that the wave function must transform according to [6]

ψ′(r, t) = exp

(

iq

~χ(r, t)

)

ψ(r, t). (24.8)

A change of the local phase of the wave function does not change the charge and currentprobability densities. Had we used another gauge in the Schrodinger equation, Eq. (24.7)would have been replaced by

JA(r, t) = −q2

mA(r, t)|ψ(r, t)|2, (24.9)

a gauge dependent quantity. The remaining part of the current density would still have theform given in Eq. (24.6), but be gauge dependent (since the canonical momentum is gaugedependent). If one transfers the part proportional to AL in Eq. (24.9) to Eq. (24.6)

JA(r, t) ⇒ Jdia(r, t), (24.10)

Jpar(r, t) ⇒ q~

2mi(ψ∗(r, t)∇ψ(r, t)− ψ(r, t)∇ψ∗(r, t))

− q2

mAL(r, t)|ψ(r, t)|2. (24.11)

Since AT is gauge invariant the new division of the total current density makes each partof J(r, t) gauge independent.

Let us assume that we have obtained the eigenfunctions, ui(r), and the associatedenergy eigenvalues, Ei, to the time-independent Schrodinger equation,

(

− ~2

2m∇2 + qφ

)

ui(r) = Eiui(r), (24.12)

which one would have in the absence of the transverse electromagnetic field, AT (r, t), and letus assume that it is sufficient to consider only the discrete part of the eigenvalue spectrum.For the discussion of the photon source problem related to electron transitions betweenbound states in the hydrogen atom the continuum part of the energy spectrum is notneeded. Taking ui(r) as a complete orthonormal set of basis functions, the general wavefunction in Eq. (24.2) can be expanded in terms of the ui(r)’s, that is

ψ(r, t) =∑

i

ci(t)ui(r), (24.13)

Transverse current density in nonrelativistic quantum mechanics 285

where the expansion coefficients ci(t) satisfy the condition

∑

i

|ci(t)|2 = 1 (24.14)

if the wave function ψ(r, t) is normalized. In principle at least, the time dependence of theci(t)’s can be determined solving the Schrodinger equation in Eq. (24.2), provided AT (r, t)is a prescribed quantity. If AT (r, t) is a dynamical (vectorial) variable the coupled set ofthe Schrodinger and Maxwell–Lorentz equations must be solved to obtain the ci(t)’s.

As we shall realize soon, much information can be obtained on the spatial extension of thephoton source domain without knowing the time dependence of the expansion coefficients.If one inserts the expansion in Eq. (24.13) into Eqs. (24.6) and (24.7) it appears that thecurrent density in Eq. (24.5) can be written in the form

J(r, t) =∑

i,j

ci(t)c∗j (t)Ji→j(r, t), (24.15)

where

Ji→j(r, t) =q~

2mi

(

u∗j (r)∇ui(r)− ui(r)∇u∗j(r))

− q2

mAT (r, t)ui(r)u

∗j (r) (24.16)

is the so-called transition current density from state i to state j. The opposite (j →i)transition current density is given by

Jj→i(r, t) = J∗i→j(r, t), (24.17)

as the reader may show from Eq. (24.16). The transition current density Ji→j(r, t) is thematrix element of the one-particle current density operator between the “initial (i)” and“final (j)” quantum states ui(r) and uj(r). For what follows, it is useful also to divide thetransition current density into para- and diamagnetic parts [Ji→j = Jpar

i→j + Jdiai→j ] given by

Jpari→j(r) =

q~

2mi

(

u∗j (r)∇ui(r) − ui(r)∇u∗j (r))

, (24.18)

and

Jdiai→j(r, t) = −q

2

mAT (r, t)ui(r)u

∗j (r). (24.19)

The paramagnetic part is always time independent, as indicated.In many situations the diamagnetic contribution to the current density is so small com-

pared to the paramagnetic contribution that it can be neglected. However, one should beaware of the fact that it is the transverse (divergence-free) part of the current density whichis identified with the source domain of the transverse photon, and it may happen that thetransverse part of the paramagnetic current density is zero even though the paramagneticcurrent density itself does not vanish. I shall discuss this situation in relation to the analysisof the photon source domain for what I call breathing mode transitions in the hydrogen atom.For these transitions the photon source domain must be identified with the transverse partof the diamagnetic probability current density. In general, the transverse photon’s sourcedomain is given by the transverse part of Eq. (24.15), i.e.,

JT (r, t) =∑

i,j

ci(t)c∗j (t)J

Ti→j(r, t), (24.20)


where, if wished, the transverse part of the transition current density, JTi→j(r, t), can be

divided into para- and diamagnetic parts. Before we specialize to the hydrogen atom, let megive a few remarks on the terms in Eq. (24.15) where i = j. The diamagnetic “transition”current density Jdia

i→i(r, t) = (−q2/m)AT (r, t)|ui|2 never vanishes. If one writes the wavefunction ui(r) as a product of its modulus |ui(r)| and a phase factor with (real) argumentαi(r), i.e.,

ui(r) = |ui(r)|eiαi(r), (24.21)

the paramagnetic “transition” current density Jpari→i(r) takes the form

Jpari→i(r) =

q~

m|ui(r)|2∇αi(r). (24.22)

It appears from Eq. (24.22) that Jpari→i(r) vanishes only if the phase is space independent.

Let us assume that the particle is in the stationary state m. (No electromagnetic fieldpresent.) In Eq. (24.15) one thus has ci(t) = 1 for i = m and ci(t) = 0 for i 6= m. The timeindependent current density in the stationary state therefore is given by J(r) = Jpar

m→m(r),and thus nonvanishing when ∇αm(r) 6= 0.

24.2 The hydrogen 1s ⇔ 2pz transition

A quantitative determination of the spatial extension of the source (sink) domain for emis-sion (absorption) of transverse photons requires a calculation of the transverse parts of thetransition current densities belonging to the relevant electronic transitions, cf. Eq. (24.20).In order to illustrate the physical insight gained by the introduction of the photon sourcedomain concept, it is sufficient to limit oneself to a single pair of levels. In the spirit of our(effective) one-particle approach, it is natural to focus on the hydrogen atom, and specif-ically the transition between the 1s and 2pz stationary states. We shall assume that theprevailing transverse vector potential is so weak that the diamagnetic transition currentdensity between these states can be neglected.

In spherical (r, θ, φ)-coordinates with the polar axis along the z-direction (polar angleθ), the 1s and 2pz wave functions are

u1s(r) =1

√

πa30exp

(

− r

a0

)

, (24.23)

u2pz(r) =

1

4√

2πa30

r cos θ

a0exp

(

− r

2a0

)

, (24.24)

a0 = 4πǫ0~2/(me2) being the Bohr radius. The electron charge is denoted by −e. By

inserting Eqs. (24.23) and (24.24) into Eq. (24.18) one obtains a paramagnetic transitioncurrent density

Jpar1s→2pz

(r) ≡J

=A

[

er

(

2 +r

a0

)

cos θ − 2eθ sin θ

]

e−br, (24.25)


where

A =e~

16i√2πma40

, (24.26)

b =3

2a0. (24.27)

As usual, er and eθ are the local radial and polar unit vectors. To lighten the notationthe superscript (par) and subscript (1s → 2pz) on the transition current density and itstransverse and longitudinal parts are omitted in this subsection.

The rotational symmetry of the orbitals around the polar axis implies that the transverseand longitudinal parts of the transition current density in Eq. (24.25) must have the form

JT,L = JT,Lr (r, θ)er + JT,L

θ (r, θ)eθ. (24.28)

A rather lengthy calculation gives the following result for the transverse coefficients [112]

JTr (r, θ) = −16A

3cos θ

e−br

[

1

br+

2

(br)2+

2

(br)3

]

− 2

(br)3

, (24.29)

and

JTθ (r, θ) = −8A

3sin θ

e−br

[

1 +1

br+

2

(br)2+

2

(br)3

]

− 2

(br)3

. (24.30)

The knowledge of the transverse coefficients allows one to obtain the longitudinal coefficientby subtraction [JL

r = Jr − JTr , JL

θ = Jθ − JTθ ]. Hence

JLr (r, θ) =

16A

3cos θ

e−br

[

3 + br

8+

1

br+

2

(br)2+

2

(br)3

]

− 2

(br)3

, (24.31)

and

JLθ (r, θ) =

8A

3sin θ

e−br

[

1

4+

1

br+

2

(br)2+

2

(br)3

]

− 2

(br)3

. (24.32)

The results for the coefficients may be obtained in three steps: (i) Carry out a spatialFourier integral transformation of J(r), J(r) ⇒ J(q). (ii) Multiply J(q) by the transverseand longitudinal delta functions representatives in Fourier space to obtain JT (q) = (U −κκ) · J(q) and JL(q) = κκ · J(q). (iii) Take these vectors back to direct space by inverseFourier integral transformation, JT (q) ⇒ JT (r) and JL(q) ⇒ JL(r). A similar scheme to theone outlined in (i)-(iii) is used in Sec. 24.3 in connection to the analysis of the photon sourcedomain responsible for emission (absorption) from (in) the 1s ⇔ 2s breathing transitionin hydrogen. In this connection more intermediate results are given. Most of these may beuseful if the reader tries to obtain Eqs. (24.29)-(24.32) herself.

The transition current density [Eq. (24.25)] essentially is exponentially confined [∼ e−br],and the characteristic decay length [b−1] is of the order of the Bohr radius, a0. Roughlyspeaking the spatial domain of the transition current density is the confinement region ofthe electron in the hydrogen atom. The transverse and longitudinal parts of the transitioncurrent density has a long-range tail with an ∼ (br)−3 profile [see Eqs. (24.29)-(24.32)].These parts also contain contributions which decay as ∼ e−br (multiplied a sum of certainpowers of br). The polar angle dependence is cos θ [for jTr and JL

r (and Jr)] and sin θ [for jTθand JL

θ (and Jθ)]. The spatial form of the source (sink) domain for emission (absorption) oftransverse photons in the 1s⇔ 2pz transition hence is given by Eqs. (24.28)-(24.30). A few


Bohr radii from the nucleus we are effectively in the rim zone, and here the T- and L-partsof the transition current density take the asymptotic (r → ∞) forms

JTr (r → ∞) = −JL

r (r → ∞) =32A

3(br)

−3cos θ, (24.33)

JTθ (r → ∞) = −JL

θ (r → ∞) =16A

3(br)

−3sin θ. (24.34)

The result in Eqs. (24.33) and (24.34) illustrates in a beautiful manner the basis for thefact that the 1s ↔ 2pz transition spectroscopically is classified as an electric dipole (ED)allowed transition.

For a given current density distribution (possibly time dependent), J(r, t), an associatedelectric dipole current density, JED(r, t), is defined by [101, 127]

JED(r, t) = J(0)(t)δ(r), (24.35)

where

J(0)(t) =

∫ ∞

−∞J(r, t)d3r (24.36)

is the spatial zero-order moment of the current density distribution. For the (time indepen-dent) transition current density in Eq. (24.25) one obtains a zero-order moment

J(0) =64π

3Ab−3ez, (24.37)

as the reader may show (use spherical coordinates for the integration). Outside the nucleus(origo), the transverse and longitudinal parts of the ED current density are given by

JTED(r) = −JL

ED(r) = δT (r) · J(0), (24.38)

where

δT (r) =3erer −U

4πr3, r 6= 0, (24.39)

is the transverse delta function in spherical contraction [see Eq. (22.4)]. Combining Eqs.(24.37)-(24.39), and using that er · ez = cos θ, one obtains for r 6= 0

JTED(r) =− JL

ED(r)

=32

3A (br)

−3

(

er cos θ +1

2eθ sin θ

)

, (24.40)

i.e., precisely the asymptotic result given in Eqs. (24.33) and (24.34), as one perhaps mighthave expected. The ED-current densities JT

ED(r) and JLED(r) [as well as JED(r)] are singular

at r = 0. Although the expressions in Eqs. (24.29)-(24.32) at first sight might appearsingular at the origo, this is not the case, of course. Physically, the formulae in Eqs. (24.29)-(24.32) are more satisfactory from a fundamental point of view than those in Eq. (24.40).

To obtain the T- and L-parts of the transition current density at the position of thenucleus we utilize that

limr→0

e−br

[

1

br+

2

(br)2+

2

(br)3

]

− 2

(br)3

= −1

3. (24.41)


The reader may prove the result in Eq. (24.40) using the first four terms in the Taylorexpansion of exp(−br) around br = 0. By means of Eq. (24.40) one readily gets

JTr (r → 0, θ)

cos θ= −J

Tθ (r → 0, θ)

sin θ=

16A

9, (24.42)

and

JLr (r → 0, θ)

cos θ= −J

Lθ (r → 0, θ)

sin θ=

2A

9. (24.43)

Addition of the T- and L-parts gives Jr(r → 0, θ) = 2A cos θ and Jθ(r → 0, θ) = −2A sin θ,in agreement with Eq. (24.25).

24.3 Breathing mode: Hydrogen 1s ⇔ 2s transition

Although the diamagnetic part of the charge probability current density [Eq. (24.7)] isneeded to ensure the gauge invariance of quantum (and semiclassical) electrodynamics, thediamagnetic current density often can be neglected, as we did in the previous section. Beinginterested in general principles for the spatial extension of a photon source, let us nowdiscuss a situation where the photon emission is caused by the diamagnetic term alone.

In a central potential the eigenfunctions of a nonrelativistic spinless particle can be cho-sen as simultaneous eigenfunctions of the mutually commuting operators H (Hamiltonian),L2 (square of the orbital angular momentum, L), and Lz (z-component of L), i.e., [52, 210],

ui(r) ≡ φk,l,m(r) = Rk,l(r)Yml (θ, φ), (24.44)

for the state i = (k, l,m) in spherical coordinates (r, θ, φ). For the Coulomb potential, theradial part of the wave function usually is written in the form Rn,l(r), where n = k + l isthe principal quantum number. States of zero angular momenta [l = 0, m = 0], for whichthe spherical harmonic is Y 0

0 (θ, φ) = (4π)−1/2, are rotationally invariant with real wavefunctions

ui(r) = (4π)− 1

2 Rn,0(r) (24.45)

in the Coulomb potential case, i = (n, 0, 0). Now, we ask ourselves the question: Is itpossible to emit/absorb a photon in a transition between rotationally invariant s-states,[i = (n, 0, 0) ⇔ f = (m, 0, 0)], and what is the spatial extension of the photon source(sink) domain? An electron oscillating, in the quantum statistical sense, between s-states,I say is in a breathing mode. Let us start with the paramagnetic transition current density[Eq. (24.18)]. For rotationally invariant states, only the radial part, er∂/∂r, of the gradientoperator, ∇, comes into play. The paramagnetic transition current density hence has thegeneral form

Jpari→j(r) = erαi→j(r). (24.46)

The symmetry of Eq. (24.46) dictates that the transition current density in wave-vectorspace has the form

Jpari→j(q) =

∫ ∞

−∞Jpari→j(r)e

−iq·rd3r = κβi→j(q), (24.47)


where, as usual, κ = q/q. If in doubt, the reader may convince herself of the correctnessof Eq. (24.47) carrying out the Fourier integral in spherical coordinates with the polar axisalong κ. Since the transverse part of Jpar

i→j(q) vanishes,

JparT,i→j(q) = (U− κκ) · κβi→j(q) = 0, (24.48)

T-photons cannot be emitted in a breathing transition via the paramagnetic mechanism.The calculation above just confirms the well-known result that transitions between s-statesare forbidden in all multipole orders (neglecting the diamagnetic contribution in a stan-dard calculation). It is not surprising that a T-photon cannot be emitted (absorbed) in abreathing mode transition, where the electron flow is in the radial direction in every spacepoint.

Let us now turn to the diamagnetic term [Eq. (24.19)], and assume, with no loss ofgenerality for the present purpose, that the prevailing transverse vector potential is constantacross the atomic domain, i.e., AT (r, t) = AT (t). With this approximation the diamagnetictransition current density between s-states (i, j) with real radial wave functions is given by

Jdiai→j(r, t) [≡ J(r, t)] = −e

2

mAT (t)ui(r)uj(r), (24.49)

where the particle (electron) charge is denoted by e (or −e, if wished). For notationalsimplicity, we denote (as indicated above) the diamagnetic transition current density byjust J(r, t), in the calculations below. Before one inserts the actual s-state wave functionsin the calculation a number of steps can be taken. In the wave-vector-time domain J isgiven by the Fourier integral

J(q; t) = −e2

mAT (t)

∫ ∞

−∞ui(r)uj(r)e

−iq·rd3r. (24.50)

Using again spherical coordinates, the integration of e−iq·r over the entire solid angle [dΩ =sin θdφdθ] gives

∫

4π

e−iq·rdΩ =4π

qrsin(qr), (24.51)

and hereafter

J(q; t) = 2πi

(

e2

m

)

AT (t)Fi→j(q), (24.52)

where

Fi→j(q) =1

q

∫ ∞

0

rui(r)uj(r)(

eiqr − e−iqr)

dr. (24.53)

To determine the T-photon source domain in the space-time representation, we take thetransverse part of J(q; t), viz.,

JT (q; t) = 2πi

(

e2

m

)

AT (t) · (U− κκ)Fi→j(q), (24.54)

back to direct space via the Fourier integral transformation

JT (r, t) = 2πi

(

e2

m

)

AT (t) ·[∫ ∞

−∞(U− κκ)Fi→j(q)e

iq·r d3q

(2π)3

]

. (24.55)


Employing spherical coordinates one can, without specifying the s-orbitals, carry out theintegration over the solid angle in q-space. With the polar axis along r = r/r one obtainsafter having carried out the integration over the azimuth angle

H(qr) ≡ 1

2π

∫

4π

(U− κκ) eiq·rdΩ

=

∫ π

0

[

(U− erer)

(

1− 1

2sin2 θ

)

+ erer sin2 θ

]

eiqr cos θ sin θdθ. (24.56)

The remaining integral is done using the substitution u = cos θ. After some elementarymanipulations one ends up with the result

H(qr) = (U− erer)

[

1

iqr

(

eiqr − e−iqr)

− 1

(iqr)2(

eiqr + e−iqr)

+1

(iqr)3(

eiqr − e−iqr)

]

+ 2erer

[

1

(iqr)2(

eiqr + e−iqr)

− 1

(iqr)3(

eiqr − e−iqr)

]

. (24.57)

With the angular integrations done, the expression for transverse transition current densityis reduced to a single integral over q, viz.,

JT (r, t) =i

π

(

e2

m

)

AT (t) ·

(U− erer)

×∫ ∞

0

[

q

rsin(qr) +

1

r2cos(qr) − 1

qr3sin(qr)

]

Fi→j(q)dq

+ 2erer

∫ ∞

0

[

− 1

r2cos(qr) +

1

qr3sin(qr)

]

Fi→j(q)dq

. (24.58)

The result above splits JT (r, t) into two parts: A part in which the vectorial component of thetransverse vector potential perpendicular to the radial direction appears [∼ AT ·(U−erer)],and a part containing the AT -component along this direction [∼ AT ·erer]. One may writeEq. (24.58) in the slightly abbreviated form

JT (r, t) =i

π

(

e2

m

)

AT (t) ·

(U− erer)

∫ ∞

0

q

rsin(qr)Fi→j (q)dq

+ (U− 3erer)

∫ ∞

0

[

1

r2cos(qr) − 1

qr3sin(qr)

]

Fi→j(q)dq

. (24.59)

The integral formula given for the transverse part of the transition current density in Eq.(24.59) holds for two (i, j) arbitrary s-states, which even may be the same (i = j).

To proceed from here, one must specify the involved states. In the following we considerthe transition from the 1s to the 2s-state. The wave function of the 1s state [u1(r) ≡ u1s(r)]was given in Eq. (24.23), and for the 2s-state one has

u2(r) ≡ u2s(r) =1

2√

2πa30

(

1− r

2a0

)

exp

(

− r

2a0

)

. (24.60)

The integration of

F1s→2s(q) =1

2π√2a30q

∫ ∞

0

r

(

1− r

2a0

)

(

eiqr − e−iqr)

e−brdr (24.61)


gives

F1s→2s(q) =4i

π√2

1

a40

q2

(q2 + b2)3, (24.62)

where, as before, b = 3/(2a0). Inserting Eq. (24.62) into Eq. (24.59), and making use of theintegrals

∫ ∞

0

q sin(qr)

(q2 + b2)3 dq =

πr

16b3(1 + br) e−br, (24.63)

∫ ∞

0

q2 cos(qr)

(q2 + b2)3 dq =

π

16b3[

1 + br − (br)2]

e−br, (24.64)

∫ ∞

0

q3 sin(qr)

(q2 + b2)3 dq =

πr

16b(3− br) e−br, (24.65)

one finally obtains the following simple result for the transverse part of the diamagnetic1s→ 2s transition current density:

JdiaT,1s→2s(r, t) =

1

6π√2

(

e2

ma30

)

e−br [br (U− erer)− 2U] ·AT (t). (24.66)

I remind the reader that Eq. (24.66) is obtained under the assumption that the transversevector potential is constant across the atomic domain. It appears from Eq. (24.66) that thesource domain of a T-photon emitted in the 1s⇔ 2s breathing mode transition is exponen-tially confined, essentially [∼ br exp(−br)]. Photon absorption from this (almost) forbiddentransition might be observable using a strong external laser field tuned to resonance withthe 1s⇔ 2s transition.

24.4 Two-level breathing mode dynamics

Let us now undertake a qualitative study of the electron dynamics associated with a givenbreathing mode transition, 1 ⇔ 2, assuming that all other atomic levels can be neglected.In such a so-called two-level approximation, the transverse current density is given by

JT (r, t) = c1(t)c∗2(t)J

diaT,1→2(r, t) + c.c., (24.67)

neglecting the diamagnetic current densities in the two levels (1 → 1, 2 → 2). If one makesthe substitutions

ci(t) = Ci(t) exp

(

− iEit

~

)

, i = 1, 2, (24.68)

in Eq. (24.67) one has

JT (r, t) = C1(t)C∗2 (t)e

iΩtJdiaT,1→2(r, t) + c.c., (24.69)

where

Ω =E2 − E1

~(24.70)


is the Bohr (or transition) angular frequency, assuming E2 > E1. For the 1s⇔ 2s transitionin hydrogen, the transition current density is given by Eq. (24.66), but even for the 1s⇔ 2sbreathing mode we still need a calculation of C1(t) and C2(t) before the transverse currentdensity has been obtained. To determine the Ci(t)’s, i = 1, 2, one utilizes that the generaltwo-level wave function

ψ(r, t) = C1(t) exp

(

− iE1t

~

)

u1(r) + C2(t) exp

(

− iE2t

~

)

u2(r) (24.71)

must satisfy the time dependent Schrodinger equation

(

H0 + HI

)

ψ(r, t) = i~∂

∂tψ(r, t), (24.72)

where H0 = p2/(2m)− eφ is the particle Hamiltonian, and

HI =e

2m(p ·AT (r, t) +AT (r, t) · p) +

e2

2mA2

T (r, t), (24.73)

is the field-particle interaction Hamiltonian, as this appears in the Coulomb gauge and withthe electron charge denoted by−e. As indicated, we work in the semiclassical approximation.If one inserts Eq. (24.71) into the Schrodinger equation, multiplies the resulting equationby u∗i (r), i = 1, 2, from the left, and finally integrates over the entire space, one obtains thefollowing coupled equations of motions for the C1(t) and C2(t)-coefficients:

i~d

dtC1(t) = C1(t)H

11I (t) + C2(t)e

−iΩtH12I (t) (24.74)

i~d

dtC2(t) = C2(t)H

22I (t) + C1(t)e

iΩtH21I (t), (24.75)

where

HijI (t) =

∫ ∞

−∞u∗i (r)HIuj(r)d

3r (24.76)

is the ij th matrix element of the interaction Hamiltonian. In deriving Eqs. (24.74) and(24.75) the orthonormality of u1(r) and u2(r) was used. In the electric-dipole limit thedynamical equations are reduced to those discussed in many textbooks on quantum optics,e.g., [214, 150].

We now divide the matrix element into its paramagnetic and diamagnetic parts, i.e.,

HijI (t) = Hij

I (t|PAR) +HijI (t|DIA). (24.77)

The hermiticity of the momentum operator (p = p†) allows one to express the paramagneticpart in terms of the paramagnetic transition current density, given in Eq. (24.18). Hence,

HijI (t|PAR) ≡

∫ ∞

−∞u∗i (r)

e

2m(p ·AT (r, t) +AT (r, t) · p)uj(r)d3r

= −∫ ∞

−∞Jparj→i(r) ·AT (r, t)d

3r. (24.78)

So far, the analysis of the two-level dynamics has been general. Now, we limit ourselvesto breathing mode transitions. For these, we showed in Sec. 24.3 that the paramagnetictransition current density is a longitudinal vector field

Jparj→i(r) = Jpar

L,j→i(r). (24.79)


This implies that

HijI (t|PAR) = −

∫ ∞

−∞JparL,j→i(r) ·AT (r, t)d

3r = 0 (24.80)

because the integral of a product of a longitudinal and a transverse vector field over allspace always is zero. The reader may convince herself of this using the Parseval–Plancherelidentity, and thereafter for the integrand in q-space

JparL,j→i(q) ·A∗

T (q; t) = Jparj→i(q) · κκ · (U− κκ) ·A∗(q; t) = 0. (24.81)

Hence, the interaction Hamiltonian for the breathing mode case is purely diamagnetic, thatis

HijI (t) = Hij

I (t|DIA) = e2

2m

∫ ∞

−∞ui(r)uj(r)A

2T (r, t)d

3r, (24.82)

remembering that s-orbitals are real and functions of r = |r| only.For transverse vector potentials with wavelengths λ ≫ a0, a long-wavelength approach

simplifies the coupled equations for the Ci(t)’s. In our calculation of the diamagnetic tran-sition current density for the 1s→ 2s transition in hydrogen, it was possible to neglect thevariation in AT (r, t) across the atomic domain; see Eqs. (24.49) and (24.66). Thus, with nosignificant loss of generality, one may put AT (r, t) = AT (0, t) ≡ AT (t) in the on-diagonalcomponents, Hii

I (t). This implies that

HiiI (t) =

e2

2mA2

T (t), i = 1, 2, (24.83)

having assumed that ui(r) is normalized. The long-wavelength limit cannot be used in thecalculation of the off-diagonal components of Hij

I (t), because the orthogonality of u1(r) and

u2(r) then would give HijI (t) = 0. In consequence, the necessary coupling between C1(t)

and C2(t) would vanish.It is instructive to study the case where the atom is coupled to a plane and monochro-

matic (incident) field, AT (r, t) = A cos(q · r− ωt), where q ·A = 0. In order to determinethe identical off-diagonal components of Hij

I (t), we write the squared potential in the form

A2T (r, t) =

A2

4

[

e2i(q·r−ωt) + e−2i(q·r−ωt) + 2]

. (24.84)

The term 2A2/4 does not contribute to HijI (t), i 6= j, because of the orthogonality of u1

and u2. Since ui(r)uj(r) is rotational invariant the two remaining integrals only depend onthe numerical magnitude of the wave vectors 2q and −2q, and hence these integrals areidentical. Altogether, one gets

H12I (t) = H21

I (t)

=e2A2

8m

(

e2iωt + e−2iωt)

∫ ∞

−∞u1(r)u2(r)e

2iq·rd3r. (24.85)

Using spherical coordinates, the integral over the entire solid angle, Ω = 4π, gives

∫

4π

e2iq·rdΩ =2π

qrsin(2qr), (24.86)


so that we are left with a radial integration

H12I (t) =

πe2A2

2mqcos(2ωt)

∫ ∞

0

ru1(r)u2(r) sin(2qr)dr. (24.87)

At this stage, we specialize to the 1s ⇔ 2s transition. With u1(r) = u1s(r) [Eq. (24.23)],u2(r) = u2s(r) [Eq. (24.60)] and sin(2qr) = [exp(i2qr) − exp(−2iqr)]/(2i), the integrationin Eq. (24.87) easily is carried out. After some efforts, one finally obtains

H1s2sI (t) =

8√2

(

e2A2

ma40

)

q2

(4q2 + b2)3 cos(2ωt)

≃ 256√2

729

(

e2A2

m

)

(qa0)2 cos(2ωt), (24.88)

where the last expression follows because qa0 ≪ 1. At long wavelengths the off-diagonalmatrix element therefore is proportional to (qa0)

2 in lowest order.Close to resonance, i.e., for

2ω ≃ Ω, (24.89)

one may neglect the “counterpropagating” terms proportional to exp[±i(Ω + 2ω)t] in Eqs.(24.74) and (24.75), as well as the on-diagonal terms H11

I and H22I . In this rotating-wave

approximation only slowly varying terms are kept in the dynamical equations. These thenare reduced to

d

dtC1(t) = − i

2ei(2ω−Ω)tΩRC2(t), (24.90)

d

dtC2(t) = − i

2ei(Ω−2ω)tΩRC1(t), (24.91)

where

ΩR =256

√2

729

(

e2A2

m~

)

(qa0)2 (24.92)

is the Rabi frequency for the breathing mode transition. If a transition is electric-dipoleallowed, the Rabi frequency is proportional to the amplitude (A) of the (external) field.Here, it is proportional to A2, as one might have expected when the coupling is diamagnetic.In the QED description two simultaneously absorbed/emitted T-photons are involved, andthe resonance condition for the atom-field coupling therefore is at Ω = 2ω, in agreementwith Eq. (24.89). The analysis of the coupled equations in (24.90) and (24.91) can be carriedout in the usual manner [150], and need not be repeated here. With a knowledge of C1(t)and C2(t), the time development of the photon source current density [Eq. (24.69)] for the1s↔ 2s breathing mode in hydrogen thus has been determined in the case where the atomis interacting with a plane monochromatic transverse electromagnetic field.

25

Spin-1/2 current density in relativistic quantum

mechanics

25.1 Dirac matrices

In the search for a relativistic wave equation for a free particle of rest mass m one mustdemand that the energy(E)-momentum (p) [dispersion] relation

E2 = m2c4 + p2c2 (25.1)

is satisfied. By the substitutions

E → i~∂

∂t, p → −i~∇ (25.2)

in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, givesus the Klein–Gordon wave equation for a spinless particle. The Klein–Gordon equation isof second order in the space (r) and time (t) coordinates. If one seeks a covariant first-orderdifferential equation, r and t must play symmetric roles. A natural starting point thereforewould be a linear dispersion relation of the form

E = βmc2 + cα · p, (25.3)

where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compat-ible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given inEqs. (10.77)-(10.79)]:

β2 = 1, (25.4)

αiβ + βαi = 0, (25.5)

αiαj + αjαi = 2δij , (25.6)

where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot benumbers, but one can find matrices which satisfy these equations. Since the Hamiltonianappearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian,the αi and β matrices also have to be Hermitian, i.e.,

α†i = αi, i = 1− 3, (25.7)

β† = β. (25.8)

One can show that the matrices must have even rank and that the rank must be at least4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin-1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs.(25.4)-(25.8). Representations of a given rank are not unique.

297


In studies where it is useful to express the Lorentz covariance of the Dirac equation inexplicit form instead of the α and β matrices one makes use of γµ-matrices defined (withupper indices) by

γ0 = β, (25.9)

γi = βαi, i = 1− 3. (25.10)

The reader may verify that γ0 is unitary and Hermitian; that is,

(γ0)−1 = (γ0)†, (γ0)† = γ0, (25.11)

and the γi’s are also unitary but anti-Hermitian

(γi)−1 = (γi)†, (γi)† = −γi. (25.12)

In compact form the properties above can be summarized in

(γµ)† = γ0γµγ0, µ = 0− 3. (25.13)

In terms of the γµ’s the information in Eqs. (24.4)-(25.6) can be expressed in the elegantform

γµ, γν = −2ηµν1, (25.14)

where · · · stands for anti-commutator, and ηµν = ηµν is the metric tensor in Minkowskianspace [Eq. (3.4)]. To emphasize the matrix character of the anti-commutator we have in Eq.(25.14) used the symbol 1 for the unit tensor. Although written with Greek indices, theγµ-matrices do not form a four-vector. They have the same value in every reference frame,and do not change under a Lorentz transformation.

In our later analysis of the spin part of the Dirac current density, the commutatorbetween the various γµ-matrices play an important role. The commutator multiplied forlater convenience by i/2 we denote by Sµν :

Sµν =i

2[γµ, γν ] . (25.15)

In terms of the flat-space metric and Sµν , the product of two γµ matrices is given by

−γµγν = 1ηµν + iSµν . (25.16)

The Sµν-matrix is antisymmetric of course, i.e.,

Sµν = −Sνµ. (25.17)

Furthermore, the reader may prove to herself by means of Eq. (25.13) that S0i is anti-hermitian:

(

S0i)†

= −S0i, i = 1− 3, (25.18)

and Sij Hermitian:

(

Sij)†

= Sij , i, j = 1− 3. (25.19)

In compact form the relations in Eqs. (25.18) and (25.19) may be written as

(Sµν)† = γ0Sµνγ0. (25.20)

Spin-1/2 current density in relativistic quantum mechanics 299

In the standard representation for the β and αi-matrices, given in Eqs. (10.80) and(10.81), the γµ-matrices have the forms

γ0 =

(

1 00 1

)

, (25.21)

γi =

(

0 σi−σi 0

)

. (25.22)

The related antisymmetric Sµν-matrices, defined by Eq. (25.15), are given by

S0i = i

(

0 σiσi 0

)

= iαi, (25.23)

and

Sij = εijk

(

σk 00 σk

)

, (25.24)

as the reader may verify. It is well-known that the quantity

Σ =

(

σ 00 σ

)

(25.25)

multiplied by ~/2 appears as a generalized spin operator in the four-component realizationof the Dirac equation [86, 76]. The components of Σ relate to the Sij -elements as follows:

Σ = (Σ1,Σ2,Σ3) =(

S23, S31, S12)

. (25.26)

25.2 Covariant form of the Dirac equation. Minimal coupling.Four-current density

When the substitutions in (25.2) are inserted in the dispersion relation in Eq. (25.3), andthe thus obtained operator acts on a wave function ψ = ψ(r, t), one is led to the free-particleDirac equation

i~∂ψ

∂t=(

−i~cα ·∇+ βmc2)

ψ, (25.27)

which we already have written down in relation to our comparison of photon and neutrinowave mechanics [Eq. (10.75)]. For what follows, it is convenient to express the Dirac equationin terms of the γµ-matrices. Thus, by multiplying Eq. (25.27) with γ0/c from the left, andremembering that (γ0)2 = 1, one obtains

(i~γµ∂µ −mc)ψ = 0. (25.28)

The Lorentz covariance of the free Dirac equation is conveniently proved starting from Eq.(25.28). Although γµ is not a four-vector, Eq. (25.28) is the so-called covariant form ofthe Dirac equation. One may rewrite the Dirac equation in terms of the covariant four-momentum operator,

pµ =~

i∂µ =

~

i

(

1

c

∂

∂t,∇

)

, (25.29)


as follows

(γµpµ +mc)ψ = 0. (25.30)

Introducing the Feynman dagger (or slash) notation [73] for an arbitrary four-vector aµby the definition

a ≡ γµaµ (25.31)

the free-particle Dirac equation takes the very compact form

(p+mc)ψ = 0. (25.32)

The coupling of a relativistic spin-1/2 particle of charge q to a classical electromagneticfield described by the four-potential Aµ is introduced making the usual minimal couplingsubstitution

pµ ⇒ pµ − qAµ (25.33)

in the free Dirac equation. Hence,

[γµ (pµ − qAµ) +mc]ψ = 0. (25.34)

Below, we shall continue to treat the electromagnetic field as a classical quantity, i.e.,Aµ ⇒ Aµ. In slash notation, Eq. (25.34) reads

(p− qA+mc)ψ = 0. (25.35)

The Dirac equation with electromagnetic interaction, written conveniently in the form

[γµ (−i~∂µ − qAµ) +mc]ψ = 0, (25.36)

implies a four-current probability conservation equation. To derive this we take the Hermi-tian conjugate of Eq. (25.36), viz.,

(i~∂µ − qAµ)ψ† (γµ)† +mcψ† = 0. (25.37)

By multiplication with γ0 from the right, and use of 1 = (γ0)2, Eq. (25.37) can be writtenas

(i~∂µ − qAµ)ψ†γ0γ0 (γµ)† γ0 +mcψ†γ0 = 0. (25.38)

At this point it is useful to introduce the Dirac conjugate wave function, ψ, defined by

ψ = ψ†γ0. (25.39)

Since γ0(γµ)†γ0 = γµ, the Hermitian conjugate of the Dirac equation may thus be writtenin the compact form

(i~∂µ − qAµ) ψγµ +mcψ = 0. (25.40)

Multiplying Eq. (25.40) with ψ from the right, and Eq. (25.36) with ψ from the left oneobtains the two equations

[

(i~∂µ − qAµ) ψ]

γµψ +mcψψ =0, (25.41)

ψγµ (−i~∂µ − qAµ)ψ +mcψψ =0. (25.42)


By subtracting these two equations one gets upon division by i~

(

∂µψ)

γµψ + ψγµ∂µψ = 0, (25.43)

or equivalently after multiplication by cq

∂µJµ = 0, (25.44)

where

Jµ = cqψγµψ (25.45)

is interpreted as the charge probability four-current density (simply called the four-currentdensity) in the charge conservation law, given in Eq. (25.44). Since γ0γ0 = 1, and γ0γi = αi,Eq. (25.45) immediately gives one the standard expressions for the relativistic probabilitycharge

ρ = qψ†ψ, (25.46)

and three-current,

J = qψ†cαψ, (25.47)

densities.

25.3 Gordon decomposition of the Dirac four-current density

It is surprising perhaps that the Dirac three-current density takes the simple form given inEq. (25.47) even when the Dirac particle is coupled to an electromagnetic field, rememberingthat the vector potential appears explicitly in the expression for J in the nonrelativisticregime. In the nonrelativistic Schrodinger equation the wave function is a scalar ψ (ψ† = ψ∗),and the charge probability current density is given by [155, 127, 53]

J =q

2m[ψ∗ (−i~∇− qA)ψ + ψ (i~∇− qA)ψ∗]

=q~

2mi(ψ∗∇ψ − ψ∇ψ∗)− q2

mA|ψ|2. (25.48)

The question now arises whether it is possible to rewrite the expression in Eq. (25.45)for the Dirac four-current density in such a manner that a four-dimensional generalizationof the three-form in Eq. (25.48) appears explicitly. In that case, the remaining part ofthe Dirac four-current density will relate directly to the generalized spin via Sµν , cf. Eqs.(25.23) and (25.24). The so-called Gordon decomposition, which we discuss below, achievesthis goal. From the spin contribution in the Gordon formula, one can derive the well-knownexpression for the spin current density in the weakly relativistic Pauli theory.

To determine the Gordon decomposition we start from the electromagnetically coupledDirac equation in (25.34), i.e.,

mcψ = γν (qAν − pν)ψ. (25.49)


Multiplying this equation by (q/m)ψγµ from the left leads to the following formula for theµth-component of the four-current density

Jµ = cqψγµψ = ψγµγν(

q2

mAν − q

mpν

)

ψ. (25.50)

An alternative expression for Jµ can be established starting from the Hermitian conjugateDirac equation in Eq. (25.40), i.e.,

mcψ = (qAν + pν) ψγν . (25.51)

Multiplication of Eq. (25.51) by (q/m)γµψ from the right gives

Jµ = cqψγµψ =

[(

q2

mAν +

q

mpν

)

ψ

]

γνγµψ. (25.52)

By addition of Eqs. (25.50) and (25.52) the following formula for the µth-component of theDirac four-current density is obtained:

Jµ =q2

2mAν ψ (γµγν + γνγµ)ψ

+q

2m

[(

pνψ)

γνγµψ − ψγµγν (pνψ)]

. (25.53)

Using the anti-commutator relations between the γµ-matrices [Eq. (25.14)] in the termcontaining the four-potential, and the form given for the product of two γµ-matrices [Eq.(25.16)] in the terms containing pν , one gets

Jµ =− q2

mAµψψ +

q

2m

[

ψ (pµψ)−(

pµψ)

ψ]

+iq

2m

[(

pνψ)

Sνµψ − ψSµν (pνψ)]

. (25.54)

Utilizing next the antisymmetry of the Sµν-matrix [Eq. (25.17)], and a slight rewriting, onefinally reaches the Gordon decomposition of the four-current density:

Jµ =q

2m

ψ (pµ − qAµ)ψ −[

(pµ + qAµ) ψ]

ψ

+iq

2mpν[

ψSµνψ]

. (25.55)

In the nonrelativistic limit, the last part of Eq. (25.55) [containing the spin via Sµν ] isabsent, as is well-known. It is manifest that the remaining part of Eq. (25.55) has a formcorresponding to a four-dimensional relativistic generalization of the three-current densitygiven in Eq. (25.48). It is sometimes useful to express Jµ in the form

Jµ =q

2m

[

ψπµψ −(

πµψ)

ψ]

+q

2mi(πν + qAν)

[

ψSµνψ]

, (25.56)

where

πµ = pµ − qAµ (25.57)

is the kinetic four-momentum operator.


25.4 Weakly relativistic spin current density

When relativistic effects are sufficiently small the four-component Dirac equation reduces tothe two-component Pauli equation, as discussed in many books, e.g., in [88, 86, 76]. Thus,if one writes the Dirac spinor in bispinor notation, viz., as

ψ =

(

φχ

)

=

(

φ0χ0

)

exp

(

−imc2

~t

)

, (25.58)

and assumes that the kinetic and potential energies of the particle are small compared tothe rest mass energy, mc2, it follows from the Dirac equation that

χ = (2mc)−1σ · πφ, (25.59)

in the lowest-order approximation. In Eq. (25.59) π is the kinetic three-momentum operator.Roughly speaking, it appears from Eq. (25.59) that the ratio between χ and φ is of the orderof the ratio between the particle speed and the speed of light, |π|/(mc), a small quantityin the weakly relativistic domain. Hence, χ and φ are called the small and large bispinorcomponent of ψ. The column bispinors

φ =

(

φ1φ2

)

, χ =

(

χ1

χ2

)

(25.60)

have the associated Hermitian conjugate row bispinors

φ† = (φ∗1, φ∗2), χ† = (χ∗

1, χ∗2). (25.61)

In the bispinor notation the Dirac adjoint wave function to ψ is given by

ψ = ψ†γ0 =(

φ†, χ†)(

1 00 −1

)

=(

φ†,−χ†) (25.62)

in the standard representation.Let us now determine the explicit form of the spin four-current density [last part of Eq.

(25.55)],

Jµspin =

iq

2mpν[

ψSµνψ]

, (25.63)

in the weakly relativistic regime, starting from an examination of the antisymmetric 4 × 4matrix

Rµν ≡ ψSµνψ. (25.64)

By combining Eqs. (25.24), (25.58) and (25.62), one obtains

Rij = εijk(

φ†,−χ†)(

σk 00 σk

)(

φχ

)

= εijk(

φ†σkφ− χ†σkχ)

. (25.65)

In view of Eq. (25.59) one hence gets the following weakly relativistic (WR) expression forRij ≡ Rij

WR in lowest order

RijWR = εijkφ

†σkφ, i, j = 1− 3. (25.66)


The remaining elements

R0j = −Rj0 = ψS0jψ, (25.67)

determined by inserting Eqs. (25.23), (25.58) and (25.62) into Eq. (25.67), are given by

R0j = i(

φ†,−χ†)(

0 σjσj 0

)(

φχ

)

= i(

φ†σjχ− χ†σjφ)

. (25.68)

A comparison of Eqs. (25.65) and (25.68) shows that the elements of R0j approximatelyare mc/|π|-times smaller than the nonvanishing elements of Rij . For consistency with Eq.(25.66), one therefore has

R0jWR = 0, j = 1− 3, (25.69)

in lowest order.The result in Eq. (25.69) implies that the µ = 0-component of the spin current density

vanishes in the WR-limit, i.e.,

J0spin,WR = 0. (25.70)

In the lowest-order approximation we may therefore conclude that no charge density isassociated with the spin dynamics, that is ρspin,WR = J0

spin,WR/c = 0.The spatial components of the weakly relativistic current density, viz.,

J ispin,WR =

iq

2mpjR

ijWR, (25.71)

are readily obtained inserting the explicit expression for RijWR, given in Eq. (25.66). Hence,

J ispin,WR =

iq

2mεijk pj(φ

†σkφ)

=iq

2m

[

p× (φ†σφ)]

i. (25.72)

The weakly relativistic spin current density three-vector thus is given by

Jspin,WR =iq

2mp× (φ†σφ)

=q~

2m∇× (φ†σφ). (25.73)

From the last member of Eq. (25.73) it immediately follows that

∇ · Jspin,WR = 0. (25.74)

In the weakly relativistic limit the spin current density hence is a transverse (superscriptT) vector field:

Jspin,WR = JTspin,WR. (25.75)

The results in Eqs. (25.70) and (25.74) are in agreement with the equation of continuity forthe spin part of the current density in the Pauli theory [113]

∂µJµspin,WR = 0. (25.76)


From Eq. (25.75) one may conclude that the source domain of a photon emitted in a purespin transition is identical to that of the WR spin current density itself. In the atomic casethe photon source domain thus is of the order of the Bohr radius, and the confinement isexponential [127, 117].

Combining the explicit expressions for the Pauli spin matrices [Eqs. (10.81)] and the firstparts of Eqs. (25.60) and (25.61), the weakly relativistic spin current density [Eq. (25.73)]takes the form

Jspin,WR =q~

2m

[

ez ×∇(

|φ2|2 − |φ1|2)

−√2e+ ×∇ (φ1φ

∗2)−

√2e− ×∇(φ∗1φ2)

]

(25.77)

with complex unit vectors

e± =1√2(ex ± iey) . (25.78)

Let us assume that the spatial dynamics freezes out. In the (S2, Sz)-spin basis the two-component spinor then is given by

(

φ1φ2

)

= φ0(r)

(

↑ (t)↓ (t)

)

, (25.79)

where ↑ (t) and ↓ (t) are the spin-up (↑) and spin-down (↓) parts of the spin wave functionat the time t, and φ0(r) is the relevant frozen spatial wave function. By inserting Eq. (25.79)into Eq. (25.77) one obtains

Jspin,WR(r, t) =q~

2m

[(

| ↓ (t)|2 − | ↑ (t)|2)

ez

−√2 ↑ (t) ↓∗ (t)e+ −

√2 ↑∗ (t) ↓ (t)e−

]

×∇φ0(r). (25.80)

The result in Eq. (25.80) shows that even in a pure spin-flip process a particle current flowsin direct space. The flow only exists in the part of space where the gradient of the spatialwave function (∇φ0(r)) is nonvanishing.

The result in Eq. (25.80) may be applied for the spin current density associated withthe hyperfine ground-state dynamics of the hydrogen atom. In the 1s state the spin currentdensity is exponentially confined to a region of linear extension comparable to the Bohrradius. Since the orbital angular momentum of the electron is zero in the 1s state, only thespin dynamics of the electron and the proton are involved in the calculation of the hyperfinestructure. Since the electron mass is much smaller than that of the nucleus the spin currentdensity associated with the dynamics of the proton spin is negligible. If one assumes thatthe electron spin is coupled not only to the nuclear spin but also to a time-independentexternal magnetic field (Zeeman effect), the four electron energy eigenfunctions are givenby [52]

|u1〉 = |1, 1〉, (25.81)

|u2〉 = |1,−1〉, (25.82)

|u±〉 = α±|1, 0〉+ β±|0, 0〉, (25.83)

where

|1, 1〉 = | ↑↑〉, (25.84)

|1, 0〉 = 1√2(| ↑↓〉+ | ↓↑〉) , (25.85)

|1,−1〉 = | ↓↓〉, (25.86)


are the triplet states of the electron-proton system, and

|0, 0〉 = 1√2(| ↑↓〉 − | ↓↑〉) , (25.87)

the singlet state. The arrows in Eqs. (25.84)-(25.87) refer to the spin states of the electronand proton. The explicit expression for the coefficients α± and β± may be found in [52],for example. With the Zeeman effect included, the weakly relativistic spin current densityof the electron (q = −e) takes the general form [123]

Jspin,WR(r, t) =e~

2m〈w(t)|σ|w(t)〉 ×∇φ0(r), (25.88)

where

|w(t)〉 = c1(t)|u1〉+ c2(t)|u2〉+ c+(t)|u+〉+ c−(t)|u−〉. (25.89)

The time-dependent c-coefficients satisfy the constraint |c1(t)|2 + |c2(t)|2 + |c+(t)|2 +|c−(t)|2 = 1, provided the spin state in Eq. (25.89) is normalized; that is, 〈w(t)|w(t)〉 = 1.In the absence of the external magnetic field one only has two hyperfine levels because thetriplet states go into a three-fold degenerated state, and in this limit Eq. (25.88) reduces toa form analogous to the one in Eq. (25.80). The involved states are the F = 1 and F = 0states of the total angular (spin) momentum

F = S+ I, (25.90)

S and I being the electron and proton spin, respectively. Hence(

φ1φ2

)

= φ0(t)

(

1(t)0(t)

)

, (25.91)

where 1(t) and 0(t) are the F = 1 and F = 0 parts of the (electron) wave function at timet.

25.5 Continuity equations for spin and space four-current densities

It appears from Eqs. (25.63), (25.64), and (25.68) that a charge density accompanies the spindynamics in general. In consequence, the spin current density will not be a transverse vectorfield beyond the weakly relativistic region. However, it turns out that the spin four-currentdensity satisfies an equation of continuity. In order to prove this we write the four-divergenceof Jµ

spin [Eq. (25.63), with the abbreviation in Eq. (25.64)] in the two forms

∂µJµspin =

~q

2m∂µ∂νR

µν =~q

2m∂ν∂µR

νµ, (25.92)

where the last member follows from the first by the dummy variable interchange µ ↔ ν.Using Eq. (25.92) and the antisymmetry of Sµν (and thus also of Rµν) [Eq. (25.17)],the four-divergence can be written as follows:

∂µJµspin =

~q

4m(∂µ∂νR

µν + ∂ν∂µRνµ)

=~q

4m(∂µ∂ν − ∂ν∂µ)R

µν . (25.93)


Since [∂µ, ∂ν ] = 0, we obtain

∂µJµspin = 0. (25.94)

We know [Eq. (25.44)] that the total charge four-current density satisfies the charge con-servation law

∂µ(

Jµspace + Jµ

spin

)

= 0. (25.95)

In view of Eq. (25.94) it thus can be concluded that the spatial part of the four-currentdensity also satisfies an equation of continuity, viz.,

∂µJµspace = 0. (25.96)

As the reader may prove to herself, the result in Eq. (25.96) may be obtained directly fromthe Gordon decomposition taking the four-divergence of the spatial part of Eq. (25.55).

26

Massless photon propagators

After having identified the source domain of the transverse photon, and discussed the struc-ture of the domain in a number of special cases, we turn our attention to an analysis of thepropagation of the photon field from a given source. In the four-momentum (∼ frequency −wave-vector) representation the Feynman photon propagator is obtained from the massivephoton propagator letting the Compton wave-number go to zero, cf. the analysis in Sec.12.5.

26.1 From the Huygens propagator to the transverse photon prop-agator

It was realized in Sec. 20.1 that the contravariant four-potential, Aµ(x), generated by acontravariant four-current density distribution, Jµ(x′) is given by

Aµ(x) = µ0

∫ ∞

−∞DR(x− x′)Jµ(x′)d4x′ (26.1)

in the Lorenz gauge. The retarded scalar propagator, DR(x−x′), multiplied by the vacuumspeed of light is just the Huygens propagator well-known from classical electrodynamics(optics). I here remind the reader of the discussion of the scalar propagator in Sec. 21.4. Infour-momentum space Eq. (26.1) takes the algebraic form

Aµ(q) = µ0Jµ(q)

q2 + iε, (26.2)

where q2 is given by Eq. (12.84). An infinitesimal quantity iε (ε > 0) has been addedto q2 = q · q − (ω/c)2 for reasons to be given later. By means of the metric tensor ofMinkowskian space, gµν, one can write Eq. (26.2) as follows:

Aµ(q) = µ0gµν

q2 + iεJν(q). (26.3)

The metric tensor (elements) can be expanded in terms of four linearly independent unitpolarization vectors εµr (q), r = 0− 3; see Eq. (12.98). The quantity

GF,µν(q) =1

q2 + iε

3∑

r=0

ζrεµr (q)ε

νr (q) (26.4)

hence is the µνth element of the (massless) Feynman (F) photon propagator. In Sec. 26.6we shall make a special choice for the polarization vectors which allows us to project out inthe subspace of the T -photons a transverse photon propagator given by

GFT (q) =

U− κκq2 + iε

=U− κκ

q · q−(

ωc

)2+ iε

(26.5)

309


in the (ω/c,q)-domain [cf. Eq. (12.121)]. I remind the reader that κ = q/q. In our subse-quent discussion of the Feynman photon propagator (Sec. 26.5) we shall derive Eq. (26.5).

In real space the transverse propagator is given by the integral representation

GFT (r, τ) =

1

c

∫ ∞

−∞

U− κκq · q−

(

ωc

)2+ iε

ei(q·R−ωτ)dωd3q

(2π)4(26.6)

Leaving out iε, the integrand has poles on the real ω-axis at ω = ±cq. These we displace bymeans of iε in the usual manner [127] in order to be able to perform the ω-integration alongthe entire real ω-axis. Often a transverse photon propagator (denoted by GT ), defined asthe transverse part of the Feynman photon propagator multiplied by the speed of light, isused, i.e.,

GT (R, τ) ≡ cGFT (R, τ). (26.7)

In the space-time domain one hence obtains the following integral relation between thetransverse vector potential and the three-current density:

AT (r, t) = µ0

∫ ∞

−∞GT (R, τ) · J(r′, t′)d3r′dt′, (26.8)

with R = r−r′, and τ = t− t′. The formal connection in Eq. (26.8) was already obtained inSec. 22.1, Eq. (22.5). Because of the central importance it might be useful for the reader tocompare Eq. (26.8) to an alternative integral expression for the transverse vector potential,namely,

AT (r, t) = µ0

∫ ∞

−∞g(R, τ)JT (r

′, t′)d3r′dt′, (26.9)

once again. The well-known form in Eq. (26.9) is readily obtained as the transverse projec-tion of Eq. (26.1), remembering that DRd(ct

′) = gdt′. In Eq. (26.9) the T-photon emissionprocess is seen as a superposition of spherical wavelets coming from each point in thetransverse current density domain. All wavelets propagate with the vacuum speed of light,and the “communication” between an observation point, x = (ct, r), and a source point,x′ = (ct′, r′) thus is confined to the light cone. In Eq. (26.8) the current density distributionitself is considered to be the source of the transverse photon. In order to obtain the sameT-photon potential at every point of observation in a perspective where the source domain istruncated from the JT -distribution to the J-distribution, the transverse photon propagatorGT (R, τ) necessarily must include a part which is nonvanishing in front of the light cone.The explicit expression for GT (R, τ) already was given in Eq. (22.11), and the importanceof this propagator in relation to near-field photon emission in 3D was emphasized in Sec.22.1.

It appears from Eqs. (26.5)-(26.7) that the transverse photon propagator in the (ω,q)-domain is given by

GT (q, ω)[

= GFT (q)

]

=U− κκ

q · q−(

ωc

)2+ iε

. (26.10)

One may close the gap between Eq. (26.10) and the expression given for the T-propagatorin Eq. (22.6) by transforming the last equation to the (ω,q)-representation. With the sub-stitution r′′ = r′′′ + r′, one obtains

GT (r− r′;ω) =∫ ∞

−∞g(r− r′ − r′′′;ω)δT (r

′′′)d3r′′′ (26.11)

Massless photon propagators 311

since the transverse delta function does not depend on time. Using the folding theorem onEq. (26.11), we get

GT (q, ω) = g(q, ω)δT (q). (26.12)

Since g(q, ω) = [q ·q− (ω/c)2+ iε]−1 is the Huygens propagator, and δT (q) = U−κκ (seeEq. (2.97)), Eqs. (26.10) and (26.12) are identical.

26.2 T-photon time-ordered correlation of events

In Sec. 15.5 a sort of mean position state for the transverse photon was introduced by thedefinition

|R(r, t)〉 ≡(

2ǫ0c

~

)1

2

A(−)T (r, t)|0〉, (26.13)

see Eq. (15.121), and it was shown in Sec. 15.6 that such position states belonging todifferent positions in space (r and r′) and taken at the same time (t) are not orthogonalbut satisfy [Eq. (15.140)]

〈R(r, t)|R(r′, t)〉 = 1

π2|r− r′|2 (26.14)

outside the singularity. In terms of the positive and negative-frequency parts of the trans-verse vector potential operator the inner product [multiplied by ~/(2ǫ0c)] is given by

~

2ǫ0c〈R(x)|R(x′)〉 = 〈0|A(+)

T (x) · A(−)T (x′)|0〉. (26.15)

Instead of forming the mean value of the scalar product A(+)T (x) · A(−)

T (x′) in the photon

vacuum state, let us try to visualize the meaning of the dyadic product A(+)T (x)A

(−)T (x′),

i.e.,

~

2ǫ0c〈R(x)||R(x′)〉 ≡ 〈0|A(+)

T (x)A(−)T (x′)|0〉. (26.16)

However, before we discuss the interpretation of Eq. (26.16), let us broaden our perspectiveand consider the T-photon correlation matrix, T(x, x′), defined by

T(x, x′) ≡ 〈0|T

AT (x)AT (x′)

|0〉, (26.17)

where T is the time-ordering symbol:

T AT (x)AT (x′) = θ(t− t′)AT (x)AT (x

′) + θ(t′ − t)AT (x′)AT (x). (26.18)

The unit step functions, θ, appearing in the time-ordered product T · · · hence arrangethe two operators so that the later time stands to the left of the earlier time. Since

AT |0〉 =(

A(+)T + A

(−)T

)

|0〉 = A(−)T |0〉, (26.19)


the T-photon correlation matrix may be written in the form

T(x, x′) = θ(t− t′)T>(x, x′) + θ(t′ − t)T<(x, x

′), (26.20)

where

T>(x, x′) = 〈0|A(+)

T (x)A(−)T (x′)|0〉, (26.21)

and

T<(x, x′) = 〈0|A(+)

T (x′)A(−)T (x)|0〉. (26.22)

A comparison of Eqs. (26.16), (26.21), and (26.22) shows that

T>(x, x′) =

~

2ǫ0c〈R(x)||R(x′)〉, t′ < t, (26.23)

T<(x, x′) =

~

2ǫ0c〈R(x′)||R(x)〉, t′ > t. (26.24)

Because the analyses of the two parts of Eq. (26.20) are identical, essentially, we need only

concentrate on T>(x, x′), say. Since A

(+)T (x)|0〉 = 0, T>(x, x

′) can be written as

T>(x, x′) = 〈0|

[

A(+)T (x), A

(−)T (x′)

]

|0〉. (26.25)

The form in Eq. (26.25) is particularly useful, as we shall realize soon. The commutator

between A(+)T (x) and A

(−)T (x′) can readily be calculated using Eqs. (2.95), (15.120), and

the equal-time commutator relation in Eq. (15.28) for t = 0. Hence,

[

A(+)T (x), A

(−)T (x′)

]

=~

2ǫ0cL3

∑

q

1

q(U− κκ) ei(q·R−cqτ)

→ ~

2ǫ0c

∫ ∞

−∞(U− κκ) ei(q·R−cqτ) d3q

(2π)3q, (26.26)

where R = r−r′ and τ = t− t′, as before. Since the commutator is a c-number, one obtains

T>(x, x′) =

~

2ǫ0c

∫ ∞

−∞(U− κκ) ei(q·R−cqτ) d3q

(2π)3q(26.27)

in the continuum limit. To visualize the meaning of T>(x, x′) we transfer Eq. (26.27) to the

(ω,q)-domain using the formula

2

ic

∫ ∞

−∞

e−iωτ

q2 −(

ωc

)2

dω

2π= q−1e−icqτ . (26.28)

The result in Eq. (26.28) is obtained by contour integration along a semicircle with a radiustending toward infinity. For τ > 0, the contour is in the lower halfplane of the complexω-plane, and encircles the displaced pole at ω = cq− i0− [127]. Combining Eqs. (26.27) and(26.28) one gets

T>(x, x′) =

~

iǫ0c2

∫ ∞

−∞

U− κκq2 −

(

ωc

)2 ei(q·R−ωτ)dωd

3q

(2π)4. (26.29)


A comparison of Eqs. (26.6) [with Eq. (26.7) inserted] and (26.29) shows that

T>(x, x′) =

~

iǫ0c2GT (R, τ). (26.30)

The T-photon correlation matrix

〈0|A(+)T (x)A

(−)T (x′)|0〉 = ~

2ǫ0c〈R(x)||R(x′)〉

=~

iǫ0c2GT (R, τ) =

~

iǫ0cGF

T (R, τ), τ > 0, (26.31)

hence is just the transverse photon propagator multiplied by the imaginary constant

~/(iǫ0c2). Thus, we can think of the vacuum expectation value 〈0|A(+)

T (x)A(−)T (x′)|0〉 as

representing a T-photon being created at the mean space-time position x′ = (ct′, r′) trav-elling forward in time (τ = t − t′ > 0) to x = (ct, r) where it is annihilated. The ex-plicit expression for GT (R, τ) was given in Eq. (22.11). Remembering that the commutator

[A(+)T (x), A

(−)T (x′)] is a c-number [cf. Eq. (26.26)], we may add the commutator

[

A(+)T (r, t), A

(−)T (r′, t′)

]

=~

iǫ0c2GT (R, τ) (26.32)

to those given in Subsec. 18.3.1. The result in Eq. (26.32) was cited in connection to ourdiscussion of the correlation supermatrix in Subsec. 17.4.1; see Eq. (17.34). The term [Eq.(26.22)]

〈0|A(+)T (x′)A(−)

T (x)|0〉 = ~

2ǫ0c〈R(x′)||R(x)〉

=~

iǫ0c2GT (−R,−τ), τ > 0, (26.33)

is interpreted as a T-photon being created at x and propagating to x′, where it is annihilated.Altogether, one may conclude that the time-ordered T-photon correlation matrix, given

by

〈0|T

A(+)T (x)A

(−)T (x′)

|0〉

=~

iǫ0c2(θ(t− t′)GT (R, τ) + θ(t′ − t)GT (−R,−τ)) , (26.34)

describes the correlation of events located at x and x′. For far-field separations the correla-tion is limited to the light cone, but at near-field distances also the fundamental limitationon the spatial T-photon localization enters the correlation matrix.

26.3 Covariant correlation matrix

In Sec. 26.2, we introduced a T-photon correlation matrix of dimension 3×3 by the definitionin Eq. (26.17). The ijth element of this matrix is given by

T ij(x, x′) = 〈0|T

AiT (x)A

jT (x

′)

|0〉, (26.35)


where i, j = 1, 2, 3. In the covariant formulation of quantum electrodynamics not only T-photons but also L and S-photons appear. In order to understand how the last two photontypes contribute to the time-ordered correlation of events at two space-time points x andx′, the T-photon correlation matrix concept now is generalized to the covariant level. Inview of Eq. (26.35) the µνth element of the 4× 4 covariant correlation matrix

F(x, x′) = Fµν(x, x′) (26.36)

thus is defined by

Fµν(x, x′) ≡ 〈0|T

Aµ(x)Aν(x′)

|0〉, (26.37)

where Aµ(x), µ = 0 − 3, is the four-potential operator associated to the free electro-magnetic field Aµ(x). In the Lorenz gauge the elements of the four-potential operatorsatisfy

Aµ(x) = 0, (26.38)

cf. Eq. (12.19). The vacuum state |0〉 appearing in Eq. (26.37) is defined as the state inwhich there are no T-, L-, and S-photons present, and as we shall see soon, one thereforehas

Aµ(+)(x)|0〉 = 0, (26.39)

for all x and µ. In conventional notation, the quantity

Aµ(+)(x) =[

Aµ(−)(x)]†

(26.40)

is the positive-frequency part of the four-potential operator

Aµ(x) = Aµ(+)(x) + Aµ(−)(x). (26.41)

In view of Eq. (26.39), it is obvious that the elements of the covariant correlation matrixmay be written in the form

Fµν(x, x′) = θ(t− t′)Fµν> (x, x′) + θ(t′ − t)Fµν

< (x, x′), (26.42)

where

Fµν> (x, x′) =〈0|Aµ(+)(x)Aν(−)(x′)|0〉

=〈0|[

Aµ(+)(x), Aν(−)(x′)]

|0〉, (26.43)

and

Fµν< (x, x′) = Fµν

> (x′, x). (26.44)

26.4 Covariant quantization of the electromagnetic field: A briefreview

To gain further insight into the structure of the covariant correlation matrix, Fµν(x, x′),one must study the commutator [Aµ(+)(x), Aν(−)(x′)]. With this aim we expand the four-


potential operator [Eq. (26.41)] in a complete set of plane waves:

Aµ(+)(x) = L− 3

2

∑

q,r

(

~

2ǫ0ωq

)1

2

ar(q)εµr e

iqx, (26.45)

Aµ(−)(x) = L− 3

2

∑

q,r

(

~

2ǫ0ωq

)1

2

a†r(q)(εµr )

∗e−iqx. (26.46)

The summation over q is as previously over wave vectors allowed by the periodic boundaryconditions, and in free space the dispersion relation is

ωq = c|q|, (26.47)

so that

qx ≡ qµxµ = q · r− c|q|t. (26.48)

The summation over r = 0, 1, 2, 3 corresponds to the fact that there exists, for each q,four linearly independent polarization states. These are described by the unit polarizationvectors εµr (q), r = 0−3, which we choose to be real in the following. The four polarizationvectors satisfy the covariant orthonormality and completeness relations

εrµ(q)εµs (q) = ζrδrs r, s = 0− 3, (26.49)

∑

r

ζrεµr (q)ε

νr (q) = gµν , (26.50)

where

ζi = −ζ0 = 1, i = 1− 3. (26.51)

The reader may recall here that unit polarizations of the type given above were introducedin relation to our study of the massive photon propagator in Sec. 12.5 [see Eqs. (12.98) and(12.99)].

The photon interpretation of the quantized field is obtained when it is assumed that theequal-time (here time-independent) operators ar(q) ≡ aqr(0) and a†r(q) ≡ a†qr(0) satisfythe commutation relations

[

ar(q), a†s(q

′)]

= ζrδrsδqq′, (26.52)

[ar(q), as(q′)] =

[

a†r(q), a†s(q

′)]

= 0. (26.53)

For r = 1 − 3, Eqs. (26.52) and (26.53) are the standard boson commutation relations.A specific choice of polarization vectors in a given inertial frame of reference leads to theusual transverse (r = 1, 2) and longitudinal photons (r = 3). For r = 0, the specific choicerelates to the scalar photons. Because ζ0 = −1 it looks as if the usual role of annihilation(a0(q)) and creation (a†0(q)) operators must be interchanged for the scalar photons. Aradical modification of the standard formalism is needed to obtain a satisfactory formalismfor the scalar photons. The modification was established by Gupta [89] and Bleuler [20]. Inthe Gupta–Bleuler theory the covariant quantization is done with an indefinite metric, andthe Lorenz condition for free fields takes the form of a condition

∂µAµ(+)(x)|Ψ〉 = 0 (26.54)

which restricts the allowed states |Ψ〉, and involves mode annihilation operators only [53].


For the present purpose it is not necessary to dive into the Gupta–Bleuler theory. Theinterested reader may find a brief account of the theory oriented toward the T-photonlocalization problem in my book on the quantum theory of near-field electrodynamics [127].

The vacuum state |0〉 is defined as the state in which there are no photons of any kindpresent, i.e.,

ar(q)|0〉 = 0, ∀q, r = 0− 3. (26.55)

In view of the expansion for Aµ(+)(x) [Eq. (26.45)], one obtains the condition in Eq. (26.39),as promised.

The Hamilton operator of the field becomes

H =∑

q,r

~ωq

(

Nr(q) +1

2

)

, (26.56)

where

Nr(q) = ζra†r(q)ar(q) (26.57)

defines the number operator for the various (q, r)-modes. One-photon states, denoted by|1qr〉, are created by operating with a†r(q) on the vacuum state, i.e.,

|1qr〉 = a†r(q)|0〉. (26.58)

The inclusion of ζr in the definition of the number operator ensures that the energy ispositive for all photon types. Thus, by utilizing Eq. (26.52) we obtain

Nr(q)|1qr〉 = ζra†r(q)ar(q)a

†r(q)|0〉

= ζra†r(q)

(

a†r(q)ar(q) + ζr)

|0 〉= ζ2r a

†r(q)|0〉 = |1qr〉, (26.59)

and then[

H − 1

2

∑

qr

~ωq

]

|1qr〉 = ~ωq|1qr〉, (26.60)

with ωq(> 0) given by Eq. (26.47).

26.5 The Feynman photon propagator

After our brief summary of aspects of the covariant quantization scheme for the electromag-netic field, we return to the analysis of the structure of the covariant correlation matrix,the elements of which are given by Eqs. (26.42)-(26.44). Using the expansions given in Eqs.(26.45) and (26.46), together with the commutator relation in Eq. (26.52), it appears that

[

Aµ(+)(x), Aν(−)(x′)]

=~

2ǫ0cL3

∑

q

[

∑

r

ζrεµr (q)ε

νr (q)

]

1

|q|eiq(x−x′)

=~

2ǫ0cgµν

1

L3

∑

q

1

|q|eiq(x−x′). (26.61)


The last member of Eq. (26.61) is obtained using the completeness relation in Eq. (26.50).I urge the reader to prove the correctness of the second member of Eq. (26.61). In thecontinuum limit the commutator above is given by

[

Aµ(+)(x), Aν(−)(x′)]

=i~

ǫ0c2gµν∆+

0 (x− x′), (26.62)

where

∆+0 (x) =

c

2i

∫ ∞

−∞eiqx

d3q

(2π)3|q| . (26.63)

The function ∆+0 (x) is the positive-frequency part of the massless Jordan–Pauli (JP) scalar

propagator [104]. The massive (and real) JP-propagator ∆JP (x) has the integral represen-tation [127, 156]

∆JP (x) = ∆+(x) + ∆−(x), (26.64)

where

∆+(x) =[

∆−(x)]∗

=c

2i

∫ ∞

−∞

eiqx

[q · q+Q2C ]

1

2

d3q

(2π)3, (26.65)

with QC = mc/~. Therefore,

∆+0 (x) = lim

m→0∆+(x), (26.66)

as claimed.Utilizing Eq. (26.28), the q-space integral representation given for ∆+

0 (x) in Eq. (26.63)can be replaced by an integral representation over four-wave vector space. Hence,

∆+0 (x) = −c

∫ ∞

−∞

eiqx

q · q−(

ωc

)2

d4q

(2π)4, t > 0, (26.67)

remembering that the contour for the q0 = ω/c-integration is located in the lower halfspacesince t > 0. Let us now introduce a new function by the definition

∆µν(x) ≡ gµν∫ ∞

−∞

eiqx

q2 + iε

d4q

(2π)4, t > 0, (26.68)

where q2 = q ·q− (ω/c)2. As before, the contour integration is in the lower halfplane of thecomplex q0-plane around the displaced pole located at q0 = |q|− i0−. By means of this newfunction the µνth element of the covariant correlation matrix can be written as follows:

Fµν(x, x′) =~

iǫ0c[θ(t− t′)∆µν(x− x′) + θ(t′ − t)∆µν(x′ − x)] , (26.69)

or in compact form

Fµν(x, x′) =~

iǫ0cGF,µν(x− x′) (26.70)

where

GF,µν(x) = gµν∫

CF

eiqx

q2 + iε

d4q

(2π)4(26.71)


is the µνth element of the Feynman photon propagator GF,µν(x). The contour CF is theso-called Feynman contour [73, 156]. For x0 = ct > 0, the contour is completed in the lowerhalf of the q0-plane. For x0 < 0, the contour is completed in the upper halfplane. Since thefour-momentum space Feynman photon propagator, GF,µν(q), is related to the configurationspace propagator, GF,µν(x), by

GF,µν(x) =

∫

CF

GF,µν(q)eiqxd4q

(2π)4, (26.72)

a comparison of Eqs. (26.71) and (26.72) shows that

GF,µν(q) =gµν

q2 + iε=

1

q2 + iε

3∑

r=0

ζrεµr (q)ε

νr (q). (26.73)

The last member of Eq. (26.73) originates in the completeness relation for the four unitpolarization vectors [Eq. (26.50)], and this member closes the gap to the analysis leadingup to Eq. (26.4).

26.6 Longitudinal and scalar photon propagators

For many purposes one only requires the general form of the Feynman photon propagatorin four-wave vector space [Eq. (26.73)]. However, a specific choice of polarization vectorsoften facilitates the interpretation. In our discussion of the massive photon propagator(Sec. 12.5) it was found useful to work in a frame in which the four-wave vector is givenby qµ = (ω/c, 0, 0, |q|), where ω/c = (q · q+Q2

C)1/2. In this frame the three-wave vector

thus is directed along the 3(z)-axis, and a suitable choice for the unit polarization vectorsis

ǫµ0 = Q−1C

(ω

c, 0, 0, |q|

)

→ (1, 0, 0, 0), (26.74)

εµ1 = (0, 1, 0, 0), (26.75)

εµ2 = (0, 0, 1, 0), (26.76)

εµ3 = Q−1C

(

|q|, 0, 0, ωc

)

→ (0, 0, 0, 1). (26.77)

The vectors after the two arrows refer to the rest frame of the massive photon. For the mass-less photon no rest frame exists, but it is obvious that an acceptable choice of polarizationvectors for the massless case is obtained setting |q| = 0 in Eqs. (26.74)-(26.77). The readermay readily verify that this choice satisfies the orthonormality and completeness relations,given in Eqs. (26.49) and (26.50). I stress that setting |q| = 0 in Eqs. (26.74)-(26.77) doesnot imply that the photon wave vector is zero (it cannot be zero for a massless particle inany inertial frame). Our ǫµ0-vector choice thus is purely time-like, i.e.,

ǫµ0 ≡ nµ = (1, 0, 0, 0). (26.78)

For the remaining purely space-like vectors, we now make a generalization correspondingto the situation where the wave vector q points in an arbitrary direction (unit vector κ) inspace. Hence, what is called the longitudinal polarization, is given by

εµ3 (q) = (0, ε3(q)) = (0,κ). (26.79)


The choice for the two remaining so-called transverse polarization vectors is

εµr (q) = (0, εr(q)), r = 1, 2, (26.80)

where, altogether

εr(q) · εsq = δrs r, s = 1− 3. (26.81)

With the polarization vectors given by Eqs. (26.78)-(26.80), the Feynman photon prop-agator in four-momentum space divides into important pieces associated with transverse,longitudinal, and scalar photons, i.e.,

GF,µν(q) = GF,ijT (q) + GF,ij

L (q)+ GF,00S (q). (26.82)

The transverse and longitudinal propagators are objects in the three-momentum subspace(i, j = 1− 3), and here given by

GF,ijT (q) = GF

T (q) =ε1ε1 + ε2ε2q2 + iε

=U− κκq2 + iε

, (26.83)

and

GF,ijL (q) = GF

L(q) =κκ

q2 + iε. (26.84)

By addition of the expressions for the T and L-propagators one obtains

GFT (q) +GT

L(q) =U

q2 + iε= g(q)U, (26.85)

i.e., the Huygens scalar propagator multiplied by the 3 × 3 unit tensor. This result wasto be expected on the basis of the analysis in Subsec. 22.1.1. The expression given for thetransverse propagator in Eq. (26.83) is just the one in Eq. (26.5).

We know that the longitudinal part of the electric field results from a certain combinationof first-order derivatives of the longitudinal and scalar potentials. Furthermore, the totallongitudinal field energy is just the electrostatic Coulomb energy of the relevant system ofpoint charges. Thus, the exchange of longitudinal and scalar photons between charges mustcorrespond to the instantaneous Coulomb interaction between these. To show this, let usstart by recalling that the interaction Hamiltonian density, HI(q), in four-momentum spaceis given by [127, 53, 156]

HI(q) = −J∗µ(q)A

µ(q), (26.86)

with the sign convenient adopted in this book for the flat-space metric components [see Eqs.(3.4) and (12.80)]. Inserting the expression given in Eq. (26.3) for Aµ(q) one obtains

HI(q) = − µ0

q2 + iεJ∗µ(q)g

µνJν(q). (26.87)

The interaction Hamiltonian density related to the longitudinal photons hence is given by

HI(q) = −µ0J∗µ(q)GF,µν

L (q)Jν(q), (26.88)

where

GF,µνL (q) =

1

q2 + iε

(

qµ − ωc n

µ) (

qν − ωc n

ν)

q2 +(

ωc

)2 (26.89)


is the longitudinal photon propagator element written in covariant form. The reader mayconvince herself of the identity of Eq. (26.89) and the components of Eq. (26.84) by notingthat (i) κiκj = qiqj/|q|2, (ii) (ω/c)nµ subtracts off the time-like component of qµ, and (iii)|q|2 = q2 + (ω/c)2. Since charge conservation implies that

qµJµ(q) = 0 (26.90)

in momentum space, it appears that only the part

GF,µνL,eff (q) =

1

q2 + iε

(

ωc

)2nµnν

q2 +(

ωc

)2 (26.91)

of the longitudinal photon propagator contributes effectively (eff) to HLI (q). One now adds

this part to the scalar photon propagator, with elements written in covariant form as follows:

GF,µνS (q) = − nµnν

q2 + iε. (26.92)

The result in Eq. (26.92) appears readily from Eqs. (26.73) and (26.78). Adding the prop-agators in Eqs. (26.91) and (26.92) one obtains

GµνC (q) ≡ GF,µν

L,eff (q) + GF,µνS (q) = −n

µnν

|q|2 . (26.93)

In configuration space one thus gets

GµνC (x) = −gµ0gν0

∫ ∞

−∞

eiq·r

|q|2d3q

(2π)3

∫ ∞

−∞e−iq0x0 dq0

2π

= −gµ0gν0 δ(ct)4πr

. (26.94)

It now appears that the effective exchange of longitudinal and scalar photons correspondsto the instantaneous [δ(ct)] Coulomb interaction [∼ r−1] between charges (therefore thesubscript C on the propagator).

Part VII

Photon vacuum and quanta inMinkowskian space

27

Photons and observers

All quantum physical phenomena are maximally closed. Thus, if a photon emitted by asource later is registered in a detector, it is not meaningful “to try to find out what mighthave happened to the photon on its travel from source to detector.” To find out means tomake an experiment. Hence, if a new detector is placed somewhere on the photon route, onehas a new phenomenon. In a double-slit experiment the wave interference disappears if onetries to determine which way the particle went. Such mutually exclusive (complementary)particle and wave experiments, represent in the words of Bohr [27], “a rational generalizationof classical physics.” To obtain human knowledge of the photon we must make observations.

The quantized electromagnetic field is more than the photons. In a photon empty spaceone still has a number of important quantum physical phenomena, e.g., the Lamb shift andspontaneous emission. Furthermore, an excited atom approaching a photon empty cavitycan be reflected provided its velocity is slow enough. The reflection of the atom takes placedue to the fact that the (almost) discontinuous change in the coupling to the vacuumfluctuations at the input of the cavity gives rise to a potential barrier for the center-of-mass wave function of the atom. The dynamics of the vacuum field can be studied via acalculation of symmetric correlation functions. It appears from such a calculation that thevacuum fluctuations have a very short correlation time [53].

In Minkowskian space inertial observers have a privileged status. For this reason it isof fundamental importance to investigate how a Lorentz boost possibly affects our viewof the presence of T-, L-, and S-photons in free space. We already know from the photonwave mechanical description given in Part III that the longitudinal [AL(q) = κ · AL(q)]and scalar [A0(q)] parts of the four-potential are identical in free space. Using the Lorenzgauge condition plus the invariance of the scalar product qµq

µ we show (Subsec. 28.2.1)that the equality of AL(q) and A0(q) is a Lorentz invariant property. By a suitable gaugetransformation within the Lorenz gauge it is even possible to make the choice AL(q) =A0(q) = 0. Starting in a given inertial frame O from the choice Aµ(x) = (0,AT (q)),where only transverse photons are present, a boost to another inertial frame O′ “generates”both longitudinal and scalar photons. Since the number of produced L and S-photons is thesame, in every inertial frame only the transverse photons contribute to the total energy ofthe free electromagnetic field.

In Subsec. 28.2.2 we study the covariant potentials plane-mode decompositions, andmake the bridge to the well-known covariant photon description. The mode functions ob-tained are studied further in Sec. 28.3. On the basis of the relativistic definition of the innerproduct, it is shown that the positive-frequency mode functions, fr(x), are orthonormal-ized in a generalized sense, and that the complex conjugate f∗s -modes are orthogonal to thefr-modes. The mode function satisfies the dynamical equation

∂

∂tfr(x) = −iωrfr(x) (27.1)

in a given inertial frame, O. A Lorentz boost to the inertial frame O′ results in a new modefunction, fr(x) ⇒ f ′r(x

′), which satisfies a dynamical equation form-identical to the one givenin Eq. (27.1) but with a relativistically shifted Doppler frequency, ω′

r = γ(ωr−v ·qr). Since

323


the frequency in the O′ frame also is positive, a Lorentz boost does not mix the positiveand negative-frequency modes in the plane-wave expansion of the four-potential. The statedescribing a collection of photons with certain momenta is boosted into a state describingthe same photons (so to speak), but with boosted momenta. Thus, the total number ofoperators in the two frames will coincide. In particular the vacuum state will coincide forthe O and O′ inertial observers.

The first-quantized description given in Secs. 28.1-28.2 is extended to the second-quantized level in Sec. 28.3 with the particular purpose of studying the vacuum. The vacuumstate is defined as the state in which there are no photons of any kind (T,L,S) present. Let usconcentrate on a single wave-vector (q) mode. For this mode one may write the VACUUMstate in the obvious notation

|0〉 = |0T 〉 ⊗ |0L〉 ⊗ |0S〉. (27.2)

In free space the most general allowed LS-Fock state, |ψLS〉, necessarily must hold the samenumber of L and S-quanta. With the near-field (NF) annihilation operator defined by

aNF (q) =i√2(aL(q)− aS(q)) (27.3)

for mode q, the most general allowed LS-state belonging to q has the form

|ψLS〉 =∞∑

n=0

cn

(

a†NF (q))n

|0LS〉. (27.4)

Since there is only a net effect of the L and S-photons in the presence of field-matterinteraction, in a PHYSICAL VACUUM state,

|0PHY S〉 = |0T 〉 ⊗ |ψLS〉, (27.5)

only the number of transverse photons needs to be zero. In analogy with the result obtainedin the first-quantized theory, it is possible in the second-quantized formalism to remove thefree-space longitudinal and scalar photons form |ψLS〉 by a suitable gauge transformationwithin the Lorenz gauge. We show this explicitly for a |ψLS〉-state containing an admixtureof one L-photon and one S-photon.

Let us assume that we have made an expansion of the four-potential operator A(x) interms of a given set of mode functions fi(x) ≡ fr(x,q), i.e.,

A(x) =∑

i

[

aifi(x) + a†i f∗i (x)

]

, i = (r,q). (27.6)

For an observer using the f -modes in the quantization procedure the global photon vacuumstate |0f 〉 is defined by

ai|0f〉 = 0, ∀i. (27.7)

Since the basis set fi(x) is not unique, a second observer may use a different basis set,

gj(x), for the expansion of A(x). Thus,

A(x) =∑

j

[

bjgj(x) + b†jg∗j (x)

]

. (27.8)

For this observer the global vacuum state, |0g〉, is defined by

bj |0g〉 = 0, ∀j. (27.9)

Photons and observers 325

Do the f and g-observers agree upon what they characterize as a photon vacuum? If notwhat is then the “correct” physical vacuum? Since the two sets of mode functions arecomplete the gj-modes can be expanded after the fi-modes (and conversely). Thus,

gj(x) =∑

i

[αjifi(x) + βjif∗i (x)] . (27.10)

The relation in Eq. (27.10) together with its inverse are known as the Bogolubov trans-formation [22, 23] (Sec. 29.1). If the two observers are to agree upon what they consideras a photon vacuum, the g-observer must find that the f -observer vacuum, |0f 〉, also is avacuum seen in her perspective. If ng,i is the mode (i) number operator for the g-observer,then it can be shown (Sec. 29.2) that the mean number of gi-particles in the f -vacuum isgiven by

〈0f |ng,i|0f〉 =∑

j

ζj |βij |2, (27.11)

where ζj = +1 for r = 1− 3, and ζj = −1 for r = 0. Thus, if any of the βij coefficients arenonvanishing the |0f〉-state is not a photon vacuum state for the g-observer.

Let us assume that the f -observer is an inertial observer (frame O) in Minkowski space,and that he uses monochromatic (ωf ) plane-wave mode functions. If the g-observer alsois an inertial observer (boosted frame O′) using monochromatic (ωg) plane-wave modefunctions, the relativistic Doppler shift, ωg = γ(ωf − v · qf ), indicates that the positiveand negative frequencies are not mixed in the f → g Bogolubov transformation. From Eq.(27.10) it then appears that all the βji’s are zero. Hence, one can conclude from Eq. (27.11)that 〈0f |nj,i|0f〉 = 0, ∀i. In Minkowski space all inertial observers agree upon what theycharacterize as a physical photon vacuum. The privileged status of inertial observers inMinkowski space, makes the Minkowski vacuum, |0M 〉(= |0f 〉 = |0g〉) a “natural” choice forthe “correct” physical vacuum.

In Sec. 29.3 we shall learn that an observer that accelerates uniformly through theMinkowski vacuum will observe a thermal (Planck) spectrum of particles [19]. This phe-nomenon often is named the Unruh effect [233, 234]. Technically, we reach this conclusionusing a so-called Rindler coordinatization [203] in Minkowski space. From a historical pointof view it is interesting that Max Born already in 1909 anticipated many features of Rindler’swork [37]. The associated particles (photons) are called Rindler particles [19]. The numberdistribution in the Planck spectrum, viz.,

nPlanck(ω) =

[

exp

(

~ω

kBT0

)

− 1

]−1

, (27.12)

is characterized by an effective absolute temperature

T0 =a

2πkB, (27.13)

a being the magnitude of the four-acceleration on a certain (ξ = 0) Rindler trajectory; seeSubsec. 29.3.1.

In general relativity inertial observers become free-falling observers, and in general de-tectors in different free falls will not agree on a definition of vacuum. The whole question ofthe meaning of the vacuum and particle (photon) concept cannot be answered without con-sidering the measurement process. The realization that the photon particle concept does nothave a universal meaning but depends on the motion of the detector raises deep questionsin quantum electrodynamics. On top of this the measurement process in quantum physics


cannot be described alone via the deterministic development of the Schrodinger (Dirac)equation. In a measurement the global wave function collapses to one of the eigenstatesfor the relevant observable, at least in theory. Since all detections inevitably are linked to(human) observations in the macroscopic world, measurement theory involves a study ofthe borderline problem between quantum and classical physics.

Since the Hamilton and momentum operators of the field can be obtained by an integra-tion of (certain components) of the energy-momentum stress tensor Tµν over all space, itmight be useful to connect the state of the electromagnetic field to the locally defined quan-tity 〈ψ|Tµν(x)|ψ〉. For a fixed field state, |ψ〉, the results obtained by different observerswould then be related by tensor transformation. Hence, if 〈ψ|Tµν |ψ〉 = 0 for one observer,the mean value of Tµν(x) will vanish for all observers. Different observers then agree uponwhat they see as a physical photon vacuum. The connection of the state-of-the-field conceptto a locally (in space-time) defined quantity also might turn out to be fruitful seen in theperspective of the spatial localization problem for T-photons.

In Sec. 12.3 we realized that transverse photons composed of sufficiently high frequen-cies acquire an effective mass in their diamagnetic interaction with a many-body electronsystem, provided the electron density can be considered as homogeneous. In Sec. 12.4 thenonrelativistic analysis of Sec. 12.3 was extended to a covariant description of the interac-tion between the photon field, Aµ(x), and a spinless charged boson particle. Provided thenorm of the boson scalar wave function is robust we were able to conclude that Aµ(x)satisfies the Proca equation.

In Chapt. 30, we shall realize that even a photon in global vacuum can acquire a mass,provided we are prepared to revise our view on the nature of the physical photon vacuum.Notwithstanding the circumstance that we have no experimental evidence for a vacuummass of the T-photon, the formalism described in Chapt. 30 in itself is potentially useful inphoton physics, e.g., in relation to nonlinear optics and near-field optics.

If the photon vacuum in some sense is analogous to the ground state of an interactingmany-body system, it is possible that the lowest energy level (ground state) of the electro-magnetic field is degenerate (non-unique) [167]. The eigenstate of a selected ground stateis not invariant under the symmetry transformations of the system Lagrangian. This asym-metric situation is precisely what is needed to generate so-called vacuum screening currents[6]. The Goldstone model [81] is the simplest example of a field theory exhibiting spon-taneous symmetry breaking [243, 156, 6]. Using this model we reach the Goldstone bosonand Higgs particle concepts. The complex Higgs field φ(x) (with charge q) [96, 97, 98] nextis coupled to the electromagnetic field, Aµ(x), by the usual replacement of the ordinaryderivatives by the gauge-covariant derivatives. On the basis of the U(1) Higgs model, oneis led to (i) a photon vacuum mass

M =|q|vc, (27.14)

where v =√2φ0, φ0 being the displaced vacuum ground state scalar field, and (ii) a vacuum

screening (sc) current density

jV AC,µsc = −

( q

~

)2(

2vH(x) +H2(x))

Aµ(x). (27.15)

Classically, the real Higgs field, H(x), represents (small) displacements in the radial planefrom the minima of the Goldstone potential. Upon quantization H(x) leads to neutral spin-zero Higgs particles (bosons). We finish Chapt. 30 with a brief discussion of the ’t Hooftgauge and the related propagator [230]. In the ’t Hooft propagator

Gµν′t Hooft = − gµν

q2 −Q2C

, (27.16)

Photons and observers 327

where QC = |q|v/~, the troublesome qµqν/Q2C-term, appearing in the analysis in Sec. 12.5,

is absent.

28

The inertial class of observers: Photon vacuum and

quanta

28.1 Transverse photon four-current density

Let us start with a division of the transverse vector potential, AT (r, t), into its positive-and negative-frequency parts, i.e.,

AT (r, t) = A(+)T (r, t) +A

(−)T (r, t). (28.1)

As we know, the positive-frequency (analytic) part,

A(+)T (r, t) ≡ F(r, t), (28.2)

relates to the transverse photon (F) concept, and the negative-frequency part,

A(−)T (r, t) ≡ FA(r, t), (28.3)

to the transverse antiphoton (FA) concept. In free space the two parts satisfy the waveequations

F(r, t) = 0, (28.4)

and

FA(r, t) = 0, (28.5)

as a consequence of the fact that AT (r, t) = 0; cf. the analysis in Sec. 2.3.A transition four-current density concept between two positive-frequency transverse

vector-potential distributions F1(r, t) and F2(r, t), satisfying the free-space wave equations

F1(r, t) = F2(r, t) = 0, (28.6)

can be introduced in the following manner: Multiply F1 by the Hermitian conjugate F†2

of F2 from the left, and subtract the Hermitian conjugate F†2 of the second member of

Eq. (28.6) multiplied from the right by F1. Thus,

F†2F1 −

(

F†2

)

F1 = 0, (28.7)

or equivalently

∇ ·[

F†2∇F1 −

(

∇F†2

)

F1

]

+∂

c∂t

[(

∂

c∂tF†

2

)

F1 − F†2

∂

c∂tF1

]

= 0. (28.8)

329


In Eq. (28.8), a notation has been used in which the ith (i = 1 − 3) component of a formα†∇β is defined by α†(∂β/∂xi), i.e., as the product of the row vector α∗ and the columnvector ∂β/∂xi. From Eq. (28.7) and onward the dot (·) used when multiplying row andcolumn vectors in Sec. 15.6 is omitted. Remembering the expressions for ∂µ [Eq. (3.42)]and ∂µ [Eq. (3.44)] it appears that Eq. (28.8) (multiplied by a certain constant K) canbe written in the covariant form

∂µJµ1→2(x) = 0, (28.9)

where

Jµ1→2(x) = K

[

F†2∂

µF1 −(

∂µF†2

)

F1

]

. (28.10)

It makes sense to consider Jµ1→2(x) as a T-photon transition four-current density from

the field distribution F1 to the distribution F2, and Eq. (28.9) as the related equation ofcontinuity. In three-vector notation Eq. (28.9) is written as

∇ · J1→2(r, t) +∂

∂t

(

c−1J01→2(r, t)

)

= 0. (28.11)

By integrating Eq. (28.11) over the entire r-space, using Gauss theorem, and assuming (asusual) that the surface integral vanishes at infinity, it follows that the quantity

〈F2|F1〉 ≡∫ ∞

−∞c−1J0

1→2(r, t)d3r (28.12)

is time independent. As suggested by our notation,

〈F2|F1〉 =K

c2

∫ ∞

−∞

[(

∂

∂tF†

2

)

F1 − F†2

∂

∂tF1

]

d3r (28.13)

is the relativistic definition of the inner product of F1 and F2 up to a constant, K. Below,this constant will be chosen in such a manner that the one-photon states are normalized,i.e., 〈F|F〉 = 〈Φ|Φ〉 = 1. I here remind the reader of the treatment given in Secs. 15.4-15.6.

It appears from Eq. (15.11) that the transverse vector-potential fields associated withtwo (i = 1, 2) monochromatic plane waves in the continuum representation are given by

Fi(r, t) ≡ fi(r, t) = (2π)− 3

2

(

~

2ǫ0ωi

)1

2

εsi(qi)ei(qi·r−ωit), i = 1, 2, (28.14)

where ωi = c|qi|. The inner product of these states is obtained from Eq. (28.13). Thus,

〈f2|f1〉 =K

c2~

2ǫ0(ω1ω2)

− 1

2 i (ω1 + ω2) ei(ω2−ω1)t

× ε†s2(q2) · εs1(q1)

(

1

2π

)3 ∫ ∞

−∞ei(q1−q2)·rd3r. (28.15)

Since the integral [divided by (2π)3] is just the Dirac delta function, the inner productbecomes

〈f2|f1〉 =K

c2i~

ǫ0δs1s2δ(q1 − q2). (28.16)

The inertial class of observers: Photon vacuum and quanta 331

By requiring that the inner product relating to the fi’s is orthonormalized in the Diracsense, we have

〈f2|f1〉 = δs1s2δ(q1 − q2). (28.17)

Consistency of the last two equations hence gives

K =ǫ0c

2

i~. (28.18)

With this choice for K the T-photon transition four-current density [Eq. (28.10)] takes theexplicit form

Jµ1→2(x) =

ǫ0c2

i~F†

2∂µF1, (28.19)

using the standard abbreviation

F†i∂

µFj ≡ F†i∂

µFj −(

∂µF†i

)

Fj . (28.20)

We now specialize our general considerations to polychromatic single-photon states. IfF(x) denotes the positive-frequency part of the vector potential associated to the T-photon,the related four-current density will be given by

Jµ(x) =2ǫ0c

2

~ℜ[

F†(x)∂µF(x)]

. (28.21)

One may prove this starting from Eq. (28.10), with K given by Eq. (28.18). For F1 = F2 =F, the two terms on the right-hand side of Eq. (28.20) divided by the imaginary unit, areeach other’s complex conjugate, that is

i−1F†∂µF = 2ℜ[

F†∂µF]

. (28.22)

This connection immediately leads to the expression given for the T-photon four-currentdensity in Eq. (28.21).

Our introduction of the single-photon wave function concept, Φ(r, t), via the meanposition state |R〉(r, t) [see Secs. 15.5 and 15.6] makes it clear that

F(x) =

(

~

2ǫ0c

)1

2

Φ(x). (28.23)

Expressed in terms of Φ(x) the four-current density takes the form

Jµ(x) = cℜ[

Φ†(x)∂µΦ(x)]

. (28.24)

For the inner product

〈F|F〉 = iǫ0c

~

∫ ∞

−∞F†(x)∂0F(x)d3r, (28.25)

Eq. (28.23) gives

〈F|F〉 = i

c

∫ ∞

−∞Φ†(x)

∂

∂tΦ(x)d3r. (28.26)


The expression of the right side of this equation is identical to the one in Eq. (15.129), andin view of Eqs. (15.90), (15.91), and (15.93), one obtains

〈F|F〉 = 〈Φ|Φ〉 = 1, (28.27)

as required.If one, with K still given by Eq. (28.18), maintains the definition of the inner product in

Eq. (28.13) also for the negative-frequency part of the transverse vector potential, namely

〈FA,2|FA,1〉 =ǫ0i~

∫ ∞

−∞

[(

∂

∂tF†

A,2

)

FA,1 − F†A,2

∂

∂tFA,1

]

d3r, (28.28)

the relation

FA(x) = F∗(x) (28.29)

leads to the following connection between the scalar products for the particle and antiparticleparts of AT (x):

〈FA,2|FA,1〉 = −〈F2|F1〉∗. (28.30)

The four-current densities of the T-photon and its antiparticle are identical, i.e.,

JµA(x) = Jµ(x), (28.31)

a result which follows immediately by combining Eqs. (28.21) and (28.29).

28.2 Boosts

28.2.1 Lorentz and Lorenz-gauge transformations of the four-potential

Let us consider the situation where a frame O′ moves with a uniform velocity v as seenfrom an inertial system O. Under the assumption that the coordinate axes in O and O′ areparallel the associated homogeneous Lorentz transformation (O → O′) of the coordinates[(ct, r) → (ct′,x′)] has the form

ct′ = γ (ct− β · x) , (28.32)

x′ = x− γβct+ (γ − 1)eβeβ · x, (28.33)

with eβ = β/β, β = v/c and γ = [1 − β2]−1/2. We know from the analysis in Sec. 10.1that the scalar (A0) and vector (A) potential form a four-vector potential Aµ(x) in allcovariant gauges. In such gauges the Lorentz transformation (O → O′) of the covariantpotential [Aµ(x) → (Aµ)′(x′)] is given by

(A0)′ = γ(

A0 − β ·A)

, (28.34)

A′ = A− γβA0 + (γ − 1)eβeβ ·A. (28.35)

The transformation given by Eqs. (28.34) and (28.35) has the same form in the (ω/c,q)-domain, Aµ(q) → (Aµ)′(q′). In the covariant Lorenz gauge the scalar potential A0(q)


and the longitudinal (rotational-free), AL(q), part of the vector potential, A(q), are relatedto one another by

|q|AL(q) =ω

cA0(q), (28.36)

where AL(q) = κ ·AL(q), κ = q/|q|. In free space, where |q| = ω/c, one obtains

AL(q) = A0(q). (28.37)

The Lorentz invariance of Eq. (28.36), together with the invariance of the scalar productqµq

µ [which in free space gives |q′| = ω′/c] show that

A′L(q

′) = (A0)′(q′). (28.38)

In free space the equality of the longitudinal (AL(q)) and scalar (A0(q)) potentials hence isa Lorentz invariant property.

For what follows it is convenient to assume that the free-space vector potential in O istransverse, i.e., A = AT , and thus AL(q) = A0(q) = 0 in the Lorenz gauge. Is it possibleto make the choice

Aµ(q) =(

A0(q),A(q))

= (0,AT (q)) (28.39)

for the four-vector potential? To answer this question let us consider a gauge transformationfrom OLD to NEW potentials:

ANEWL (q) =AOLD

L (q) + i|q|χ(q), (28.40)

(A0)NEW (q) =(A0)OLD(q) + iω

cχ(q). (28.41)

In free space Eq. (28.41) may in the Lorenz gauge be written as

(A0)NEW (q) =AOLDL (q) + i|q|χ(q) (28.42)

in view of Eq. (28.37), and the dispersion relation |q| = ω/c. A choice

χ(q) =i

|q|AOLDL (q) (28.43)

for the gauge function, hence gives

ANEWL (q) =

(

A0)NEW

(q) = 0. (28.44)

A gauge transformation with the χ(q) in Eq. (28.43) is a gauge transformation within theLorenz gauge for the free potentials since the result in Eq. (28.44) is in agreement with thegauge invariant Lorenz condition [Eq. (28.36)].

28.2.2 Plane-mode decomposition of the covariant potential

It is of fundamental importance to understand how a given Lorentz boost of the four-potential affects our view on the presence of T-, L-, and S-photons. A quantitative study ofthis question turns out to be quite comprehensive, and below we shall limit ourselves to afew qualitative considerations.


We start from the general Fourier integral expansion of the µth component of the co-variant four-potential, viz.,

Aµ(x) = (2π)−4

∫ ∞

−∞Aµ(q)eiqνx

ν

d4q. (28.45)

Since the scalar product qνxν and the four-dimensional volume element d4q are Lorentz

invariants, the Fourier amplitude Aµ(q) must be a four-vector. In free space Aµ(q) canonly be nonvanishing if the squared dispersion relation qµq

µ = |q|2 − (q0)2 = 0 is satisfied.In view of this Aµ(q) can be expressed as

Aµ(q) = 4πδ[

(

q0)2 − |q|2

]

Cµ(q0,q)

=2π

|q|[

δ(

q0 − |q|)

+ δ(

q0 + |q|)]

Cµ(q0,q). (28.46)

The Lorentz invariance of the argument of the Dirac delta function implies that alsoCµ(q0,q) must be a four-vector. When inserting the last member of Eq. (28.46) intoEq. (28.45) the presence of the delta functions makes it possible to carry out the integra-tion over q0 immediately with the result that

Aµ(x) =

∫ ∞

−∞

[

Cµ(|q|,q)ei(q·r−c|q|t)

+Cµ(−|q|,q)ei(q·r+c|q|t)] d3q

(2π)3|q| . (28.47)

A variable interchange q → −q in the integral containing Cµ(−|q|,q) allows one to expressAµ(x) in the integral form

Aµ(x) =

∫ ∞

−∞

[

Cµ(|q|,q)ei(q·r−c|q|t)

+Cµ(−|q|,−q)e−i(q·r−c|q|t)] d3q

(2π)3|q| . (28.48)

Since Aµ(x) is real the Cµ-coefficients are related as follows:

Cµ(−|q|,−q) = (Cµ(|q|,q))∗ . (28.49)

It appears from Eq. (3.75) that a Lorentz boost does not mix the positive-frequency partof the four-potential,

Aµ(+)(x) =

∫ ∞

−∞Cµ(|q|,q)ei(q·r−c|q|t) d3q

(2π)3|q| , (28.50)

and the negative-frequency part

Aµ(−)(x) =(

Aµ(+)(x))∗. (28.51)

To make the bridge to the covariant photon description we introduce new coefficientsαµ(|q|,q) related to Cµ(|q|,q) via

Cµ (|q|,q) =[

(2π)3~|q|2ǫ0c

]1

2

αµ (|q|,q) . (28.52)


The quantity

αµ (|q|,q) ≡ αµ(q) (28.53)

is not a four-vector, but the scaled quantity |q|1/2αµ(q) is because Cµ (|q|,q) is a four-vector. In terms of the αµ(q)-coefficients the µ-component of the four-vector potential hasthe integral representation

Aµ(x) =

[

~

2ǫ0c(2π)3

]1

2∫ ∞

−∞|q|− 1

2

(

αµ(q)eiqνxν

+ c.c.)

d3q, (28.54)

remembering that

qν = (−|q|,q) (28.55)

in Eq. (28.54). If one expands αµ(q) after the four (r = 0 − 3) linearly independentpolarization unit vectors εµr [satisfying Eqs. (26.49) and (26.50)], i.e.,

αµ(q) =

3∑

r=0

αr(q)εµr (q), (28.56)


Aµ(x) =

3∑

r=0

∫ ∞

−∞(αr(q)f

µr (x) + c.c.) d3q, (28.57)

where

fµr (x) =

[

~

2ǫ0c|q|(2π)3]

1

2

εµr (q)eiqνx

ν

. (28.58)

On the basis of the expansion given in Eq. (28.57) it can be shown [127, 53] that the totalenergy (H) in the electromagnetic field can be written in the form

H =

3∑

r=0

∫ ∞

−∞~c|q|ζrα∗

r(q)αr(q)d3q. (28.59)

In an inertial frame where the free-space vector potential is purely transverse, the plane-wave expansion of AT (x) is given by Eq. (15.11). In the continuum limit (V → ∞) the ithcomponent of the three-vector potential thus becomes

AiT =

2∑

s=1

∫ ∞

−∞

(

αs(q)fis(x) + c.c.

)

d3q, (28.60)

where

f is(x) =

[

~

2ǫ0c|q|(2π)3]

1

2

εis(q)eiqνx

ν

. (28.61)

The index s refers to the two orthogonal unit polarization vectors [ε1(q), ε2(q)] used in thegiven expansion. If the choice given in Eqs. (26.78)-(26.80) is used in the expansion in Eq.(28.57) the contributions to Aµ

r (x) from r = 1, 2 become identical to the s = 1, 2 parts of


the transverse vector potential AiT (x). The remaining scalar (Aµ

S(x)) and longitudinal(Aµ

L(x)) contributions to (Aµ(x)) do not vanish even in free space. A boost from aninertial frame where only transverse photons are present thus “generates” both longitudinaland scalar photons. It must be remembered however that the dynamical variables αL(q)[≡α3(q)] and αS(q)[≡ α0(q)] belonging to these photon types are identical in free space, i.e.,

α0(q) = α3(q). (28.62)

In consequence only the transverse photons contribute to the total energy of the free elec-tromagnetic field, a physically satisfactory conclusion, cf. the analysis in Sec. 11.2.

28.2.3 Mode functions

It appears from Eq. (28.58) that the four-vector mode function belonging to a given polar-ization state, r, and a given wave vector, qr, is

fr(x) ≡ fµr (x) =

[

~

2ǫ0c|qr |(2π)3]

1

2

εµr (qr)eiqνrxν . (28.63)

On the basis of the relativistic definition of the inner product, namely,

〈fr |fs〉 =ǫ0i~

∫ ∞

−∞

[(

∂

∂tf†r

)

fs − f†r∂

∂tfs

]

d3r, (28.64)

[which is just the generalization of Eq. (28.13) from three- to four-vectors], and the or-thonormality condition for the unit polarization vectors [Eq. (12.99)] one obtains

〈fr|fs〉 = ζrδrsδ(qr − qs). (28.65)

The reader may easily verify this generalized orthonormalization, remembering Eqs. (28.14)-(28.18). The mode functions satisfy the dynamical equation

∂

∂tfr(x) = −iωrfr(x), (28.66)

where ωr = c|qr|(> 0). The fr-modes are said to be positive-frequency modes; cf. thestudy of complex analytical signals in Sec. 2.3. The complex conjugate (negative-frequency)f∗r -modes satisfy the equation

∂

∂tf∗r (x) = iωrf

∗r (x), (28.67)

and are orthonormal, but with norm which has the opposite sign than that of the correspond-ing fr-modes, i.e.,

〈f∗r |f∗s 〉 = −ζrδrsδ(qr − qs). (28.68)

The results in Eqs. (28.65) and (28.68) represent a generalization of the three-vector resultsin Eqs. (28.17) and (28.30) [with Fi = fi]. I leave it to the reader to show that the complexconjugate f∗s -modes are orthogonal to the fr-modes:

〈fr|f∗s 〉 = 0. (28.69)

The dynamical equation for the positive-frequency mode fr(x), given in Eq. (28.66),


refers to a given inertial frame, O. A Lorentz boost to the inertial frame O′ (same as inSec. 28.2.1) results in a new mode function f ′r(x

′) satisfying the dynamical equation

∂

∂t′f ′r(x

′) = −iω′rf

′r(x

′), (28.70)

where

ω′r = γ (ωr − v · qr) (28.71)

is the relativistic Doppler shifted frequency. Since the frequency in the boosted frame is pos-itive, a Lorentz boost will not mix positive- and negative-frequency modes in the expansionof Aµ(x) [Eq. (28.57)].

28.3 Physical (T-photon) vacuum

The first-quantized field description is extended to the second-quantized level by elevatingthe expansion coefficient αr(q) and α∗

r(q) to operators: αr(q) ⇒ ar(q), α∗r(q) ⇒ a†r(q).

The annihilation and creation operators satisfy the commutation relations in Eqs. (26.52)and (26.53). The extension of Eq. (28.57) thus gives one the plane-mode expansion of thefour-potential operator, viz.,

Aµ(x) =

3∑

r=0

∫ ∞

−∞

[

ar(q)fµr (x) + a†r(q) (f

µr (x))

∗]d3q. (28.72)

The expansion in Eq. (28.72) is just the continuum version of the discrete expansion givenby Eqs. (26.41), (26.45), and (26.46). The Hamilton operator of the covariant photon theory(with the vacuum energy part omitted) takes the form

H =

3∑

r=0

∫ ∞

−∞~c|q|ζra†r(q)ar(q)d3q (28.73)

in the continuum limit [extension of Eq. (28.59)], and its discrete mode version is given byEqs. (26.56) [vacuum contribution omitted] and (26.57).

The vacuum state |0〉 is defined as the state in which there are no photons of any kindpresent, i.e., ar(q)|0〉 = 0, ∀q, r = 0− 3 [Eq. (26.55)]. We know from the classical analysisgiven in Sec. 28.2.1 that a Lorentz boost in general will change the transverse, longitudinaland scalar parts of the four-potential. The four-component potential is a four-potentialonly in the covariant family of Lorenz gauges. In these gauges the longitudinal and scalarpotentials are linked. In the Lorenz gauge, the relation is in q-space given by Eq. (28.36),and in direct space by Eq. (11.4). In the field-quantized theory the classical Lorenz conditionis replaced by Eq. (26.54), which is a restriction on the states |Ψ〉 allowed by the theory.By inserting the positive-frequency part of Aµ(x) [Eq. (28.72)] into Eq. (26.54) one obtainsin free space the condition

[

3∑

r=0

∫ ∞

−∞ar(q)qµf

µr (x)d

3q

]

|Ψ〉 = 0. (28.74)


This condition is satisfied for all r and ct if

3∑

r=0

ar(q)qµεµr (q)|Ψ〉 = 0, ∀q. (28.75)

To relate the condition in Eq. (28.75) to the longitudinal and scalar modes, we make thechoice given in Eqs. (26.78)-(26.81) for the unit polarization vectors. With qµ = (−|q|,q)the free-field Gupta–Bleuler–Lorenz condition in Eq. (26.54) takes the form [127, 53]

[aL(q) − aS(q)] |Ψ〉 = 0, ∀q, (28.76)

with the notation a0(q) = aS(q), a3(q) = aL(q).For what follows it is sufficient to consider the L and S-photons belonging to a single

wave-vector mode (q). Quantum states in the related subspace of Hilbert space we denoteby |ΨLS〉. The LS-vacuum state,

|0LS〉 = |0L, 0S〉 = |0L〉⊗

|0S〉, (28.77)

in which there are no photons present satisfies

ar(q)|0LS〉 = 0, r = L, S. (28.78)

In near-field (NF) electrodynamics the annihilation operator

aNF (q) =i√2(aL(q) − aS(q)) , (28.79)

and its related photon variable, play an important role [127, 125, 129]. In terms of aNF (q)the free-field subsidiary condition on the LS-subspace reads

aNF (q)|ΨLS〉 = 0. (28.80)

Hence, in free space there are no near-field photons, in agreement with the fact that the NF-photon concept is connected to rim-zone electrodynamics [127, 125, 129]. In the presenceof field-matter interactions the Gupta–Bleuler–Lorenz condition takes the form [127]

aNF (q)|ΨLS〉 =∆(q)

i√2|ΨLS〉, (28.81)

where

∆(q) =c

~ωq

(

~

2ǫ0ωq(2π)3

)1

2

ρ(q) (28.82)

is a quantity proportional to the charge density in reciprocal space, ρ(q).It is pretty clear that the most general allowed LS-Fock state must contain the same

number of L and S-photons in free space. In consequence, the state which contains the mostgeneral allowed admixture of longitudinal and scalar photons in a given q-mode has theform

|ΨLS〉 =∞∑

n=0

cn

(

a†NF

)n

|0LS〉, (28.83)


where the cn’s are complex constants. To prove that the state in Eq. (28.83) is allowed bythe theory we make use of the commutator relation

[

aNF (q), a†NF (q)

]

= 0. (28.84)

The reader may readily verify Eq. (28.84) starting from the commutator relation [Eq. (26.52)for q = q′]

[

ar(q), a†s(q)

]

= ζrδrs, (28.85)

among the L and S operators. Thus,

aNF (q)|ΨLS〉 =∞∑

n=0

cnaNF

(

a†NF

)n

|0LS〉

=∞∑

n=0

cn

(

a†NF

)n

aNF |0LS〉 = 0, (28.86)

and the state in Eq. (28.83) therefore satisfies the Gupta–Bleuler–Lorenz condition. Thenorm of this state is

〈ΨLS |ΨLS〉 =∞∑

m=0

∞∑

n=0

〈0LS|c∗mcnamNF

(

a†NF

)n

|0LS〉

=

∞∑

m=0

∞∑

n=0

c∗mcn〈0LS|(

a†NF

)n

amNF |0LS〉

= |c0|2〈0LS |0LS〉, (28.87)

and provided the LS-vacuum state is normalized we have

〈ΨLS|ΨLS〉 = |c0|2. (28.88)

It might come as a surprise to the reader that the norm ||ΨLS〉|| depends on |c0|, only.However, the reason for this can be traced back to the fact that states with negative andzero norms appear in our formalism. At first sight this looks like a serious difficulty sincethe probability interpretation of quantum mechanics relates to states having positive norms.Since a net effect of longitudinal and scalar photons only appears in field-matter interac-tions, and here only in intermediate states the “difficulty” does not affect any observablequantities. In the Gupta–Bleuler formalism states with negative (zero) norms occur in aselfconsistent manner in a state space with an indefinite metric [127, 53]. What might becalled the PHYSICAL VACUUM, i.e., the state where there are no transverse photons, thushas the general form

|0PHY S〉 = |0T 〉 ⊗ |ΨLS〉. (28.89)

The VACUUM state

|0〉 = |0T 〉 ⊗ |0L〉 ⊗ |0S〉 (28.90)

is the state in which there are no photons of any kind present.The occurrence of negative and zero norms originates in the unusual form of one of the

commutator relations for S-photons belonging to the same q-mode, viz.,[

aS(q), a†S(q)

]

= −1, (28.91)


see Eq. (26.52). From the spectrum of normalized Fock states belonging to the scalar modesrelated to a given q, namely,

|nS(q)〉 =(aS(q))

nS

√nS !

|0S〉, (28.92)

nS = 0, 1, 2, · · · , one obtains by means of Eq. (28.91) the unusual relation

a†S(q)|nS(q)〉 = − (nS(q) + 1)1

2 |nS(q) + 1〉, (28.93)

and the usual relation

aS(q)|nS(q)〉 = (nS(q))1

2 |nS(q) − 1〉. (28.94)

As an example of negative norm, let us consider a scalar one-photon wave packet, |ΦS〉.In the continuum limit

|ΦS〉 =[

(2π)−3

2

∫ ∞

−∞φS(q)a

†S(q)d

3q

]

|0S〉, (28.95)

where φS(q) are the appropriate amplitude weight factors for a given polychromatic scalarphoton. The expression for the norm of this state is given by the double-integral

〈ΦS |ΦS〉 = (2π)−3

∫ ∞

−∞φS(q

′)φS(q)〈0S |aS(q′)a†S(q)|0S〉d3q′d3q. (28.96)

By using the commutation relation

[

aS(q′), a†S(q)

]

= −δ(q− q′), (28.97)

which is just the continuum version of Eq. (26.52) for scalar photons [r, s = 0], the matrixelement in Eq. (28.96) becomes

〈0S|aS(q′)a†S(q)|0S〉 = 〈0S |a†S(q)aS(q′)− δ(q− q′)|0S〉= −〈0S |0S〉δ(q− q′). (28.98)

By combining Eqs. (28.96) and (28.98), one gets

〈ΦS |ΦS〉 = −[∫ ∞

−∞|φS(q)|2

d3q

(2π)3

]

〈0S |0S〉. (28.99)

The integral in Eq. (28.99) is positive, and by adjusting the amplitude coefficients so that

∫ ∞

−∞|φS(q)|2

d3q

(2π)3= 1, (28.100)

we obtain

〈ΦS |ΦS〉 = −〈0S|0S〉. (28.101)

It appears then that the scalar vacuum, |0S〉, and the polychromatic one-photon state, |ΦS〉,have norms with opposite signs. If |0S〉 is normalized to 〈0S |0S〉 = +1, the norm of |ΦS〉 isnegative. The adjustment in Eq. (28.100) gives it the value 〈ΦS |ΦS〉 = −1.


We have already come across states with zero norms, namely

|Ψ(n)LS (q)〉 =

(

a†NF (q))n

|0LS〉, ∀n 6= 0. (28.102)

Thus, it is obvious that

〈Ψ(n)LS (q)|Ψ

(n)LS (q)〉 = 0, ∀n 6= 0, (28.103)

since aNF (q) and a†NF (q) commute [Eq. (28.84)]. It is the result in Eq. (28.103), whichresulted in the value |c0|2 for the inner product of the |ΨLS〉-state given in Eq. (28.83).

The effect of the Gupta–Bleuler–Lorenz condition [Eq. (28.76)] becomes apparent if onecalculates the expectation value of the energy of an allowed state |Ψ〉. From Eq. (28.76) andits adjoint, viz.,

〈Ψ|a†S(q) = 〈Ψ|a†L(q), (28.104)

one has

〈Ψ|a†L(q)aL(q)− a†S(q)aS(q)|Ψ〉= 〈Ψ|a†L(q) (aL(q)− aS(q)) |Ψ〉 = 0, ∀q. (28.105)

It is obvious now that the mean value of H [Eq. (28.73)] is given by

〈Ψ|H |Ψ〉 =2∑

r=1

∫ ∞

−∞~c|q|a†r(q)ar(q)d3q. (28.106)

As a consequence of the subsidiary condition only transverse photons contribute to thequantum mechanical mean value of the field energy in free space, the expected result. Thesame is true for all other observables.

It was realized in Sec. 28.2 that it was possible by a gauge transformation within theLorenz gauge to remove the longitudinal and scalar potentials from a classical free-spacefour-vector potential:

AOLDL (q), (A0)OLD(q)

⇒

ANEWL (q), (A0)NEW (q)

= 0, 0. (28.107)

In the field-quantized theory an altering of an allowed admixture of longitudinal and scalarphotons is equivalent to a gauge transformation of the four-potential within the Lorenzgauge. Let us illustrate this by a simple example, in which an admixture of L and S-photons is removed by a suitable gauge transformation. Hence, let |ΨT 〉 be a free-field statecontaining transverse photons only, and

|Ψ〉 = 1 + c1

[

a†L(q) − a†S(q)]

|ΨT 〉 (28.108)

a superposition of |ΨT 〉 with two states in which, respectively, one longitudinal photon andone scalar photon is added to |ΨT 〉 in an allowed admixture. For such an admixture onlyone constant, here denoted by c1, is needed. Our goal now is to calculate the mean value ofthe four-potential operator Aµ(x) in the state |Ψ〉:

〈Ψ|Aµ(x)|Ψ〉 = 〈Ψ|Aµ(+)(x)|Ψ〉+ c.c, (28.109)

where, with discrete quantization [Eq. (26.45)],

Aµ(+)(x) = L− 3

2

∑

q

3∑

r=0

(

~

2ǫ0ωq

)1

2

εµr (q)ar(q)eiqνxν . (28.110)


By means of Eq. (28.108) and its adjoint, viz.,

〈Ψ| = 〈ΨT | 1 + c∗1 [aL(q)− aS(q)] (28.111)

one obtains

〈Ψ|Aµ(+)(x)|Ψ〉 = 〈ΨT |Aµ(+)(x)|ΨT 〉+ c1〈ΨT |Aµ(+)(x)

[

a†L(q)− a†S(q)]

|ΨT 〉

+ c∗1〈ΨT | [aL(q)− aS(q)] Aµ(+)(x)|ΨT 〉

+ |c1|2〈ΨT | [aL(q) − aS(q)] Aµ(+)(x)

[


|ΨT 〉. (28.112)

In Eq. (28.112) the terms proportional to c∗1 and |c1|2 both vanish. This is so becauseAµ(+)(x) only contains mode annihilation operators and the aL(q) − aS(q) and Aµ(+)(x)therefore commute. An interchange of these operators, in combination with the fact that

[aL(q)− aS(q)] |ΨT 〉 = 0 and [aL(q)− aS(q)][


|ΨT 〉 = 0 (since the state

|ΨT 〉 contains no L and S-photons). In the term proportional to c1 only the q-mode con-taining the L and S-photons contributes. Reverting to the notation aL(q) ≡ a3(q) andaS(q) ≡ a0(q), we obtain

c1〈ΨT |Aµ(+)(x)[

a†L(q) − a†S(q)]

|ΨT 〉

=

(

~

2ǫ0L3ωq

)1

2

c1〈ΨT |eiqx [ǫµ0 (q)a0(q) + εµ3 (q)a3(q)][

a†3(q) − a†0(q)]

|ΨT 〉

=

(

~

2ǫ0L3ωq

)1

2

c1〈ΨT |eiqx[

εµ3 (q)a3(q)a†3(q)− ǫµ0 (q)a0(q)a

†0(q)

]

|ΨT 〉

=

(

~

2ǫ0L3ωq

)1

2

c1〈ΨT | [εµ3 (q) + ǫµ0 (q)] eiqx|ΨT 〉

=

(

~

2ǫ0L3ωq

)1

2

c1〈ΨT |1

i|q|∂µ(eiqx)|ΨT 〉

=

(

~c2

2ǫ0L3ω3q

)1

2

〈ΨT |c1i∂µ(eiqx)|ΨT 〉. (28.113)

Altogether, we have reached the connection

〈Ψ|Aµ(+)(x)|Ψ〉 = 〈ΨT |Aµ(+)(x)|ΨT 〉

+

(

~c2

2ǫ0L3ω3q

)1

2

〈ΨT |c1i∂µ(eiqx)|ΨT 〉. (28.114)

The mean value of the four-potential operator Aµ(x) in the state |Ψ〉 [Eq. (28.109)] isobtained by addition of Eq. (28.114) and its complex conjugate. Hence,

〈Ψ|Aµ(x)|Ψ〉 = 〈ΨT |Aµ(x) + ∂µΛ(x)|ΨT 〉, (28.115)

where

Λ(x) =

(

2~c2

ǫ0L3ω3q

)1

2

ℜ(c1ieiqx

)

. (28.116)


We can now conclude that replacing |Ψ〉 [which contains one longitudinal photon and onescalar photon in the allowed admixture] by |ΨT 〉 [which contains no L and S-photons]corresponds to a gauge transformation of the four-potential. Since the gauge function Λ(x)satisfies

∂µ∂µΛ(x) = 0, (28.117)

the gauge transformation is a transformation within the Lorenz gauge, a satisfactory con-clusion.

29

The non-inertial class of observers: The nebulous

particle concept

29.1 Bogolubov transformation. Vacuum states

Let us now return to the plane-mode decomposition of the covariant four-potential alreadydiscussed in Subsec. 28.2.2 and and Sec. 28.3, and let the spatial mode spectrum be discrete.It appears from Eqs. (28.57) and (28.63) that the classical four-potential expanded afterthe complete set of orthonormalized fr(x,q) and f∗r (x,q) mode functions now reads

A(x) ≡ Aµ(x) =∑

r,q

[αr(q)fr(x,q) + α∗r(q)f

∗r (x,q)]

≡∑

i

[αifi(x) + α∗i f

∗i (x)] . (29.1)

In the last member of Eq. (29.1) the abbreviated notation i = (r,q) has been used. Themode functions satisfy a generalized orthonormalization condition, viz.,

〈fi|fj〉 = ζiδij , (29.2)

〈f∗i |f∗j 〉 = −ζiδij , (29.3)

〈fi|f∗j 〉 = 0, (29.4)

where, for all q, ζi = +1 for r = 1− 3, and ζi = −1 for r = 0. In a mode expansion of A(x)the choice of mode functions is not unique. Hence, one may consider the expansion in Eq.(29.1) in a generalized sense: There exists a complete set of positive [fi(x)]- and negative[f∗i (x)]-frequency mode solutions to the free-space wave equation ∂µ∂

µA(x) = 0, and themodes are orthonormal in the relativistic scalar product [Eq. (28.13), with K given by Eq.(28.18)].

The classical expansion in Eq. (29.1) is promoted to the quantum level by the elevation

of the expansion coefficients to operators: αi ⇒ ai, α∗i ⇒ a†i . The annihilation and creation

operators satisfy the usual commutation relations [Eqs. (26.52) and (26.52)]:[

ai, a†j

]

= ζiδij , (29.5)

etc. In terms of the fi-modes, the four-potential operator A(x) has the expansion

A(x) =∑

i

[

aifi(x) + a†i f∗i (x)

]

. (29.6)

The construction of a vacuum state, Fock space, etc., can be done in the standard mannerstarting from Eq. (29.5). The vacuum state, |0f〉, is defined by

ai|0f〉 = 0, ∀i. (29.7)

345


The subscript f on the vacuum state reminds us that the vacuum is defined with respectto the set of fi-modes. For each mode we have a number operator

nf,i = a†i ai, (29.8)

and a single-mode Fock state with nf,i excitations is created by repeated action by a†i inthe usual manner, i.e.,

|nf,i〉 =1

√

nf,i!

(

a†i

)nf,i

|0f 〉. (29.9)

Consider now a second complete set of positive [gj(x)]- and negative [g∗j (x)]-frequency

modes, satisfying the generalized orthonormalization condition

〈gi|gj〉 = ζiδij , (29.10)

〈g∗i |g∗

j 〉 = −ζiδij , (29.11)

〈gi|g∗j 〉 = 0. (29.12)

The classical four-potential (A(x)) and the related quantum-field operator (A(x)) may beexpanded in the new set also:

A(x) =∑

j

[

βjgj(x) + β∗j g

∗j (x)

]

, (29.13)

and

A(x) =∑

j

[

bjgj(x) + b†jg∗j (x)

]

. (29.14)

Since the two sets (fi, f∗i ) and (gj ,g

∗j ) are complete the new modes can be expanded in

terms of the old, and conversely:

gj =∑

i

[αjifi + βjif∗i ] , (29.15)

fi =∑

j

[

σijgj + τijg∗j

]

. (29.16)

The αji and βji coefficients are obtained by taking the inner product of gj with fi and f∗i ,respectively, and utilizing Eqs. (29.2)-(29.4). Hence,

αji = ζi〈fi|gj〉, (29.17)

βji = −ζi〈f∗i |gj〉. (29.18)

The σij and τij coefficients are determined in an analogous fashion with the help of Eqs.(29.10)-(29.12). Thus,

σij = ζj〈gj |fi〉, (29.19)

τij = −ζj〈g∗j |fi〉. (29.20)

Since the scalar product satisfies the relation

〈F2|F1〉 = 〈F1|F2〉∗, (29.21)

The non-inertial class of observers: The nebulous particle concept 347

as the reader easily may prove to herself on the basis of the definition in Eq. (28.13), σijand τij can be related to αji and βji. Thus,

σij = ζiζjα∗ji, (29.22)

τij = −ζiζjβ∗ji. (29.23)

By inserting the results in Eqs. (29.22) and (29.23) into Eq. (29.16) one obtains the expan-sion

fi =∑

j

ζiζj[

α∗jigj − β∗

jig∗j

]

. (29.24)

The relations in Eqs. (29.15) and (29.24) are known as Bogolubov transformations [47, 22,23, 19], and the quantities αji and βji are called Bogolubov coefficients.

The αji- and βji-coefficients satisfy two important relations which we now shall establish.Let us start by transferring Eq. (29.15) to Dirac notation (F1 → |F1〉):

|gj〉 =∑

k

[αjk|fk〉+ βjk|f∗k 〉]

=∑

k

ζk [|fk〉〈fk| − |f∗k 〉〈f∗k |] |gj〉, (29.25)

where the last expression follows from Eqs. (29.17) and (29.18). It appears from Eq. (29.25)that

∑

k

ζk [|fk〉〈fk| − |f∗k 〉〈f∗k |] = 1, (29.26)

where 1 is the identity operator. The linear functional 〈F2| associating the number 〈F2|F1〉to |F1〉 relates to the relativistic scalar product, i.e.,

ǫ0i~

∫ ∞

−∞

[

∂

∂tF†

2 − F†2

∂

∂t

]

(· · · ) d3r ⇒ 〈F2|. (29.27)

The first relation between the Bogolubov coefficients are derived as follows:

ζiδij = 〈gj |gi〉= 〈gj |1|gi〉=∑

k

ζk [〈gj |fk〉〈fk|gi〉 − 〈gj |f∗k 〉〈f∗k |gi〉]

=∑

k

ζk

[

ζ2kαikα∗jk − (−ζk)2 βikβ∗

jk

]

, (29.28)

and hence

∑

k

ζk[

αikα∗jk − βikβ

∗jk

]

= ζiδij . (29.29)

The second relation between the coefficients may be obtained using the bra and ket versions


of the transformation in Eq. (29.15). Thus,

0 = 〈gi|g∗j 〉

=∑

k,l

〈(αikfk + βikf∗k ) |(

α∗jlf

∗l + β∗

jlfl)

〉

=∑

k,l

[

αikβ∗jl〈fk|fl〉+ βikα

∗jl〈f∗k |f∗l 〉

]

=∑

k,l

[

αikβ∗jlζkδkl + βikα

∗jl (−ζk) δkl

]

, (29.30)

and then finally

∑

k

ζk[

αikβ∗jk − βikα

∗jk

]

= 0. (29.31)

Since we consider the basis sets (fi, f∗i ) and (gj ,g

∗j ) as equivalent, the annihilation (bj)

and creation (b†j) operators associated with the expansion of the four-potential operatorafter the gi-modes necessarily must satisfy the same (Bose–Einstein) type of commutation

relations as the ai and a†i -operators, viz.,

[

bi, b†j

]

= ζjδij , (29.32)

etc. Related to the gj-modes we have a vacuum state, |0g〉, defined by

bj |0g〉 = 0, ∀j, (29.33)

a mode number operator

ng,j = b†j bj , (29.34)

and a mode Fock state with ng,j excitations given by

|ng,j〉 =1

√

ng,j !

(

b†j

)ng,j

|0g〉. (29.35)

29.2 The non-unique vacuum

Let us now imagine that the electromagnetic field is in the f -vacuum state, |0f 〉, for anobserver using the f -modes in her quantization procedure, and let us then ask the question:How will an observer using the g-modes in his description characterize the |0f 〉-state? Inparticular, we would like to calculate the expectation value of the g number operator ng,i

in the f -vacuum. Since the bj and b†j-operators do not act directly on the f -Fock states, wemust find the relation between the annihilation and creation operators in the f and g-modedescriptions. Since the four-vector potential, A(x), and therefore also the quantum-field

operator A(x), is the same in both mode decompositions it appears from Eqs. (29.6) and(29.14) that

∑

i

[

aifi + a†i f∗i

]

=∑

j

[

bjgj + b†jg∗j

]

. (29.36)


The Bogolubov transformation in Eq. (29.15) allows one to express the right-hand side ofEq. (29.36) in terms of the f -modes. Thus,

∑

j

[

bjgj + b†jg∗j

]

=∑

j,i

[

bj (αjifi + βjif∗i ) + b†j

(

α∗jif

∗i + β∗

jifi)

]

. (29.37)

A comparison of Eqs. (29.36) and (29.37) then implies that

ai =∑

j

[

αji bj + β∗jib

†j

]

, (29.38)

a†i =∑

j

[

βjibj + α∗jib

†j

]

, (29.39)

since the coefficients to fi, respectively f∗i , must match. The reader may check that ai and a†i

indeed are each other’s adjoint. By inserting the Bogolubov transformation in Eq. (29.24),and its complex conjugate, on the left side of Eq. (29.36) and identifying afterward thecoefficients to gj, respectively g∗

j , one obtains

bj =∑

i

ζiζj

[

α∗jiai − βjia

†i

]

, (29.40)

b†j =∑

i

ζiζj

[

−β∗jiai + αjia

†i

]

. (29.41)

The quantum mechanical mean value of the gj-mode number operator, ng,j , in the f -vacuumnow can be calculated easily. By means of the transformations in Eqs. (29.40) and (29.41)we obtain

〈0f |ng,j |0f 〉 = 〈0f |b†j bj |0f〉

=∑

k,l

〈0f |ζkζj(

−β∗jk ak + αjk a

†k

)

ζlζj

(

α∗jlal − βjla

†l

)

|0f〉

=∑

k,l

ζkζlζ2j β

∗jkβjl〈0f |aka†l |0f 〉

=∑

k,l

ζkζlβ∗jkβjl〈0f |a†l ak + ζkδkl|0f 〉

=∑

k,l

ζ2kζlβ∗jkβjlδkl. (29.42)

The mean number of gi-particles in the f -vacuum thus is given by

〈0f |ng,i|0f〉 =∑

j

ζj |βij |2. (29.43)

We now can conclude that what looks like a vacuum (|0f 〉) from the f -observer perspectivewill in the ith mode contain an average number of particles (

∑

j ζj |βij |2) from the perspec-tive of the g-observer. If any of the βij coefficients are nonvanishing the vacuum states |0f〉and |0g〉 will not coincide. The reason for this originates in the fact that βij describes anadmixture of creation (annihilation) operators from one basis into annihilation (creation)


operators in the other basis. For example, the state |0g〉 which is annihilated by all bi’s, i.e.,

bi|0g〉 = 0, ∀i, will not be annihilated by ai, unless βji = 0 ∀j:

ai|0g〉 =∑

j

β∗ji b

†j|0g〉 =

∑

j

β∗ji|1g,j〉. (29.44)

Acting with the ai-annihilation operator on the g-vacuum thus results in a linear superpo-sition of one-photon states for an observer using the g-mode description.

In Minkowski space there is a “natural” set of mode functions, namely the monochro-matic and plane fr(x)

′s, given in Eq. (28.63), and the conventional physical vacuum statedefined in terms of these mode functions is the agreed vacuum for all inertial observers.Although an accelerated observer in Minkowski space will observe a (thermal) spectrum ofphotons in the inertial (Minkowski (M)) vacuum, |0〉 ≡ |0M 〉, as we shall understand whenwe study the Unruh effect in the next subsection, the vacuum |0M 〉 is a strong candidatefor the “correct” (“physical”) vacuum. This is so because inertial observers have a specialstatus in flat space-time.

In general relativity the situation is complicated because when gravitational fields arepresent, inertial observers become free-falling observers, and in general detectors in differentfree falls will not agree on a definition (choice) of vacuum [19].

In a sense, it appears from the foregoing discussion that the whole issue concerning the“best” description of the physical photon vacuum cannot be answered without consideringthe measurement process in quantum physics. The question of the meaning of the particle(physical photon, here) concept thus cannot be answered without addressing the role ofthe photon detector, emphasizing once more that according to Niels Bohr, a phenomenonis only a phenomenon when it is a registered phenomenon [29, 30, 31]. If a detector istravelling along some trajectory in a possibly curved space-time, the detector measures theproper time τ along its world line, and positive and negative frequencies will be defined withrespect to τ . Hence, if a set of modes fi can be found which satisfy Dfi/Dτ = −iωifi, whereD/Dτ is the directional covariant derivative along the trajectory, perhaps one may use thesemodes to calculate how many physical photons the detector will register. I remind the readerthat by a “physical” photon, according to our earlier analysis, we mean a T-photon (seeSec. 28.3).

In Secs. 16.1 and 16.2 we discussed single-photon wave packets in Minkowskian space-time. Since these photon types are constructed from a superposition of positive-frequencyplane-wave photons, it is obvious that in all transformations between basis sets belongingto different t-matrices no mixing of positive and negative mode functions appears. That allβij ’s are zero is manifest from the operator relations in Eqs. (16.13) and (16.15). Observersreferring to different wave packet basis sets thus agree on the physical vacuum.

The variance of the gj-mode number operator, ng,j , in the f-vacuum, is expressed bythe squared standard deviation

(∆ng,j)2= 〈0f | (ng,j − 〈0f |ng,j |0f〉)2 |0f 〉= 〈0f |n2

g,j |0f 〉 − (〈0f |ng,j |0f〉)2 . (29.45)

To determine the term

〈0f |n2g,j |0f〉 = 〈0f |b†j bj b†j bj |0f〉 (29.46)

one just needs the inner product of

b†j bj |0f〉 =∑

k

|βjk|2|0f 〉 −∑

k,l

ζkζlαjlβjka†l a

†k|0f 〉 (29.47)


with its adjoint

〈0f |b†j bj = 〈0f |∑

m

|βjm|2 − 〈0f |∑

m,n

ζmζnα∗jnβ

∗jmaman. (29.48)

Obviously, one obtains

〈0f |b†j bj b†j bj |0f〉

=

(

∑

k

|βjk|2)2

+∑

k,l,m,n

ζmζnζkζlαjnβjmα∗jlβ

∗jk〈0f |amana†l a

†k|0f〉. (29.49)

In view of Eq. (29.43) it is realized that

(∆ng,j)2=

∑

k,l,m,n

ζmζnζkζlαjnβjmα∗jlβ

∗jk〈0f |amana†l a

†k|0f〉. (29.50)

The matrix element in Eq. (29.50) is calculated by utilizing the commutation relation inEq. (29.5) and the definition of the f-vacuum state [Eq. (29.7)]. Hence,

〈0f |amana†l a†k|0f 〉 = 〈0f |am

(

a†l an + ζnδnl

)

a†k|0f 〉

= 〈0f |ama†l ana†k|0f〉+ ζnδnl〈0f |ama†k|0f〉

= 〈0f |ama†l(

a†kan + ζnδnk

)

|0f 〉+ ζnζmδnlδmk

= ζnζm (δnkδml + δnlδmk) . (29.51)

The squared standard deviation now becomes

(∆ng,j)2=

∑

k,l,m,n

ζkζlαjnβjmα∗jlβ

∗jk (δnkδml + δnlδmk)

=∑

k,l

ζkζl[

αjkα∗jlβjlβ

∗jk + αjlβjkα

∗jlβ

∗jk

]

=

(

∑

k

ζkαjkβ∗jk

)(

∑

l

ζlα∗jlβjl

)

+

(

∑

k

ζkβjkβ∗jk

)(

∑

l

ζlαjlα∗jl

)

=

∣

∣

∣

∣

∣

∑

k

ζkαjkβ∗jk

∣

∣

∣

∣

∣

2

+

(

∑

k

ζk|αjk|2)(

∑

k

ζk|βjk|2)

. (29.52)

Finally, it thus appears that the squared standard deviation is given by

(∆ng,i)2 =

∣

∣

∣

∣

∣

∣

∑

j

ζjαijβ∗ij

∣

∣

∣

∣

∣

∣

2

+

∑

j

ζj |αij |2

∑

j

ζj |βij |2

(29.53)

after the index changes j → i, k → j. The variance of the gi-mode number operator ng,i

in the f-vacuum thus depends on both the αij- and βij -coefficients. If wished, the results inEq. (29.53) can be expressed in alternative forms using the relevant relations between theBogolubov coefficients [Eqs. (29.29) and (29.31) for i = j] .


29.3 The Unruh effect

In this section we consider the experiences of an observer that accelerates uniformly throughthe Minkowski vacuum state. Our example illustrates the idea that observers with differentnotations of positive- and negative-frequency modes will disagree on the particle content ina given state. The Unruh effect [19, 233, 234] states that an uniformly accelerating observerin a Minkowski vacuum state will observe a thermal (Planck) spectrum of particles. Theabsolute temperature of the Planck spectrum turns out to equal the magnitude of thefour-acceleration (divided by 2π).

29.3.1 Rindler space and observer

The qualitative physics underlying the Unruh effect can be understood by considering amassless scalar field in just two space-time dimensions. In the context of the electromagneticfield this relates to a study of the quantized wave equation for free plane fields propagatingin a given direction (coordinate x) with a specific polarization. Let the massless scalar field

be denoted by φ(x, t). The associated field operator φ(x, t) obeys the free wave equation

(

∂2

∂x2− ∂2

∂t2

)

φ(x, t) = 0. (29.54)

Note that throughout Sec. 29.3 we shall choose units in which the speed of light

c = 1. (29.55)

In the (x, t)-coordinates the metric is given by

ds2 = dx2 − dt2. (29.56)

A Rindler coordinatization (η, ξ) of Minkowski space is associated with the transforma-tion [47, 19, 167]

t = a−1eaξ sinh(aη), (29.57)

x = a−1eaξ cosh(aη), (29.58)

where a is a positive constant. The new coordinates have ranges

−∞ < η, ξ <∞. (29.59)

Only positive x’s are covered by the transformation, and since cosh(aη) > | sinh(aη)| onlya quadrant of Minkowski space, namely the wedge x > |t|, is covered by Eqs. (29.57) and(29.58), as shown in Fig. 29.1. The region given by x > |t|, and denoted by I in the figure, iscalled Rindler space, even though it is just a part of the two-dimensional Minkowski space.Since

dx = eaξ [sinh(aη)dη + cosh(aη)dξ] , (29.60)

dt = eaξ [cosh(aη)dη + sinh(aη)dξ] , (29.61)

the metric is given by

ds2 = e2aξ(

dξ2 − dη2)

(29.62)


>

>

>

>

>

>

>t

x

h

FIGURE 29.1Rindler coordinatization of Minkowski space. Coordinates with η = constant (C) arestraight lines through the origin, and coordinates with ξ = constant (K) are hyperbo-lae. Uniformly accelerated observers move on world lines ξ = K. Region I (IV) is accessibleto observers undergoing constant acceleration in the +x-direction (−x-direction). RegionsII and III are the remaining future and past regions, respectively.

in Rindler coordinates. We can also define (ηIV , ξIV ) ≡ (η, ξ)-coordinates in the wedgex < −|t| (region IV in Fig. 29.1) by

t = −a−1eaξ sinh(aη), (29.63)

x = −a−1eaξ cosh(aη). (29.64)

With the same labelling of the Rindler coordinates in the two wedges [(ηI , ξI) ≡ (η, ξ),(ηIV , ξIV ) ≡ (η, ξ)] the metric [Eq. (29.62)] is the same in regions I and IV.

Consider an observer propagating on the trajectory xµ(τ) [µ = 0, 1]

t(τ) = α−1 sinh(ατ), (29.65)

x(τ) = α−1 cosh(ατ), (29.66)

τ being the proper time, and α a positive constant which physical role we now shall examine.The components of the proper acceleration

aµ =d2

dτ2xµ (29.67)

are given by

at = α sinh(ατ), (29.68)

ax = α cosh(ατ), (29.69)

and α therefore is the magnitude of the proper acceleration:

√

aµaµ =(

(ax)2 −

(

at)2)

1

2

= α. (29.70)


Since

x2 − t2 = α−2, (29.71)

it appears that trajectories of constant α are hyperbolae in the (t, x)-plane. Returning tothe wedge I Rindler transformation in Eqs. (29.57) and (29.58) we see that lines of constantξ are hyperbolae (Fig. 29.1)

x2 − t2 = a−2e2aξ. (29.72)

Observers moving on these trajectories have the proper acceleration

α = ae−aξ. (29.73)

On hyperbolae of large positive ξ (lying far from x = t = 0) the observers are weaklyaccelerated. On trajectories that closely approach x = t = 0, ξ has large negative values, andhence, according to Eq. (29.73), high proper accelerations. Asymptotically the hyperbolaeapproach the straight (light) lines t = ±x, and these lines act as event horizons. Lines ofconstant η are straight,

t = x tanh(aη), (29.74)

and approach the light lines (t = ±x) for η → ±∞.

29.3.2 Rindler particles in Minkowski vacuum

Consider now the scalar wave equation

(

∂2

∂x2− ∂2

∂t2

)

φ(x, t) = 0. (29.75)

It follows from the analysis in Secs. 28.1 and 28.2 that this equation in the continuum limitpossesses standard orthonormal mode solutions

fq(x, t) =

(

~

4πǫ0ω

)1

2

ei(qx−ωt) (29.76)

with ω = |q| > 0 and −∞ < q <∞. Setting

~ = ǫ0 = 1 (29.77)

in what follows, we have

fq(x, t) = (4πω)−1

2 ei(qx−ωt), ω > 0. (29.78)

The modes with q > 0, i.e.,

fq(u) = (4πω)− 1

2 e−iωu, u ≡ t− x, (29.79)

are right-moving waves (along u = constant), and the modes with q < 0, namely

fq(v) = (4πω)−1

2 e−iωv, v ≡ t+ x, (29.80)

are left-moving waves (along v = constant). Both mode types, fq(u) and fq(v), have positivefrequencies (ω > 0). The Minkowski vacuum state, |0M 〉, and the associated Fock space are


constructed by (i) expanding φ(x, t) in terms of fq(x, t) and its complex conjugate, f∗q (x, t),

and (ii) elevation of the classical expansion coefficients to the operator level [φ(x, t) ⇒φ(x, t)].

In the Rindler regions I and IV one may adopt an alternative prescription for the quan-tization based on the Rindler coordinatization. Since

∂

∂ξ= a

(

t∂

∂t+ x

∂

∂x

)

, (29.81)

∂

∂η= a

(

x∂

∂t+ t

∂

∂x

)

, (29.82)

in both regions, one obtains

∂2

∂ξ2− ∂2

∂η2= a2

(

x2 − t2)

(

∂2

∂x2− ∂2

∂t2

)

, (29.83)

and then by means of Eq. (29.72), which holds in both region I and IV

∂2

∂x2− ∂2

∂t2= e−2aξ

(

∂2

∂ξ2− ∂2

∂η2

)

. (29.84)

The wave equation in Eq. (29.75) thus transforms into(

∂2

∂ξ2− ∂2

∂η2

)

φ(ξ, η) = 0 (29.85)

in Rindler coordinates.We now define two sets of positive-frequency modes, one [g

(1)q (ξ, η)] with support in

region I and the other [g(2)q (ξ, η)] with support in region IV:

g(1)q (ξ, η) =

(4πω)− 1

2 ei(qξ−ωη), (ω > 0), I0, IV

(29.86)

and

g(2)q (ξ, η) =

0, I

(4πω)− 1

2 ei(qξ+ωη), (ω > 0) IV. (29.87)

The set in Eq. (29.86), along with its conjugate set [g(1)q (ξ, η)]∗, form a complete set of basis

modes in the Rindler region I. Because

∂

∂ηg(1)q = −iωg(1)q , (29.88)

g(1)q certainly has positive frequency in the usual sense with respect to η. In region I, wherex > |t|, increasing η corresponds to increasing t for fixed ξ since

∂

∂η=a

x

(

x2 − t2) ∂

∂t, ξ = constant. (29.89)

The modes g(1)q hence are of positive frequency with respect to future-directed propagation.

The set in Eq. (29.87), together with its conjugate set, [g(2)q (ξ, η)]∗, form a complete basis

in region IV. It appears that

∂

∂(−η)g(2)q = −iωg(2)q , (29.90)


so that g(2)q is a positive-frequency mode with respect to −η. However, since

∂

∂(−η) =a

x

(

t2 − x2) ∂

∂t, ξ = constant, (29.91)

increasing −η corresponds to increasing t in wedge IV [x < −|t|] where (t2 − x2)/x < 0

the g(2)q -modes also are of positive frequency with respect to future-directed propagation.

Neither of the sets (g(1)q , (g

(1)q )∗) and (g

(2)q , (g

(2)q )∗) is complete on the entire Minkowski

space of course, but the sets together form a complete set for expansion of fields extendingover the whole space-time [19].

The findings above imply that a given field operator may be expanded in eitherMinkowski modes,

φ(x, t) =

∫ ∞

−∞

[

aqfq(x, t) + a†qf∗q (x, t)

]

dq, (29.92)

or Rindler modes

φ(ξ, η) =

∫ ∞

−∞

[

b(1)q g(1)q (ξ, η) +(

b(1)q

)† (g(1)q (ξ, η)

)∗

+ b(2)q g(2)q (ξ, η) +(

b(2)q

)† (g(2)q (ξ, η)

)∗]

dq. (29.93)

Although the Hilbert space for the theory is the same in both representations, the Fockspaces will be different. In particular the Minkowski (M) and Rindler (R) vacuum states,|0M 〉 and |0R〉, defined by

aq|0M 〉 = 0, ∀q, (29.94)

and

b(1)q |0R〉 = b(2)q |0R〉 = 0, ∀q, (29.95)

will be different, as we shall soon see.We know from the analysis in Sec. 29.2 that the two vacuum states |0M 〉 and |0R〉

must be different if an annihilation (creation) operator from one basis set turns into anadmixture of annihilation (creation) and creation (annihilation) operators in another basisset. The conclusion above originates in the circumstance that a given, say positive-frequencymode function gj, from one basis set when expanded after another (fi, f

∗i )-basis set involves

both positive- and negative-frequency modes if just one of the Bogolubov coefficients βji isnonvanishing.

We can conclude that the vacuum states |0M 〉 and |0R〉 cannot be equivalent by inspec-tion of the structure of the positive-frequency Rindler modes, given by Eqs. (29.86) and(29.87). Let us examine the case with right-moving waves (q > 0). In region I, the trans-formation in Eqs. (29.57) and (29.58) gives the following relations between the Minkowskicoordinates (t, x) and Rindler coordinates (η, ξ):

a(x− t) = ea(ξ−η), I. (29.96)

If we rewrite Eq. (29.96) in the form

a(x − t) = exp [iω (ξ − η)] aiω , (29.97)


it appears that

eiω(ξ−η) = [a (x− t)]iωa . (29.98)

For q > 0 [q = ω], the left side of Eq. (29.98), divided by (4πω)1/2, is just g(1)q (ξ, η) in region

I. Thus, in terms of the Minkowski coordinates

(4πω)1

2 g(1)q (x, t) = aiωa (x− t)

iωa , I. (29.99)

Following the procedure above, we also can express g(2)q in region IV in Minkowski coordi-

nates. From the transformation in Eqs. (29.63) and (29.64), one obtains

a (−t− x) = ea(ξ+η), (29.100)

and

eiω(ξ+η) =[

ea(ξ+η)]

iωa

= aiωa (−t− x)

iωa . (29.101)

Remembering that we are considering the case q > 0, a combination of Eqs. (29.87) and(29.101) leads to

(4πω)1

2 g(2)q (x, t) = aiωa (−x− t)

iωa , IV. (29.102)

It appears from Eqs. (29.99) and Eq. (29.102) that the right-moving Rindler modes do notjoin smoothly as one passes from I to IV. The right-moving modes therefore are non-analyticat x = t = 0. In contrast the positive-frequency right-moving Minkowski modes [Eq. (29.79)]are analytic. This analyticity property remains true for any linear superposition of theseMinkowski modes. Hence none of the right-moving positive-frequency Rindler modes can bea combination of pure positive-frequency Minkowski modes. In consequence |0M 〉 and |0R〉cannot be equivalent.

To determine the distribution of Rindler particles present in the Minkowski vacuum,one must determine the Bogolubov transformation connecting the Minkowski and Rindlermodes. The Bogolubov coefficients may be calculated directly from Eqs. (29.17) and (29.18),as these appear in the scalar case. Here we follow an elegant method due to Unruh [233, 234].The basic idea is to seek a linear combination of the Rindler modes in Eqs. (29.99) and(29.102) which is analytic throughout space. The analytic extension of the function in Eq.(29.99) to the whole space-time is straightforward: Use this function for any values of (x, t)!

The task now is to replace g(2)q (x, t) in region IV by a “modified” function which corresponds

to the above-mentioned analytic extension of q(1)q (x, t) to region IV. If we reverse the wave

number and take the complex conjugate of g(2)q , we obtain (remembering q = ω > 0)

(4πω)1

2

[

g(2)−q

]∗= eiω(ξ−η), IV. (29.103)

The expression in Eq. (29.103) can be expressed in Minkowski coordinates by utilizing thatthe transformation in Eqs. (29.63) and (29.64) implies that

a (t− x) = ea(ξ−η), IV, (29.104)

and hence

eiω(ξ−η) = [a (t− x)]iωa , IV. (29.105)


By combining Eqs. (29.103) and (29.105), one gets

(4πω)1

2

[

g(2)−q(x, t)

]∗= a

iωa (t− x)

iωa , IV. (29.106)

A comparison of Eqs. (29.99) and (29.106) shows that one just needs to change t − x tox− t in Eq. (29.106) to have obtained an analytic extension of the expression in region I toregion IV. Setting

−1 = e−iπ, (29.107)


(4πω)1

2

[

g(2)−q(x, t)

]∗= a

iωa e

πωa (x− t)

iωa , (29.108)

so that

(4πω)1

2 e−πωa

[

g(2)−q(x, t)

]∗= a

iωa (x − t)

iωa , IV. (29.109)

The combination

(4πω)1

2

g(1)q (x, t) + e−πωa

[

g(2)−q(x, t)

]∗= a

iωa (x − t)

iωa (29.110)

therefore is the mode we have sought for q > 0. The right-moving positive-frequencyMinkowski mode, given in Eq. (29.79) is analytic and bounded for complex (x, t) as longas ℑ(t − x) ≤ 0. The same holds for the combination in Eq. (29.110) when we choose thebranch cut for the imaginary power (x − t)iω/a to lie in the upper-half of the complex(t− x)-plane, i.e., ln(−1) = −iπ [19]. This is consistent with the setting in Eq. (29.107). Aproperly normalized version of the combination in Eq. (29.110) is given by

h(1)q (x, t) =[

2 sinh(πω

a

)]− 1

2

exp(πω

2a

)

g(1)q (x, t)

+ exp(

−πω2a

) [

g(2)−q(x, t)

]∗, ω = q > 0. (29.111)

Let us verify that the q-modes given by Eq. (29.111) are orthonormalized. Since the Rindler

modes g(1)q and g

(2)−q only have support in regions I and IV, respectively, one has

〈g(1)q1 |[

g(2)−q2

]∗〉 = 0 (29.112)

for all q1- and q2-modes. The inner product of two h(1)q -modes hence becomes

〈h(1)q1 |h(1)q2 〉 =1

2

[

sinh(πω1

a

)

sinh(πω2

a

)]− 1

2

×

exp[ π

2a(ω1 + ω2)

]

〈g(1)q1 |g(1)q2 〉+ exp[

− π

2a(ω1 + ω2)

]

〈[

g(2)−q1

]∗|[

g(2)−q2

]∗〉

.

(29.113)

The calculation of the inner products in Eq. (29.113) is done on the basis of the relativisticdefinition of the inner product [the scalar version of Eq. (28.64) with ǫ0 = ~ = 1]. Thus,one finds

〈g(1)q1 |g(1)q2 〉 = δ (q1 − q2) , (29.114)

〈[

g(2)−q1

]∗|[

g(2)−q2

]∗〉 = −δ (q2 − q1) , (29.115)


as the reader may show. By inserting Eqs. (29.114) and (29.115) into Eq. (29.113) oneobtains

〈h(1)q1 |h(1)q2 〉 = exp(

πω1

a

)

− exp(

−πω1

a

)

2 sinh(

πω1

a

) δ (q1 − q2) , (29.116)

i.e., the postulated Dirac orthonormalization

〈h(1)q1 |h(1)q2 〉 = δ (q1 − q2) . (29.117)

To get a complete set, one must supplement the right-moving (q > 0) positive-frequency

modes, h(1)q , by a set of left-moving (q < 0) positive-frequency modes, h

(2)q . These are

obtained by an analytic extension of the g(2)q -modes [Eq. (29.102)] from region IV to region

I. Following a procedure analogous to the one leading to the h1q(x, t)-mode spectrum [Eq.(29.111)] leads to the following result:

h(2)q (x, t) =[

2 sinh(πω

a

)]− 1

2

exp(πω

2a

)

g(2)q (x, t)

+ exp(

−πω2a

) [

g(1)−q(x, t)

]∗, ω = −q > 0. (29.118)

The modes above are orthonormalized in the Dirac sense, i.e.,

〈h(2)q1 |h(2)q2 〉 = δ (q1 − q2) . (29.119)

The right- and left-moving positive-frequency modes are orthogonal, that is

〈h(1)q1 |h(2)q2 〉 = 0, (29.120)

as the reader may prove.It is obvious by now that the given field operator [Eq. (29.93)] can be expanded in the

(h(1)q , h

(2)q )-modes and their complex conjugates, [(h

(1)q )∗, (h(2)q )∗]:

φ =

∫ ∞

−∞

[

c(1)q h(1)q +(

c(1)q

)† (h(1)q

)∗+ c(2)q h

(2)2 +

(

c(2)q

)† (h(2)q

)∗]

dq. (29.121)

Since the modes h(1)q and h

(2)q can be expressed alone in terms of positive-frequency

Minkowski modes, fq [Eq. (28.78)], we know that the vacuum state for an observer us-ing the h-modes, will be identical to that of a Minkowski observer, i.e.,

c(1)q |0M 〉 = c(2)q |0M 〉 = 0. (29.122)

The excited states will not coincide. The Rindler annihilation operators b(1)q and b

(2)q can

be related to the c-operators by taking the scalar products (φ, g(1)q ) and (φ, g

(2)q ) first with

φ given by the expansion in Eq. (29.93) and then with φ from Eq. (29.121). Hence, oneobtains after some straightforward calculations

b(1)q =[

2 sinh(πω

a

)]− 1

2

[

exp(πω

2a

)

c(1)q + exp(

−πω2a

)(

c(2)−q

)†]

, (29.123)

and

b(2)q =[

2 sinh(πω

a

)]− 1

2

[

exp(πω

2a

)

c(2)q + exp(

−πω2a

)(

c(1)−q

)†]

. (29.124)


The Bogolubov transformation given by Eqs. (29.123) and (29.124) allows one to obtainthe required relation between the Rindler and Minkowski vacuum states, cf. the analysis inSec. 29.2.

We have learned from the development in this chapter that the particle concept doesnot generally have universal significance. A given detector will either detect particles, ornot, and the meaning of the particle concept cannot be answered without a discussion ofthe quantum measurement process. In Part VIII we shall analyze the photon measurementproblem in quantum physics. We have seen that the particle concept is closely related to thedivision of the field into its positive- and negative-frequency parts. A detector moving on agiven world line measures the proper time τ along the trajectory, and an observer attachedto the detector will define positive and negative frequencies with respect to the proper time.If we assume that a set of modes fi that obey

dfidτ

= −iωifi (29.125)

can be found, these modes can be used to calculate how many particles the observer detects.It appears from Eq. (29.62) that a Rindler observer moving on a trajectory of constant

ξ measures a proper time

τ = eaξη, (29.126)

assuming τ = 0 for η = 0. This observer travels with a uniform four-acceleration given byα = a exp(−aξ) [Eq. (29.73)], and modes which are of positive frequency with respect to ηalso has positive frequency with respect to the proper time. Rindler observers moving withconstant four-acceleration hence will count detected particles by the mode-number operators

(b(1)q )†b(1)q [region I] and (b

(2)q )†b(2)q [region IV]. In the Minkowski vacuum, a Rindler observer

in region I detects in mode q a mean number of particles (n(1)R (q)) given by

n(1)R (q) = 〈0M |(b(1)q )†b(1)q |0M 〉. (29.127)

Using Eq. (29.123) and its Hermitian conjugate, one obtains in view of Eq. (29.122)

n(1)R (q) =

[

2 sinh(πω

a

)]−1

〈0M | exp(

−πωa

)

c(1)−q

(

c(1)−q

)†|0M 〉

=exp

(

−πωa

)

2 sinh(

πωa

) 〈0M |0M 〉 = δ(0)

exp(

2πωa

)

− 1, (29.128)

remembering that our modes are normalized in the Dirac sense in the continuum description,〈0M |0M 〉 = δ(0). We have thus come to the conclusion that the result in Eq. (29.128) is aPlanck spectrum

nPlanck(ω) = [exp

(

~ω

kBT0

)

− 1]−1 (29.129)

with absolute temperature

T0 =a

2πkB, (29.130)

remembering that ~ = 1, here. The quantity kB is Boltzmann’s constant. In view of Eq.(29.126) one would expect that T0 is the temperature seen by an observer travelling on the


path ξ = 0. The absolute temperature T seen by an observer moving on any other trajectorywith ξ = constant 6= 0 will be given by

T =a exp (−aξ)

2πkB=

α

2πkB, (29.131)

α being the proper acceleration [Eq. (29.73)]. When ξ → ∞ the observer acceleration goesto zero, and T → 0, in agreement with the fact that the Rindler observer in this limitbecomes an inertial (Minkowski) observer.

The reader may show that a Rindler observer in region IV will detect the same meannumber of particles as an observer in region I, provided the two observers move on trajec-tories with the same a:

n(2)R (q) = 〈0M |(b(2)q )†b(2)q |0M 〉

=δ(0)

exp(

2πωa

)

− 1. (29.132)

Above, we have briefly discussed certain aspects of the particle concept in flat space-time.In curved space-time it will generally not be possible to find positive-frequency modes ofall of space-time. Readers interested in the particle concept in the presence of gravitationalfields may consult the excellent book by Birrell and Davies [19].

30

Photon mass and hidden gauge invariance

In Sec. 12.3 we briefly discussed the linear interaction between the transverse part of aclassical electromagnetic field and a system of charged particles (electrons). At high fre-quencies we concluded that the main contribution to the many-body conductivity tensorcame from the diamagnetic interaction. In the superconducting state the diamagnetic field-electron interaction is the dominating one in the weak-field limit [212, 45, 8, 107, 108].The diamagnetic interaction gives a spatially local relation between the microscopic elec-tron current density and the transverse vector potential, the proportionality factor being−(e2/m)N0(r), where N0(r) is the local many-body electron density. If one assumes thatthis density is homogeneous, the transverse photon acquires an effective mass (proportionalto the plasma frequency of the electron system), and the transverse vector potential obeysthe Proca equation since the transverse vector potential is gauge invariant.

In Sec. 12.4 we extended the considerations to a covariant description of the interactionbetween the electromagnetic potential, Aµ(x) and a spinless charged boson particle. Inthis case we came to the conclusion that the four-component vector potential satisfies theProca equation provided the norm of the scalar wave function of our charged boson is robust(constant in space-time). However, it turned out that the Lorenz condition is a constrainton the formalism. We shall now address the following question: Is it possible for a photonto acquire a mass in vacuum without destroying the gauge invariance freedom?

30.1 Physical vacuum: Spontaneous symmetry breaking

The answer to the question above is “yes” provided we are prepared to revise our view on“the nature of the physical vacuum.”

In the covariant description of the quantized electromagnetic field, the vacuum state |0〉is defined as the state in which there are no photons of any kind present, i.e.,

ar(q)|0〉 = 0, ∀q, r = 0− 3; (30.1)

as discussed in our brief review of covariant field quantization [Sec. 26.4]. The vacuum stateabove is a state of minimum field energy, it is unique (non-degenerate), and the expectationvalues of the field operators are zero, 〈0|ar(q)|0〉. Our road to the introduction of a photonmass goes via screening (diamagnetic) currents associated with the presence of massiveparticles (electrons, spinless charged bosons). In 1960, it was suggested by Nambu andcoworkers [167] that the physical vacuum of a quantum field is in some sense analogous tothe ground state of an interacting many-body system [45, 152]. Although such a groundstate is a stable state of minimum energy it need not be one in which all quantum fields havezero average value. If the lowest energy level of our system is non-degenerate the groundstate is unique, and possesses the symmetry of the system Lagrangian, L. If, however,the lowest energy level is degenerate (non-unique) the corresponding eigenstates are notinvariant under the symmetry transformations of L. If one selects one of the degenerate

363


ground states as the ground state, the ground state no longer shares the symmetries ofL. This way of obtaining an asymmetric ground state is known as spontaneous symmetrybreaking [209, 243, 156, 6]. The asymmetric situation of our physical vacuum is preciselywhat is needed to generate so-called vacuum screening currents.

In the following we shall assume the presence of a complex scalar field, the Higgs fieldφ(x), which associated field operator φ(x) has an expectation value (c) different from zeroin the vacuum, i.e.,

〈0|φ(x)|0〉 = c 6= 0. (30.2)

The Higgs field must be a scalar field and c a constant if we require that the vacuum state(s)is invariant under homogeneous Lorentz transformations and translations.

The simplest example of a field theory exhibiting spontaneous symmetry breaking is theGoldstone model [209, 88, 156, 6, 81]. In this model it is assumed that the Higgs field hasa potential energy density

V(φ) = µ2|φ(x)|2 + λ|φ(x)|4, (30.3)

where µ2 and λ (so far) are arbitrary real parameters. As we require the energy of the fieldto be bounded from below, λ must be positive. If µ2 > 0, the minimum in V(φ) is at |φ| = 0[φ = 0], V(φ) = 0, and this case in turn leads to a non-degenerate vacuum ground state. Aninteresting situation occurs for µ2 < 0. Now, V(φ) possesses a local maximum for |φ| = 0,and an absolute minimum (M) at

|φ(x)| = φM =

(

−µ2

2λ

)1

2

. (30.4)

Remembering that φ(x) is a complex scalar field, i.e.,

φ(x) =1√2(φ1(x) + iφ2(x)) , (30.5)

where φ1 and φ2 are real, it appears that V(x) possesses infinitely many absolute minimain the complex φ-plane. With φ written in polar coordinates, φ = |φ| exp(iθ), it is seen thatthe minima lie on the circle

φ0 = φMeiθ, 0 ≤ θ < 2π. (30.6)

A sketch of the potential energy density V(φ) for µ2 < 0 is shown in Fig. 30.1. The Goldstonepotential is invariant under global [U(1)] phase transformations of the Higgs field

V(φ′(x)) = V(φ(x)), (30.7)

where

φ(x) → φ′(x) = φ(x)eiα (30.8)

with a constant (x-independent) phase, α. The Lagrangian density for the field φ(x) (tobe written down below) is also invariant under U(1) transformation. The transformation inEq. (30.8) may be conceived as a rotation of φ in the complex (φ1, φ2)-plane:

φ′1 = φ1 cosα− φ2 sinα, (30.9)

φ′2 = φ1 sinα+ φ2 cosα. (30.10)

Photon mass and hidden gauge invariance 365

>

>

f

f

)

>

>q h

0 f

V

n

n

f

)

FIGURE 30.1Upper figure: Goldstone potential (energy density) V(φ) = µ2|φ(x)|2 + λ|φ(x)|4 for µ2 < 0.The potential possesses an absolute minimum at |φ| = [−µ2/(2λ)]1/2 ≡ v/

√2, and a local

maximum for |φ| = 0. Lower figure: In the complex φ-plane, the minima lie on a circle ofradius v/

√2, and the continuous set of related vacuum eigenstates are the degenerate ground

states (G) of the Higgs field, φ(x). With the choice θ = 0, one selects a particular state(G0) as the ground state. Small deviations of the Higgs field (multiplied by

√2) from the

ground state (G0) value in the radial direction, H(x), and along the valley of minimum V ,η(x), lead upon quantization to massive neutral scalar Higgs bosons and massless Goldstonebosons (never observed in nature, and eliminated by a certain gauge transformation).

The symmetry in Eq. (30.7) is called an internal symmetry, and the rotation given by Eqs.(30.9) and (30.10) an internal space rotation. The set of all global phase transformationsforms a group U(1), meaning the group of all unitary (U) matrices of rank one (1). Thegroup U(1) is Abelian because different transformations commute.

Spontaneous symmetry breaking occurs if one chooses one particular θ-direction to rep-resent the vacuum ground state. The choice for θ is not significant, of course, so let ustherefore just take θ = 0. With this choice, the now real ground state scalar field is

φ0 = φM ≡ v√2, v > 0. (30.11)

comparing Eqs. (30.4) and (30.11) we see that v = (−µ2/λ)1/2.


30.2 Goldstone bosons

In the remaining part of this chapter it will be useful to change the definition of the metrictensor of flat space-time from our old one gµνOLD, given by Eq. (6.1), to

gµνNEW = −gµνOLD =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (30.12)

With the new definition the Lagrangian density of the Goldstone model is

L(x) = [∂µφ∗(x)] [∂µφ(x)] − µ2|φ(x)|2 − λ|φ(x)|4, (30.13)

where φ(x) and φ∗(x) are regarded as independent fields. From the Lagrange equation, viz.,

∂µ[

∂L∂ (∂µφ∗(x))

]

− ∂L(x)∂φ∗

= 0, (30.14)

we now obtain the following dynamical equation for φ(x) in the Goldstone model

∂µ∂µφ(x) + µ2φ(x) + 2λ|φ(x)|2φ(x) = 0, (30.15)

with

∂µ∂µ =1

c2∂2

∂t2−∇2 ≡ − (30.16)

for our new choice of metric tensor. Via the conjugate momenta [209, 242, 53, 156]

π(x) ≡ ∂L(x)∂ (∂0φ∗(x))

= ∂0φ(x), (30.17)

π∗(x) ≡ ∂L(x)∂ (∂0φ(x))

= ∂0φ∗(x), (30.18)

and the general expression for the Hamiltonian density [209, 242, 53, 156]

H(x) = π∗(x)∂0φ(x) + π(x)∂0φ∗(x) − L(x), (30.19)

one gets the following Goldstone Hamiltonian density:

H(x) =[

∂0φ∗(x)]

[∂0φ(x)] + [∇φ∗(x)] · [∇φ(x)]

+ µ2|φ(x)|2 + λ|φ(x)|4. (30.20)

Let us return to the dynamical equation for φ(x), Eq. (30.15). It is a nonlinear equationwhen λ 6= 0, and it has a form which is well-known in connection with a number of third-order phenomena in nonlinear optics; see, e.g., [21, 216, 213, 44, 41]. Because of this formalanalogy theoretical methods employed in studies of third-order susceptibility effects mightbe useful in further analyses of the dynamical equation for φ(x). We shall not follow up onthis point here. Near the origo of the complex φ-plane the dynamical Goldstone equationapproaches the form

(

− µ2)

φ(x) = 0, (30.21)


which for µ2 > 0 is just the Klein–Gordon wave equation for a scalar (boson) field, withCompton wave number QC = µ(> 0), and related mass

Mφ =~µ

c, (30.22)

cf. the discussion in Sec. 12.1. This finding is in agreement with the fact that the potentialV(φ) is a harmonic oscillator potential in the vicinity of φ = 0. For µ2 < 0, the pointφ(x) = 0 relates to an unstable equilibrium configuration (the harmonic potential bends“downward,” as shown in Fig. 29.1). In the “particle language,” the situation µ2 < 0 wouldcorrespond to particles of imaginary mass!

Being interested in the case µ2 < 0, we now introduce two real classical fields H(x) [theHiggs field [96, 97, 98]] and η(x) through the equation

φ(x) =1√2(v +H(x) + iη(x)) . (30.23)

The two fields H(x) and η(x) describe the deviations of the field φ(x) from the displacedground state φ0 = v/

√2; see Fig. 29.1. Expressed in terms of the new fields the Goldstone

Lagrangian density takes the form

L(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x)

+1

2[∂µη(x)] [∂µη(x)]

− λvH(x)(

H2(x) + η2(x))

− λ

4

(

H2(x) + η2(x))2, (30.24)

where the relation −µ2 = λv2 has been used. Let us divide L(x) into three pieces

L(x) = LH(x) + Lη(x) + LHη(x). (30.25)

The first two, namely,

LH(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x), (30.26)

and

Lη(x) =1

2[∂µη(x)] [∂µη(x)] , (30.27)

are quadratic in the fields H(x) and η(x). There is no coupling between the fields in theseterms. The piece

LHη(x) = −λvH(x)(

H2(x) + η2(x))

− λ

4

(

H2(x) + η2(x))2

(30.28)

is of third and fourth order in the deviations from the equilibrium state, and it containscoupling terms between H(x) and η(x), as well as terms of third (H3(x)) and fourth(H4(x), η4(x)) order in either of the deviations. The part LHη(x) may be called an in-teraction Langrangian density, containing mutual and self-field interaction terms. For smalldeviations from the displaced equilibrium at φM = v/

√2, it will usually be possible to treat

the dynamics associated to LHη(x) as a perturbation added to the dynamics following fromLH(x)+Lη(x). Neglecting the LHη(x) part of L(x), the Langrange equation for the system


separates into two Klein–Gordon equations, viz., one for each of the real fields η(x) andH(x):

∂µ∂µη(x) = 0, (30.29)(

∂µ∂µ − 2λv2)

H(x) = 0. (30.30)

Upon quantization,

η(x) ⇒ η(x) = η(+)(x) + η(−)(x), (30.31)

H(x) ⇒ H(x) = H(+)(x) + H(−)(x), (30.32)

both equations lead to neutral spin-zero particles (bosons). The η-particle has zero mass,and the H-particle has mass v

√2λ. In the global vacuum state |0〉 there are no particles

present:

η(+)(x)|0〉 = H(+)(x)|0〉 = 0. (30.33)

Since

φ(x) =1√2

(

v + H(x) + iη(x))

, (30.34)

one obtains in the presence of spontaneous symmetry breaking

〈0|φ(x)|0〉 = v√2, (30.35)

cf. Eq. (30.2). Classically, the quantity H(x) represents a (small) displacement in the radialplane (φ2 = 0) from the minimum. This displacement thus is related to a quadratic increasein the potential from its minimum, and this is the reason that the H-particle is massive. Thequantity η(x) represents a displacement along the valley of minimum potential energy, andthe η-particle therefore becomes massless. A valley of constant (minimum) V(φ) is just whatcame from our model of spontaneous symmetry breaking, and the presence of zero-mass η-particles hence is a consequence of the degeneracy of the vacuum. The zero-mass scalarbosons are known as Goldstone bosons. No Goldstone bosons are observed in nature, andbelow we shall see how one can get rid of these using the U(1) gauge invariance principle.

30.3 The U(1) Higgs model

In order to couple the complex charged Higgs field to the electromagnetic field, Aµ(x),we replace the ordinary derivatives in the Goldstone Langrangian density [Eq. (30.13)] bythe gauge-covariant derivatives, i.e.,

∂µ ⇒ ∇Gµ ≡ Dµ = ∂µ − iq

~Aµ(x), (30.36)

cf. the discussion in Secs. 8.6 and 12.4, and specifically Eqs. (8.133) and (12.48), if we alsoadd the Lagrangian density of the free electromagnetic field [Eq. (3.87), with omission of theunimportant factor µ−1

0 = ǫ0c2. Leaving out this factor we still obtain the classical Maxwell

equations if the interaction Lagrangian −JµAµ is multiplied by µ0], namely,

LF (x) = −1

4Fµν(x)F

µν (x), (30.37)


where [Eq. (8.100)]

Fµν (x) = ∂µAν(x) − ∂νAµ(x), (30.38)

we obtain the full Lagrangian density of the coupled Higgs and electromagnetic fields:

L(x) = [Dµφ(x)]∗[Dµφ(x)] − µ2|φ(x)|2

− λ|φ(x)|4 − 1

4Fµν(x)F

µν (x). (30.39)

Note that the replacement ∂µφ∗ = (∂µφ(x))∗ ⇒ (Dµφ(x))∗ was used. The L(x) in Eq.(30.39) is the Langrangian of the Abelian Higgs model. Eq. (30.39) is manifest invariantunder the U(1) gauge transformation [Eqs. (8.130) and (8.131)]

φ(x) ⇒ φ(x) = φ(x) exp

(

iq

~χ(x)

)

, (30.40)

Aµ(x) → Aµ(x) = Aµ(x) + ∂µχ(x), (30.41)

remembering that the covariant derivative of φ(x) transforms as φ(x), i.e.,

Dµφ(x) → Dµφ(x) = exp

(

iq

~χ(x)

)

Dµφ(x). (30.42)

To rewrite the Lagrangian in terms of H(x) and η(x) let us begin from

(Dµφ)∗ (Dµφ) =

[(

∂µ − iq

~Aµ

)

φ

]∗ [(

∂µ − iq

~Aµ

)

φ

]

= (∂µφ)∗(∂µφ) +

( q

~

)2

AµAµ|φ|2

+iq

~[(Aµφ∗) (∂µφ) − (∂µφ∗) (Aµφ)] . (30.43)

With φ(x) given by Eq. (30.23) we have

( q

~

)2

AµAµ|φ|2 =1

2

(qv

~

)2

AµAµ

+1

2

( q

~

)2(

H2 + 2vH + η2)

AµAµ, (30.44)

and

iq

~[(Aµφ∗) (∂µφ)− (∂µφ∗) (Aµφ)]

=iq

~Aµ [φ∗∂µφ− φ∂µφ

∗]

= −qv~Aµ∂µη +

q

~Aµ (η∂µH −H∂µη) . (30.45)

The two contributions in Eqs. (30.44) and (30.45), plus the free-field term in Eq. (30.37),are the ones we need to add to the Goldstone Langrangian of the Abelian Higgs model.


Hence,

L(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x)

+1

2[∂µη(x)] [∂µη(x)]

− 1

4Fµν(x)F

µν(x) +1

2

(qv

~

)2

Aµ(x)Aµ(x)

− qv

~Aµ(x)∂µη(x)

− λvH(x)(

H2(x) + η2(x))

− λ

4

(

H2(x) + η2(x))2

+1

2

( q

~

)2

Aµ(x)Aµ(x)(

H2(x) + 2vH(x) + η2(x))

+q

~Aµ (η∂µH −H∂µη) . (30.46)

The last three lines in Eq. (30.46) contain terms of third and fourth order in the fields(Aµ, H, η), and these couplings may be treated by perturbation theory. If the second-orderterm (−qv/~)Aµ∂µη [line four] was absent, the three first lines would describe uncoupled(free) H , η, and Aµ-fields. The term proportional to Aµ(x)∂µη(x) couples the Goldstoneand electromagnetic fields, and in consequence Aµ(x) and η(x) are not independent normalvariables. Such a coupling must necessarily be present because the Lagrangian density inEq. (30.39) has four degrees of freedom [two from the complex scalar field φ(x), and twofrom the massless free photon field], whereas the Lagrangian density in Eq. (30.46) has five[one from H(x), one from η(x), and three from the massive free photon field], neglectingthe Aµη-coupling.

To reduce the apparent degrees of freedom in Eq. (30.46) we now seek to eliminate theGoldstone field, η(x), by a gauge transformation. By writing φ(x) [Eq. (30.23)] in polarform

φ(x) =1√2

[

(v +H(x))2 + η2(x)]

1

2

exp [iα(x)] , (30.47)

where tanα(x) = η(x)/(v +H(x)), a gauge transformation with local gauge function

χ(x) = −~

qα(x) (30.48)

transforms the complex field into a real one

φ(x) =1√2

[

v2 + 2vH(x) +H2(x) + η2(x)]

1

2 . (30.49)

To first order in the deviations

φ(x) =1√2(v +H(x)) , (30.50)

a real quantity, and

χ(x) = − ~

qvη(x). (30.51)

The gauge in which the transformed field has the form in Eq. (30.50) is called the unitary


gauge. The gauge function in Eq. (30.51) depends on η(x) only, and in the unitary gaugethe Goldstone field is eliminated. In turn this means that the Langrangian density of theAbelian Higgs model [Eq. (30.46)] takes the form [in a notation where for simplicity weleave out the tilde (∼) on the top of all transformed quantities]

L(x) = LH(x) + LF (x) + LI(x), (30.52)

where

LH(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x) (30.53)

LF (x) = −1

4Fµν(x)F

µν (x) +1

2

(qv

~

)2

Aµ(x)Aµ(x), (30.54)

and

LI(x) =− λvH3(x) − λ

4H4(x)

+1

2

( q

~

)2

Aµ(x)Aµ(x) [2v +H(x)]H(x). (30.55)

We can interpret

L0(x) = LH(x) + LF (x) (30.56)

as the free-field Lagrangian density of a real scalar Klein–Gordon field (Higgs field), H(x),and a real massive vector field (photon field), Aµ(x), and LI(x) as the interaction La-grangian density. The LI(x)-part of L(x) includes mutual and self-field terms.

The dynamical equation for the free Higgs field is obtained from the Lagrange equation

∂µ[

∂LH(x)

∂ (∂µH(x))

]

− ∂LH(x)

∂H(x)= 0. (30.57)

Since the canonical momentum is

PHµ (x) ≡ ∂LH(x)

∂ (∂µH(x))= ∂µH(x), (30.58)

we find the Klein–Gordon equation

[

∂µ∂µ + 2λv2]

H(x) = 0. (30.59)

On quantization, H → H , LH(x) gives rise to neutral scalar Higgs bosons of mass

MH =~

c

(

2λv2)

1

2 . (30.60)

In the next section we shall study the dynamical equation for the electromagnetic vectorpotential Aµ(x). We shall see that Aµ(x) upon quantization gives rise to neutral massivevector bosons (massive photons) of mass |q|v/c, a result the reader perhaps already mayhave anticipated by a glance at LF (x) [Eq. (30.54)], remembering that −(1/4)Fµν(x)F

µν (x)is the Lagrangian density for the free massless electromagnetic field [Eq. (3.87)].


30.4 Photon mass and vacuum screening current

The dynamical equation for the free photon field is obtained from the Lagrange equation

∂ν

(

∂LF (x)

∂ (∂νAµ)

)

− ∂LF (x)

∂Aµ= 0, (30.61)

with LF (x) given by Eq. (30.54). From the relation between the field tensor and the covariantvector potential, written in covariant tensor form in Eq. (8.101), one has

−1

4Fµν(x)F

µν(x) = −1

4(∂µAν − ∂νAµ) (∂

µAν − ∂νAµ)

= −1

2(∂νAµ) (∂

νAµ − ∂µAν) , (30.62)

and hence

∂LF (x)

∂ (∂νAµ)= −1

2(∂νAµ − ∂µAν) . (30.63)

Furthermore, one gets

∂LF (x)

∂Aµ=

1

2

(qv

~

)2

Aµ. (30.64)

By combining Eqs.(30.61), (30.63), and (30.64), we obtain the Proca equation

∂ν∂νAµ(x) − ∂µ (∂νA

ν(x)) +Q2CA

µ(x) = 0, (30.65)

where

QC =|q|v~. (30.66)

With the new metric choice in Eq. (30.12), one has

∂µNEW =

1

c

∂

∂t,−∇

= −∂µOLD, (30.67)

and since ∂NEW,ν∂νNEW = −, it is obvious that Eqs. (12.79) and (30.65) are equivalent in

form. For a physical vacuum with spontaneous symmetry breaking, the Goldstone potentialenergy density in Eq. (30.3), hence leads to a photon vacuum mass

M =|q|vc. (30.68)

The dynamical equation for Aµ(x) in the presence of interaction with the Higgs fieldis determined by the Lagrange equation

∂ν

[

∂ (LF (x) + LI(x))

∂ (∂νAµ)

]

− ∂ (LF (x) + LI(x))

∂Aµ= 0. (30.69)

Since the coupling to the Higgs field is proportional to Aµ(x)Aµ(x), one only needs to addthe term

∂LI(x)

∂Aµ=

1

2

( q

~

)2(

2vH(x) +H2(x))

Aµ(x), (30.70)


multiplied by 2, to the right-hand side of Eq. (30.64). The equation

(

∂ν∂ν +Q2

C

)

Aµ(x) − ∂µ (∂νAν(x))

= −( q

~

)2(

2vH(x) +H2(x))

Aµ(x), (30.71)

thus is the dynamical equation of motion for the µth component of a massive vector fieldAµ(x) coupled in a specific way to the neutral real scalar Higgs field, H(x). The couplingis proportional to Aµ(x) if the Higgs field is approximated by its free form. In a sense, onemay say that the massive electromagnetic field in the displaced vacuum is subjected to avacuum screening (sc) current density

jV AC,µsc = −

( q

~

)2(

2vH(x) +H2(x))

Aµ(x), (30.72)

which in form resembles a four-component version of the diamagnetic current density dis-cussed in Sec. 12.3. Note that the contribution in Eq. (30.72) occurs in addition to the“diamagnetic” current density −|q|v/~ making the photon massive.

30.5 ’t Hooft gauge and propagator

It appears from the analysis in Sec. 12.5 that a massive photon propagator

Gµν =1

q2 −Q2C

(

−gµν + qµqν

Q2C

)

(30.73)

can be related to the Proca equation in Eq. (30.65), cf. Eq. (12.94). The somewhat differentforms in Eqs. (12.94) and (30.73) is a consequence of our use of two different definitions ofthe metric tensor. Thus, with gµνNEW = −gµνOLD [Eq. (30.12)], one obtains q2NEW = −q2OLD,and since one always has (qµqν)NEW = (qµqν)OLD, the two forms are in agreement witheach other.

The expression given in Eq. (30.73) for the massive photon propagator was reachedfrom the Proca equation in Eq. (30.65). Since this equation relates to a specific gauge,viz., the unitary gauge [6], it can be concluded that the “massive photon propagator” is agauge dependent concept. In order to elaborate on this point, and compare the situationwith spontaneous symmetry breaking of the degenerate vacuum to that of a relativisticsuperconductor [Sec.12.4], let us reintroduce the notation with a tilde (∼) on top of thetransformed four-component potential. To first order in the deviations of the complex field,φ(x), from the displaced ground state, the gauge function in Eq. (30.51) transforms φ(x) toa real field [Eq. (30.50)], and the contravariant potential into

Aµ(x) = Aµ(x) − ~

qv∂µη(x). (30.74)

The transformed potential satisfies the Proca equation

(

∂ν∂ν +Q2

C

)

Aµ(x) − ∂µ(

∂νAν(x)

)

= 0, (30.75)

or equivalently

(

∂ν∂ν +Q2

C

)

Aµ(x) = 0, (30.76)


since the Lorenz condition

∂µAµ(x) = 0 (30.77)

is a constraint for QC 6= 0. By taking the four-divergence of Eq. (30.74), and using Eq.(30.77), one sees that our original aµ(x) is in a gauge specified by

∂µAµ = − ~

qvη(x). (30.78)

This is the unitary gauge. A comparison to Eq. (12.78), shows the formal analogy between arelativistic superconductor in which the norm of the wave function is constant in space-time[Eq. (12.65)]. Since it was the gauge choice with the constraint in Eq. (30.78) which allowedus to eliminate the Goldstone bosons, we expect that other choices of gauge will reintroducethese virtual massless scalar bosons. In a sense the properties of these “ghost particles” areanalogous to those of the longitudinal and scalar photons in quantum electrodynamics.

Let us now return to the Lagrangian density in Eq. (30.46), and let us leave out thethird- and fourth-order interaction terms. The remaining part, denoted by L0(x) for reasonswhich will become clear soon, is given by

L0(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x)

+1

2[∂µη(x)] [∂µη(x)] −

qv

~Aµ(x)∂µη(x)

− 1

4Fµν(x)F

µν (x) +1

2

(qv

~

)2

Aµ(x)Aµ(x), (30.79)

and it contains the bilinear coupling term −(qv/~)Aµ(x)∂µη(x). If this term can be elimi-

nated by a gauge choice, L0(x) becomes a free-field Lagrangian density.Before we turn to the relevant gauge choice we add to the Lagrangian density a cer-

tain four-divergence. From Lagrangian field theory follows that two Lagrangian densities(L′(x),L(x)) which differ by a four-divergence

L′(x) = L(x) + ∂µΩµ(x), (30.80)

have the same action integral (taken over all space-time), i.e.,

I =

∫ ∞

−∞L′(x)d4x =

∫ ∞

−∞L(x)d4x. (30.81)

This is so because the action integral of ∂µΩµ(x) can be transformed into a surface inte-

gral by Gauss’s theorem in four dimensions, and the integrand vanishes on that infinitelydistant surface. To the Lagrangian density in Eq. (30.79) one may therefore add a term(qv/~)∂µ[A

µ(x)η(x)] without entering any predictions for observable quantities. Since

∂µ [Aµ(x)η(x)] −Aµ(x)∂µη(x) = η(x)∂µA

µ(x), (30.82)

our modified Lagrangian density [L′0(x) ≡ L0(x)] hence is

L0(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x)

+1

2[∂µη(x)] [∂µη(x)] +

qv

~η(x)∂µA

µ(x)

− 1

4Fµν(x)F

µν (x) +1

2

(qv

~

)2

Aµ(x)Aµ(x). (30.83)


It is now easy to eliminate the coupling term (qv/~)η(x)∂µAµ(x) from Eq. (30.83) by im-

posing the ’t Hooft gauge condition

∂µAµ(x) =

qv

~η(x). (30.84)

With this choice one may add the term

−1

2

[

∂µAµ(x) − qv

~η(x)

]2

= 0 (30.85)

to L0(x). The L0(x)-Lagrangian density now takes its final form, viz.,

L0(x) =1

2[∂µH(x)] [∂µH(x)]− 1

2

(

2λv2)

H2(x)

+1

2[∂µη(x)] [∂µη(x)] −

1

2

(qv

~

)2

η2(x)

− 1

4Fµν(x)F

µν(x) +1

2

(qv

~

)2

Aµ(x)Aµ(x) −1

2[∂µA

µ(x)]2 . (30.86)

The Lagrangian density in Eq. (30.86) now refers to three real (independent) free fields: TheHiggs field (H(x)), a massive Goldstone field (η(x)) [mass: |q|v/c], and a massive photonfield (Aµ(x)).

Since

∂ν

∂[

− 12 (∂

αAα)2]

∂ (∂νAµ)

= ∂ν[

(−∂αAα)∂ (∂αAα)

∂ (∂νAµ)

]

= ∂µ (−∂αAα) = −∂µ (∂νAν) , (30.87)

it appears (taking into account the calculation in the first part of Sec. 30.4) that Aµ(x)satisfies the Proca equation

(

−Q2C

)

Aµ(x) = 0 (30.88)

in the ’t Hooft gauge [Eq. (30.84)]. The related ’t Hooft propagator is

Gµν′tHooft = − gµν

q2 −Q2C

, (30.89)

cf. the analysis in Sec. 12.5. The troublesome term qµqν/Q2C which occurred in the propa-

gator in Eq. (30.73) is absent from Eq. (30.89), and in the limit QC → 0 the massless Feyn-man photon propagator is obtained [see Eq. (12.96), and remember that gµνNEW = −gµνOLD,q2NEW = −q2OLD]. This suggests that the Abelian Higgs model is renormalizable, like QED[209, 6].

Part VIII

Two-photon entanglement inspace-time

31

The quantal photon gas

Let us consider a gas of N non-interacting identical quantum particles, with discrete single-particle states 1, 2, · · · , i, · · · , possessing the eigenenergies ε1 ≤ ε2 ≤ · · · εi ≤ · · · . The quan-tum state of the whole gas is specified by the set of occupation numbers n1, n2, · · · , ni, · · · ,where ni is the number of particles in the single-particle state i. According to quantummechanics two mutually exclusive classes of (elementary) particles exist. From a quantumstatistical point of view, the classes are distinguished on the basis of the possible values ofthe occupation numbers. In the first class there is as such no restriction on the occupationnumbers ni, i.e.,

ni = 0, 1, 2, · · · , ∀i. (31.1)

The kind of particles which belong to this class are named bosons, and the quantum statisticsthey obey is known as Bose–Einstein (BE) statistics. In the second class the occupationnumbers are restricted to the values

ni = 0, 1, ∀i, (31.2)

so that at most one particle can be in any state. The particles belonging to the secondclass are named fermions, and the quantum statistics they obey is known as Fermi–Dirac(FD) statistics. The restriction in (31.2) states the Pauli exclusion principle: Two identicalfermions cannot be in the same single-particle state. The general formulation of the exclu-sion principle was first given by Pauli in 1925 [182, 183, 184]. There is a most remarkableconnection between the spin (intrinsic angular momentum) of a quantum particle and thestatistics: Bosons possess integral spin, and fermions half-integral spin. For the elemen-tary particles the spin is (in units of ~) 0 (Higgs), 1/2 (electron, muon, tauon, neutrinos,quarks; and their antiparticles), or 1 (photons, vector bosons [W+, W−, Z0], gluons). Forthe unobserved graviton the spin is predicted to be 2.

There is a further remarkable connection which relates the statistics to the symmetryproperties of the quantum state of the total particle system. Thus, for identical bosons thestate must be symmetric if two bosons are interchanged, whereas it for identical fermionsmust be antisymmetric. The symmetry and antisymmetry requirements are general in thesense that they hold not only for non-interacting particles but also in the presence of particleinteraction. Thus, for two-particle (1, 2) systems one has in general

|ψBE(2, 1)〉(t) = +|ψBE(1, 2)〉(t), (31.3)

and

|ψFD(2, 1)〉(t) = −|ψFD(1, 2)〉(t). (31.4)

If the two identical particles do not interact with each other, and we assume that theparticles are in the single-particle eigenstates i and j, with associated state vectors |ψi〉 and|ψj〉, the two-boson state is

|ψBE(1, 2)〉 =1√2[|ψi(1)〉 ⊗ |ψj(2)〉+ |ψi(2)〉 ⊗ |ψj(1)〉] , (31.5)

379


and the two-fermion state

|ψFD(1, 2)〉 = 1√2[|ψi(1)〉 ⊗ |ψj(2)〉 − |ψi(2)〉 ⊗ |ψj(1)〉] . (31.6)

The factor 1/√2 in the equations above takes care of the state normalization. In agreement

with the exclusion principle, we see that |ψFD(1, 2)〉 vanishes identically if i = j, indicatingthat both fermions would be in the same single-particle state (ni = 2). For bosons we canhave i = j (ni = 2), and consequently (in normalized form) |ψBE(1, 2)〉 = |ψi(1)〉 ⊗ |ψi(2)〉.

Although, as such, there is no restriction on the ni’s for bosons, the condition∑

i

ni = N (31.7)

must be satisfied for a gas of N particles (bosons or fermions). The total energy E of ourN non-interacting particles is given by

E(n1, n2, · · · , ni, · · · ) =∑

i

niεi. (31.8)

Since the totality of sets of occupation numbers gives one all possible microstates of thegas, the partition function of the gas, in equilibrium at temperature T in an enclosure ofvolume V , is given by

Z(T, V,N) =∑

n1,n2,···exp

(

−β∑

i

niεi

)

, (31.9)

where β = 1/(kBT ).After the general summary given above. we now consider the quantal photon gas. As we

have discussed in detail in previous parts of the book, photons are spin-1 particles, and assuch they obey BE statistics. Furthermore, to an extremely good approximation, they do notinteract with each other. This follows from the essential linearity of the Maxwell equationsin global vacuum. The weak quantum-mechanical nonlinearity, which enables photons to bescattered by photons, stems from the fact that electron-positron pairs can be created (andafterward destroyed) from the vacuum in the presence of sufficiently strong electromagneticfields. A particle vacuum hence behaves like a nonlinear medium (with an extremely smallnonlinear microscopic conductivity tensor). The photon gas hence is considered as a perfectquantal boson gas in what follows.

Since the observation of a T-photon requires photon-matter interaction, and since pho-tons are destroyed (created) in their interaction with massive particles, it is in studiesinvolving a quantal photon gas usually not a good approximation to assume that the totalphoton number (N) is conserved. To illustrate this aspect it is fruitful to consider the well-known black-body radiation, i.e., the radiation within an opaque enclosure whose walls aremaintained at a uniform temperature, T . The continual emission and absorption of pho-tons by the particles (atoms) of the walls imply that the number of photons in the cavityfluctuates about a mean value, determined by the thermal equilibrium conditions. Conse-quently, the partition function [Eq. (31.9)] for the photons in the cavity is not restricted bythe condition in Eq. (31.7): Each of the single-particle occupations numbers can assume allvalues from zero to infinity. For black-body radiation, the photon (ph) partition function,Zph, thus is a function of T and V , and given by

Zph(T, V ) =

( ∞∑

n1=0

∞∑

n2=0

· · ·)

exp

(

−β∑

i

niεi

)

=Π∞i=1 [1− exp (−βεi)]−1 . (31.10)

The quantal photon gas 381

From Eq. (31.10) one can calculate the mean photon occupation number (ni) in the statei in the usual manner [162]. Thus

ni = − 1

β

∂

∂εilnZph(T, V ) =

1

exp (βεi)− 1. (31.11)

The result in Eq. (31.11) is in agreement with what would have been obtained starting fromthe grand canonical ensemble, setting the chemical potential (µ) to zero at the end of thecalculation. The reason that the black-body photon gas has µ = 0 may be associated to thefact that the partition function is a function of T and V , only.

Now, let us turn our attention toward a classification of the single-photon states in thecavity. We assume that the photons in the enclosure are free, neglecting thus all esotericeffects associated to the field-matter interaction in the rim zone of the cavity walls, and tothe birth and death processes of the photon. In a classical electromagnetic approach, thefield modes usually are taken as standing (or running) monochromatic plane waves for abox-formed enclosure (of volume V ). With this in mind, let the photon wave functions bemonochromatic plane waves. The energies and momenta of the photons then are given bythe de Broglie relations εi = ~ωi and pi = ~qi, ∀i. Under the assumption that the boxvolume is sufficiently large (V → ∞ ⇒ ωi → ω, pi → p) the magnitude of the photonmomentum is given by p = ~ω/c.

In quantum statistics the number of microstates (Ω) in the phase space volume VqVp isgiven by

Ω = h−3VqVp (31.12)

for a system (particle) with three degrees of freedom. The correspondence in Eq. (31.12),originates in the Heisenberg uncertainty relations between position and momentum for aquantum particle, viz., ∆x∆px & h, ∆y∆py & h, ∆z∆pz & h. Qualitatively, the locationand momentum cannot be measured simultaneously with an accuracy greater than allowedby the Heisenberg uncertainty relations above. In view of this, phase space volume elements(cells) of smaller size than

∆Vq∆Vp = ∆x∆y∆z∆px∆py∆pz = h3 (31.13)

have no physical (observational) importance. Our studies of the spatial localization problemfor the T-photon have made it clear that the volume ∆Vq = ∆x∆y∆z cannot be madearbitrarily small, but this fact does not affect the considerations above. Under thermalequilibrium conditions there is no preferential direction of photon propagation in a largecavity. The number of microstates for photons in a spherical shell in momentum space (radiip and p+ dp) and the direct-space volume V therefore is given by

dΩ = 2

(

V

h3

)

4πp2dp. (31.14)

The factor 2 in Eq. (31.14) arises from the fact that one to each mode has two independentphoton polarization states. Since p = ~ω/c, Eq. (31.14) can be rewritten in the form dΩ/V =π−2c−3ω2dω. A classical field calculation leads exactly to the same expression for the modedensity, dΩ/V . Perhaps not unexpected, the particle and field approaches thus agree. Sincethe mean photon energy at ω is given by ~ωn(ω), and the mode density by per unit frequencyinterval is V −1dΩ/dω = π−2c−3ω2, the mean energy density at ω is given by

u(ω, T ) =~ω3

π2c3

[

exp

(

~ω

kBT

)

− 1

]−1

. (31.15)


The expression in Eq. (31.15) is Planck’s formula for black-body radiation. Planck came tohis formula in the autumn of 1900 [189, 190, 85], and his achievement started the quantumphysical revolution.

Experimental studies of a quantal photon gas require field-matter interaction, and inChapt. 32 we briefly present elements of the general theory for quantum measurements.Starting from the union, H = H1 ⊗ H2, of two Hilbert spaces H1 and H2 we realize thatthere exist states (state vectors) in H which cannot be written as a single tensor productof two state vectors in H1 and H2, respectively. Such states are called entangled states.After having introduced the definition of an observable, and the postulate that a so-called“reduction of the wave packet” takes place in a measurement, we analyze the situationwhere a measurement is carried out on only one part of a two-part physical system.

In the wake of the general discussion of quantum measurement theory and entanglement,we turn our attention toward the photons, starting with a brief account of polarization(spin) entanglement for a two-photon system (Sec. 32.5). Typically, an entangled two-photonpolarization state has the form

|ψ〉 = 1√2(|R(1)〉 ⊗ |R(2)〉+ |L(1)〉 ⊗ |L(2)〉) , (31.16)

where |R(i)〉 and |L(i)〉 refer to single-photon states of positive (R) and negative (L) helicityfor particle number i (i = 1, 2). Entanglement may occur even if the two particles do notinteract (in the sense that the interaction Hamiltonian vanishes). In this context one mustnot forget the lesson of Niels Bohr [28]: “... that in such (an entangled) quantum stateone cannot speak about the individual properties of each of the particles, even if they arefar apart (and space-like separated).” Note that the two-boson state given in Eq. (31.5) isan entangled state among non-interacting particles. The entanglement here associates tothe fact that the two bosons are identical, and their common state therefore unchanged(symmetric) upon particle interchange (1 ↔ 2). Thus, it is a general aspect of quantumphysics that symmetrization (antisymmetrization) of boson (fermion) n-body states amongidentical particles inevitably results in state entanglement.

Measurements are carried out in Minkowskian (or Riemannian) space and not in Hilbertspace, and the observer moves on a certain world line with his detector (laboratory). In thepolychromatic single-photon case the link between the Hilbert and Minkowski descriptionswas made by the T-photon mean position state, |R〉(r, t) [Eq. (15.121)]. Thus, the innerproduct

Φ(r, t) = 〈R|Φ〉(r, t) (31.17)

gives us the single-photon wave function in Minkowskian space. To understand the mostfundamental aspects of spatial entanglement between photon wave packets in direct spaceit is sufficient to study two-photon systems. In Chapt. 33 we establish the general theory fortwo-photon wave mechanics and the associated first and second-order correlation matrices,and in Chapt. 34 the theory is applied to a study of spatial entanglement in spontaneoustwo-photon cascade emission. By an extension of the definition given for |R〉(r, t), thetransverse two-photon mean position state (|R2〉) is defined by

|R2〉(r, r′, t, t′) ≡2ǫ0c

~A

(−)T (r, t)A

(−)T (r′, t′)|0〉. (31.18)

On the basis of this dyadic quantity one is led to the so-called tensorial transverse two-photon two-times “wave function” in configuration, viz.,

Φ2(r, r′, t, t′) ≡ 〈R2|Φ2〉(r, r′, t, t′). (31.19)

The quantal photon gas 383

The dyadic quantity Φ2(r, r′, t) ≡ Φ2(r, r

′, t, t) is our two-particle wave function, and weshow in Sec. 33.2 that Φ2 satisfies a dynamical Schrodinger-like two-particle wave equation.

The first-order two-photon correlation matrix, G(1)2 (x1, x2), is obtained making the re-

placement |Φ〉 ⇒ |Φ2〉 (〈Φ| ⇒ 〈Φ2|) in Eq. (16.88), and in Subsec. 33.3.1 we prove that

G(1)2 (x1, x2) = L−3

∑

j

[

Φ(j)1 (x1)

]∗Φ

(j)1 (x2), (31.20)

where Φ(j)1 is a one-photon wave function constructed on the basis of the amplitude weight

factors, φij , for the given polychromatic two-photon state in Hilbert space. To each j is

associated a single-photon amplitude weight factor φ(j)i ≡ φij , and by means of these the

one-photon wave function Φ(j)1 (x) is constructed in the usual manner. The second-order

correlation matrix of a transverse two-photon state, viz.,

G(2)2 (x1, x2, x3, x4) =

(

2ǫ0c

~

)2

〈Φ2|A(−)T (x1)A

(−)T (x2)A

(+)T (x3)A

(+)T (x4)|Φ2〉, (31.21)

can be expressed solely by means of a tensor product of dyadic two-photon two-times wavefunctions. Hence,

G(2)2 (x1, x2, x3, x4) = Φ∗

2(x1, x2)Φ2(x3, x4). (31.22)

The second-order correlation matrix of a single-photon state vanishes because a single pho-ton cannot be detected, and thus destroyed, at two different space-time points.

Our study of the spatial entanglement associated with spontaneous two-photon cascadeemission in a three-level atom is based on the Weisskopf–Wigner (WW) theory of spon-taneous emission. We prepare ourselves for the general analysis by a brief account of thedescription of spontaneous photon emission in a two-level atom (Sec. 34.1). On the basis ofthe electric-dipole and rotating-wave approximations, and under the assumption that theprocess is memory-less (Markow approximation), we end up with a well-known expressionfor the global state vector of the coupled atom-field system. In Sec. 34.2 we calculate thewave function of the spontaneously emitted photon, first in q-space and then in r-space.The result obtained is valid at all distances from the atom, and thus represents an extensionof a previously calculated expression for the far-field part of the T-photon wave function[214, 165, 166]. From the outgoing wave function, we project the genuine transverse pho-ton wave function out. Finally, it is shown that the near-field part of the T-photon wavefunction contains a non-propagating evanescent part, which exhibits exponentially damped(with the spontaneous decay constant of the upper atomic level) harmonic oscillations (withthe Bohr transition frequency). The evanescent component of the T-photon wave functionobeys microcausality, as it must, and is inevitably present since the spatial T-photon embryolocalization is not perfect during the spontaneous emission process.

In Sec. 34.3 we study the spontaneous two-photon cascade emission from a three-levelatom with the aim of describing the spatial two-photon entanglement. In the framework ofthe WW-theory the two-photon state vector is determined (Subsec. 34.3.1). From the well-known [214] final result for |Φ2〉 [Eq. (34.135)], we obtain (Subsec. 34.3.2) the two-photontwo-times wave function following the scheme in Eqs. (31.18) and (31.19). From the outgoingtwo-photon two-times wave function, Φ2,OUT (r1, r2, t1, t2), one obtains the transverse part,Φ2,T (r1, r2, t1, t2) , by replacing the dyadic Green function G (appearing twice in Φ2,OUT )by its transverse (propagator) part, GT . The result obtained holds in all distances from theatom, and has to the best of my knowledge not been given in the literature, previously. Thejoint probability density for transverse photon detection at the space-time points x1 and


x2, essentially, is given by the second-order correlation matrix

G(2)2 (x1, x2, x2, x1) = Φ∗

2,T (x1, x2)Φ2,T (x2, x1), (31.23)

and an analysis of the structure of Φ2,T (x1, x2) reveals the details of the physical observa-tion process; see Subsec. 34.3.3. If we denote the decay constants of the upper and lowertransitions by Γa and Γb, respectively, it is physically obvious that the entire decay pro-cess for Γa ≫ Γb can be seen as an emission process in which the two photons are emittedindependently. This conclusion agrees with the fact that our calculation shows that the two-photon two-times wave function is built from the single-photon wave functions belongingto the upper (a→ b) and lower (b→ c) transitions. Hence,

Φ2,T (x1, x2) = ΦabT (x1)⊗Φbc

T (x2) +ΦbcT (x1)⊗Φab

T (x2), Γa ≫ Γb. (31.24)

Although the two photons are non-interacting they are still in an entangled state, cf. Eq.(31.5) [The Hilbert state in Eq. (31.5) is represented in direct space by the inner product〈r1, r2|ψBE(1, 2)〉, |r1, r2〉 = |r1〉 ⊗ |r2〉, being the tensor product of the relevant positionstates. In the photon case these are |R1〉 and |R2〉.] A T-photon is a T-photon, that is, alltransverse photons are identical as particles, but in Eq. (31.24) they are in different quantumstates [Φab

T ,ΦbcT ]. In the general case, where the condition Γa ≫ Γb is not satisfied, the two-

photon wave function, Φ2,T (x1, x2) cannot be written as a sum of tensor products involvingonly one-photon wave functions. The entanglement in the two-photon cascade emissionthus is between interacting T-photons. The interaction between the photons arises from thecorrelations of the two (a → b, b → c) atomic transition current densities in the three-levelatom.

The presence of both near (NF) and far-field (FF) parts in the transverse propagatorimplies that the structure of Φ2,T (x1, x2) contains contributions of the types FF-FF, FF-NF(NF-FF) and NF-NF, and various time-like and evanescent contributions.

32

Quantum measurements

32.1 Tensor product space (discrete case)

Starting from two Hilbert spaces H1 and H2 of dimension N1 and N2, respectively, one canconstruct a larger Hilbert space

H = H1 ⊗H2, (32.1)

a sort of union of the two. The space H is called the tensor product of H1 and H2. Statevectors and operators of these spaces will be denoted by an index, (1) or (2). Associatedwith a pair of state vectors |φ(1)〉 and |χ(2)〉 belongs a state vector

|φ(1)〉 ⊗ |χ(2)〉 ≡ |φ(1)〉|χ(2)〉 ≡ |φ(1)χ(2)〉 (32.2)

in the combined Hilbert space. Let us choose basis sets |un(1)〉 and |uk(2)〉 for H1 andH2. In terms of these one can express the arbitrary vectors |φ(1)〉 and |χ(2)〉 as follows:

|φ(1)〉 =∑

n

an|un(1)〉, (32.3)

|χ(2)〉 =∑

k

bk|vk(2)〉. (32.4)

In the Hilbert space H one may use a basis |un(1)⊗ |vk(2)〉〉 for expansion of the tensorproduct in Eq. (32.2). Thus,

|φ(1)〉 ⊗ |χ(2)〉 =∑

n,k

anbk|un(1)〉 ⊗ |vk(2)〉. (32.5)

The components of a tensor product state vector in H hence are products, anbk, of com-ponents of the two state vectors in the Hilbert spaces H1 and H2. The most general statevector in H, |ψ〉, can in the basis |un(1)〉 ⊗ |vk(2)〉 be expanded as follows:

|ψ〉 =∑

n,k

cnk|un(1)〉 ⊗ |vk(2)〉, (32.6)

where the double index nk on the expansion coefficient cnk specifies the vector axes in H.Although an arbitrary vector |ψ〉 in H cannot be written as a single vector product of twovectors in H1 and H2 [Eq. (32.5)], |ψ〉 can always be decomposed into a linear combinationof tensor product vectors [Eq. (32.6)]. This result relates to the fact that the elements ofan arbitrary matrix cnk cannot in general be given as the elements anbk of a dyadicproduct of two vectors with components an and bk. The scalar product in H of twoarbitrary basis vectors, |un(1)〉 ⊗ |vk(2)〉 and |un′(1)〉 ⊗ |vk′(2)〉, is given by

〈un′(1)vk′(2)|un(1)vk(2)〉 = 〈un′(1)|un(1)〉〈vk′ (2)|vk(2)〉 = δnn′δkk′ . (32.7)

385


The last member of Eq. (32.7) is obtained provided each of the bases |un(1)〉 and |vk(2)〉is orthonormal in its respective Hilbert space. If this is the case, the basis in H, i.e., |un〉⊗|vk(2)〉, is orthonormal, too.

Let A(1) and B(2) be two linear operators acting respectively in H1 and H2. The tensorproduct operator

A(1)⊗ B(2) ≡ A(1)B(2) (32.8)

is a linear operator in H, defined by its action on a tensor product |φ(1)〉 ⊗ |χ(2)〉:[

A(1)⊗ B(2)]

[|φ(1)〉 ⊗ |χ(2)〉] =[

A(1)|φ(1)〉]

⊗[

B(2)|χ(2)〉]

. (32.9)

For what follows, it is useful to introduce the so-called extension concept for a linear operatoracting in H1 (or H2). Thus, if B(2) in Eq. (32.8) is the identity operator 1(2) in H2,B(2) = 1(2), then the operator

A = A(1)⊗ 1(2) (32.10)

is called the extension of A(1) into H.

32.2 Definition of an observable (discrete case)

Consider a Hermitian operator O = O†, and the associated discrete eigenvalue equation

O|ψin〉 = λn|ψi

n〉, i = 1, 2, · · · , gn. (32.11)

The eigenvalues of O, λn, are real and the degree of degeneracy of the nth eigenvalue isdenoted by gn. The eigenvectors belonging to a given n, |ψi

n〉, i = 1, 2, · · · , gn, can alwaysbe chosen orthonormal, 〈ψi

n|ψjn〉 = δij . With such a choice, and in view of the fact that

eigenstates belonging to different n’s are orthogonal, and can be normalized, the eigenvectorsfor O form an orthonormal system:

〈ψin|ψj

m〉 = δnmδij . (32.12)

By definition the Hermitian operator O is an observable if the orthonormal system of eigen-vectors forms a basis in the given state (Hilbert) space. The closure relation

∑

n

gn∑

i=1

|ψin〉〈ψi

n| = 1 (32.13)

expresses the completeness of the set of eigenvectors |ψin〉. The projector Pn onto the

subspace formed by the gn eigenvectors belonging to a given n is given by

Pn =

gn∑

i=1

|ψin〉〈ψi

n|, (32.14)

and it is manifest Hermitian, Pn = P †n.

The completeness of the set of orthonormal eigenvectors, |ψin〉, allows one to expand

every state vector as follows:

|ψ〉 =∑

n

gn∑

i=1

cin|ψin〉, (32.15)

where cin = 〈ψin|ψ〉.

Quantum measurements 387

32.3 Reduction of the wave packet (discrete case)

It is a postulate in quantum mechanics that every measurable physical quantity A is de-scribed by an observable A acting in the system’s state space. It is further postulated thatthe only possible result of a measurement of A is one of the eigenvalues of A, and that theprobability for obtaining the eigenvalue λn is given by

P(λn) =

gn∑

i=1

|〈ψin|ψ〉|2 (32.16)

for a system which “immediately before” [251] the measurement is in the normalized quan-tum state |ψ〉. If the eigenvalue λn is obtained in the measurement quantum mechanicspostulates that the state of the system “immediately after” [251] the measurement is in thestate

|ψ′〉 =[

gn∑

i=1

|cin|2]− 1

2 gn∑

i=1

cin|ψin〉. (32.17)

The quantity [∑

i |cin|2]−1/2 ensures that the state is normalized, 〈ψ′|ψ′〉 = 1. The changefrom |ψ〉 to |ψ′〉 as a result of the measurement often is called “the reduction of the wavepacket.” In terms of the projection operator in Eq. (32.14), the modification of the statevector as a result of a measurement process (giving the result λn) may be written as follows:

|ψ〉 ⇒ |ψ′〉 = Pn|ψ〉[

〈ψ|Pn|ψ〉]

1

2

, (32.18)

as the reader may show using that P 2n = Pn.

32.4 Measurements on only one part of a two-part physical system

We now return to the tensor product space in Sec. 32.1, and imagine a measurement carriedout (bearing) on part (1), only. Let the physical quantity we measure in H1 be describedby the observable A(1), which we assume only has a discrete set of eigenvalues, an. If theeigenvalue an is gn-fold degenerate in H1, the eigenvalue equation takes the form

A(1)|uin(1)〉 = an|uin(1)〉, i = 1, 2, · · · , gn, (32.19)

where |uin(1)〉 is the eigenvector. The set of eigenvectors, |uin(1)〉, forms a basis in H1,and with no loss of generality, we assume that the set is orthonormalized, i.e.,

〈uin(1)|ujm(1)〉 = δijδnm. (32.20)

The extension of A(1) into H = H1 ⊗H2 is given by

A = A(1)⊗ 1(2), (32.21)


and from this it is obvious that the spectrum (set of eigenvalues) of A in H1 ⊗ H2 is thesame as that of A(1) in H1. However, the degree of degeneracy of a given eigenvalue, sayan, will be N2gn, where N2 is the dimension of H2. For N2 > 1, the eigenvalues in H areall degenerate even if A(1) has a nondegenerate spectrum in H1.

We now assume that a measurement of the observable A(1) was carried out, and hasgiven the value an. If the global (H) state of the system just before the measurement is |ψ〉[assumed to be normalized, 〈ψ|ψ〉 = 1], we seek to determine the (normalized) state |ψ′〉 ofthe global system immediately after the measurement. To calculate |ψ′〉 it is useful to makeuse of the projection operator concept. In H1 the projector onto the subspace formed bythe gn eigenvectors related to an is

Pn(1) =

gn∑

i=1

|uin(1)〉〈uin(1)|, (32.22)

cf. Eq. (32.14). To make predictions for the global system one extends Pn(1) into H1 ⊗H2.Thus, with

Pn = Pn(1)⊗ 1(2), (32.23)

it appears from the modification in Eq. (32.18) that the state of the global system just afterthe measurement is given by

|ψ′〉 = Pn|ψ〉[

〈ψ|Pn|ψ〉]

1

2

. (32.24)

In general, the state vector |ψ′〉 is a complicated object, and its physical meaning maybe difficult to unravel. Therefore, it is useful to see what |ψ′〉 represents physically in caseswhere the state before the measurement is particularly simple.

Let us assume that the initial state is a tensor product:

|ψ〉 = |φ(1)〉 ⊗ |χ(2)〉. (32.25)

To realize such a state one can imagine two separate systems, (1) and (2), prepared in thenormalized states |φ(1)〉 and |χ(2)〉, and then united without interacting. For the state inEq. (32.25), one obtains

Pn|ψ〉 =[

Pn(1)|φ(1)〉]

⊗[

1(2)|χ(2)〉]

=[

Pn(1)|φ(1)〉]

⊗ |χ(2)〉, (32.26)

and then, since P 2n(1) = Pn,

〈ψ|Pn|ψ〉 = 〈φ(1)|Pn(1)|φ(1)〉〈χ(2)|χ(2)〉

=∥

∥

∥Pn(1)|φ(1)〉

∥

∥

∥

2

, (32.27)

where ‖| · · · 〉‖ denotes the norm of | · · · 〉. Combining Eqs. (32.24), (32.26), and (32.27), itappears that the state after the measurement still is tensor product state, viz.,

|ψ′〉 = Pn(1)|φ(1)〉∥

∥

∥Pn(1)|φ(1)〉∥

∥

∥

⊗ |χ(2)〉. (32.28)


The state of system (2) is the same before and after the measurement, and the state ofsystem (1) has changed in a manner analogous to the situation discussed in Sec. 32.4, asone might have anticipated. When the global state has a tensor product form [Eq. (32.25)]all physical observations relating to only one part of the system [(1) and (2)] do not dependon the state of the other part. In a product state [|φ(1)〉 ⊗ |χ(2)〉] the two parts, (1) and(2), are said to be uncorrelated.

If the measured eigenvalue an is nondegenerate in H1, the extended projector is givenby

Pn = [|un(1)〉〈un(1)|]⊗ 1(2). (32.29)

The closure theorem for an arbitrary choice of an orthonormal basis, |uk(2)〉, in H2, viz.,∑

k

|vk(2)〉〈vk(2)| = 1(2), (32.30)

(including the summation over states related to possible degenerate eigenvalues in k) allowsus to write the extended projector in the form

Pn =∑

k

|un(1)vk(2)〉〈un(1)vk(2)|

= |un(1)〉 ⊗∑

k

|vk(2)〉〈un(1)vk(2)|, (32.31)

cf. the notation used in Eq. (32.2). Using this form of the projector in Eq. (32.24), thereader may verify that the state after the measurement is the tensor product state

|ψ′〉 = |un(1)〉 ⊗ |χ′(2)〉, (32.32)

where

|χ′(2)〉 =∑

k |vk(2)〉〈un(1)vk(2)|ψ〉[∑

k |〈un(1)vk(2)|ψ〉|2]1

2

. (32.33)

It appears from the expression in Eq. (32.33) that the state of system (2), |χ′(2)〉, dependson the result of the measurement carried out on system (1) [and given the nondegenerateeigenvalue an], even if the measurement is carried out when the two parts of the systemare very far away from each other. This “paradox” is the reason why Einstein, Podolsky,and Rosen (EPR) in an article entitled “Can quantum mechanical description of physicalreality be considered complete” [70] reasoned that quantum mechanics is incompatible withany reasonable idea of reality. In an article with the same title, Bohr replied that the EPRconcept of reality was too limited [28], and that the quantum mechanical formalism alwaysrefers to closed phenomena and must be considered as a rational generalization of classicalphysics. Bohr’s point of view agrees with the conclusion reached above.

In the general case, where the measured eigenvalue an of A(1) in H1 is degenerate, theextended projector takes the form

Pn =

gn∑

i=1

∑

k

|uin(1)vk(2)〉〈uin(1)vk(2)|, (32.34)

and from Eq. (32.24) it follows that the state of the global system just after the measurementis given by

|ψ′〉 =∑gn

i=1

∑

k |uin(1)vk(2)〉〈uin(1)vk(2)|ψ〉[∑gn

i=1

∑

k |〈uin(1)vk(2)|ψ〉|2]1

2

. (32.35)


It is clear from inspection of Eqs. (32.22), (32.24), and (32.30) that the state |ψ′〉 cannotdepend on the choice of basis in H2. When the initial state |ψ〉 is not a tensor productstate, the final state |ψ′〉 exhibits correlations between the systems (1) and (2), and |ψ′〉hence cannot be written as a tensor product of states in (1) and (2). A comparison of Eqs.(32.6) and (32.34) (with the normalization factor written as [〈ψ|Pn|ψ〉]1/2) shows that theexpansion coefficient (here cik) is given by

cik =〈uin(1)vk(2)|ψ〉[

〈ψ|Pn|ψ〉]

1

2

, (32.36)

remembering that the value of n is fixed by the measured eigenvalue an.A quantum state in H = H1⊗H2, which cannot be written as a single tensor product of

states in H1 and H2 is said to be an entangled quantum state, or a non-factorizable quantumstate.

32.5 Entangled photon polarization states

To illustrate the principle of entanglement and a quantum measurement process relating toonly one part of a physical system, it is sufficient to consider a system consisting of justtwo photons. Let the Hilbert spaces H1 and H2 refer to photons (1) and (2), respectively.The Hilbert space for a single photon [(1) or (2)] is the tensor product of the orbital andspin state spaces for the particle. If we assume that the orbital and spin dynamics (in a firstapproximation) are uncorrelated, the state vector of the photon is a simple tensor productof some orbital state vector and a spin state vector. According to the general analysis in Sec.32.4, a measurement of the polarization of this photon cannot change the (initial) spatialpart of the photon state vector. In relation to photon polarization measurements it is thussufficient to identify H1 and H2 with the spin state spaces of the two particles.

There are several experimental schemes allowing studies of spin correlations of photons.In spontaneous two-photon cascade emission a three-level atom proceeds, for example, froman upper level with total spin J = 0 to an intermediate J = 1 level, and terminates in a J = 0level, the ground state [155, 214]. Often a certain three-level scheme in Ca40 is employed.The two emitted photons are distinguishable because they have different frequencies (ν1, ν2).Due to angular momentum conservation, there is a strong correlation in the polarizationof the two photons. In the transition J = 0 → J = 1 → J = 0, the initial and finalstates of the atom have zero angular momentum, and therefore the two-photon state mustalso have zero angular momentum. Let us suppose that we consider only pairs of plane-wave photons that leave the atom in opposite directions, along the z-axis of a Cartesiancoordinate system. By optical frequency filters we can arrange it so that photons of frequencyν1 are transmitted in the +z-direction, and photons of frequency ν2 in the −z-direction.The entangled polarization state of the two-photon system thus has the form

|ψ〉 = 1√2(|R(1)〉 ⊗ |R(2)〉+ |L(1)〉 ⊗ |L(2)〉) , (32.37)

where, with i = 1, 2,

|R(i)〉 = a†+(i)|0〉, (32.38)


and

|L(i)〉 = a†−(i)|0〉, (32.39)

refer to single-photon states of positive (R) and negative (L) helicity, respectively. Thecreation operators creating these right(R) and left(L)-hand circular polarized states outof the global vacuum bear subscripts + and −. The two-photon state in Eq. (32.37) isnormalized, 〈ψ|ψ〉 = 1. A change of basis to linear polarization states [Eqs. (13.39), (13.40)and (13.47)] transforms Eq. (32.37) into

|ψ〉 = 1

2√2[(|x(1)〉 + i|y(1)〉)⊗ (|x(2)〉 − i|y(2)〉)

+ (|x(1)〉 − i|y(1)〉)⊗ (|x(2)〉+ i|y(2)〉)]

=1√2(|x(1)〉 ⊗ |x(2)〉+ |y(1)〉 ⊗ |y(2)〉) , (32.40)

remembering that photon (2) propagates in the −z-direction.Let us assume that we by means of a polarizer have carried out a polarization measure-

ment on photon (1) and found that this photon was polarized along the x-axis. After themeasurement photon (1) is in the state |x(1)〉. Since

〈x(1)x(2)|ψ〉 = 1√2, (32.41)

〈x(1)y(2)|ψ〉 = 0, (32.42)

it appears from Eq. (32.33) that the state of photon (2) is

|χ′(2)〉 = |x(2)〉. (32.43)

Without observing photon (2) we thus know that it is polarized along the x-axis after themeasurement on photon (1). The two-photon state |ψ〉 hence is reduced to

|ψ′〉 = |x(1)〉 ⊗ |x(2)〉. (32.44)

If a measurement on photon (1) shows that this photon is polarized along the y-axis, thepolarization state of the two-photon system would be

|ψ′〉 = |y(1)〉 ⊗ |y(2)〉 (32.45)

upon the measurement.It is good to show by an explicit calculation that the state in Eq. (32.37) is non-

factorizable. The general states for photon (1) and (2) are given by

|φ(1)〉 = a1|R(1)〉+ a2|L(1)〉, (32.46)

|χ(2)〉 = b1|R(2)〉+ b2|L(2)〉, (32.47)

where the conditions |a1|2+|a2|2 = |b1|2+|b2|2 = 1 ensure that the two states are normalized,〈φ(1)|φ(1)〉 = 〈χ(2)|χ(2)〉 = 1. Expanded in the helicity bases the most general tensorproduct state therefore has the form

|φ(1)〉 ⊗ |χ(2)〉 =a1b1|R(1)〉 ⊗ |R(2)〉+ a1b2|R(1)〉 ⊗ |L(2)〉+ a2b1|L(1)〉 ⊗ |R(2)〉+ a2b2|L(1)〉 ⊗ |L(2)〉. (32.48)


This uncorrelated state we now compare with the most general two-photon state, viz.,

|ψ〉 =c11|R(1)〉 ⊗ |R(2)〉+ c12|R(1)〉 ⊗ |L(2)〉+ c21|L(1)〉 ⊗ |R(2)〉+ c22|L(1)〉 ⊗ |L(2)〉, (32.49)

where the constraint |c11|2 + |c12|2 + |c21|2 + |c22|2 = 1 normalizes the state, 〈ψ|ψ〉 = 1. Inpassing, the reader might notice that Eqs. (32.48) and (32.49) are special cases of the statesgiven in Eqs. (32.5) and (32.6). The Hilbert spaces H1, H2 and H = H1 ⊗ H2 relating tothe photon polarizations have the dimensions N1 = N2 = 2, and N = N1N2 = 4. The statein Eq. (32.49) is non-factorizable unless cij = aibj for all i = 1, 2, j = 1, 2 pairs. Hence, thetwo-photon state factorizes only if

c11c22 = c12c21. (32.50)

For the state given in Eq. (32.37) one has c11 = c22 = 1/√2, and c12 = c21 = 0, and this

state therefore certainly is an entangled two-photon polarization state.

33

Two-photon wave mechanics and correlation matrices

33.1 Two-photon two times wave function

It appears from the description in Sec. 15.3 that a global state containing a single photonin each of the modes i = (q, s) and j = (q′, s′) has the form

|ψ〉 = |ψij〉 ⊗∏

k 6=i,j

|0k〉, (33.1)

where

|ψij〉 = |1i〉 ⊗ |1j〉 = a†i (0)a†j(0)|0i〉 ⊗ |0j〉. (33.2)

A Fock state containing precisely two photons also can be constructed by putting the twoquanta into the same mode (say i), that is

|ψij〉 = |2i〉 ⊗ |0j〉 =1√2

[

a†i (0)]2

|0i〉 ⊗ |0j〉. (33.3)

By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatictwo-photon state, |Φ2〉, can be formed:

|Φ2〉 =1

2L−3

∑

i,j

φij a†i (0)a

†j(0)|0〉. (33.4)

The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relationbetween the two-photon wave functions in direct and reciprocal space having the samestructural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitudeweight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq.(33.4) shows that only the sum φij + φji can play a physical role, and without loss ofgenerality one therefore may set

φji = φij . (33.5)

The inner product of |Φ2〉 with itself is readily obtained utilizing that

〈0|akala†i a†j|0〉 = δkiδlj + δkjδli. (33.6)

393


Thus,

〈Φ2|Φ2〉 =L−6

4

∑

i,j,k,l

φijφ∗kl〈0|akala†i a†j |0〉

=L−6

4

∑

i,j,k,l

φijφ∗kl (δkiδlj + δkjδli)

=L−6

4

∑

i,j

φij(

φ∗ij + φ∗ji)

=L−6

2

∑

i,j

|φ∗ij |2. (33.7)

The two-photon state is normalized, i.e.,

〈Φ2|Φ2〉 = 1, (33.8)

provided the weight factors satisfy the condition

L−6

2

∑

i,j

|φij |2

→ 1

2

∑

s,s′

∫ ∞

−∞|φs,s′(q,q′)|2 d3q

(2π)3d3q′

(2π)3

= 1. (33.9)

By acting on |Φ2〉 with the global number operator one obtains

N |Φ2〉 =1

2L−3

∑

i,k,l

φkla†i aia

†ka

†l |0〉

=1

2L−3

∑

i,k,l

φkla†i

(

a†kai + δik

)

a†l |0〉

=1

2L−3

∑

i,k,l

φkl

[

a†i a†l δik + a†i a

†k

(

a†l ai + δil

)]

|0〉

=1

2L−3

∑

i,l

φila†i a

†l |0〉+

∑

i,k

φkia†i a

†k|0〉

=1

2L−3

∑

i,j

(φij + φji) a†i a

†j |0〉. (33.10)

In view of Eqs. (33.4) and (33.5), one finally has

N |Φ2〉 = 2|Φ2〉. (33.11)

The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator witheigenvalue 2, a result which of course follows from the construction in Eq. (33.4).

In Sec. 15.5 a T-photon “mean” position state in Hilbert space was introduced letting

the negative-frequency part of the transverse vector potential operator, A(−)T (r, t), act on

the global vacuum state. By a natural extension of the definition given in Eq. (15.121) forthe single-photon case, we define the transverse two-photon “mean” position state as

|R2〉(r, r′, t, t′) ≡2ǫ0c

~A

(−)T (r, t)A

(−)T (r′, t′)|0〉. (33.12)

Two-photon wave mechanics and correlation matrices 395

As indicated by the notation |R2〉 is a dyadic quantity. Resolved in the helicity basis for

the polarization, where A(−)T is given by Eq. (15.120), |R2〉 takes the form

|R2〉(r, r′, t, t′) = L−3∑

i,j

(qiqj)− 1

2 e∗i e∗je

−i(qi·r−ωit)e−i(qj ·r′−ωjt′)a†i (0)a

†j(0)|0〉. (33.13)

A tensorial (dyadic) transverse two-photon two-times wave function in configurationspace now is defined as follows:

Φ2(r, r′, t, t′) ≡ 〈R2|Φ2〉(r, r′, t, t′). (33.14)

It is instructive to give an explicit expression for Φ2(r, r′, t, t′). Using the helicity basis in

the mode expansion of 〈R2| and |Φ2〉 one has

Φ2(r, r′, t, t′) =

1

2L−6

∑

i,j,k,l

φij (qkql)− 1

2 ekel

× exp [i (qk · r+ ql · r′ − ωkt− ωlt′)] 〈0|akala†i a†j |0〉. (33.15)

By means of Eq. (33.6) the expression above reduces to

Φ2(r, r′, t, t′) =

1

2L−6

∑

i,j

φij (qiqj)− 1

2

×[

eiejei(qi·r+qj ·r′−ωit−ωjt

′) + ejeiei(qj·r+qi·r′−ωjt−ωit

′)]

. (33.16)

If one interchanges i and j in the part of Eq. (33.16) which contains the dyadic productejei and makes use of Eq. (33.5), one obtains

Φ2(r, r′, t, t′) = L−6

∑

i,j

(qiqj)− 1

2 φijeiejei(qi·r+qj·r′−ωit−ωjt

′). (33.17)

Let us now introduce the quantity

Φij ≡ (qiqj)1

2 φij . (33.18)

In the continuum limit Eq. (33.18) takes the form

Φss′ (q,q′) = (qq′)

1

2 φss′ (q,q′), (33.19)

and Φss′ (q,q′) is the two-photon scalar wave function in the wave-vector representation.

This is so because the dyadic two-photon two-times wave function [Eq. (33.17)] in thecontinuum limit has the following plane-wave expansion (here given in the helicity basis forthe polarization):

Φ2(r, r′, t, t′) =

∑

s,s′

∫ ∞

−∞Φss′(q,q

′)es(κ)es′(κ′)

× ei(q·r−ωqt)ei(q′·r′−ωq′ t

′) d3q

(2π)3q

d3q′

(2π)3q′. (33.20)

The form in Eq. (33.20) is in full agreement with the principle of Lorentz-invariant inte-gration on the light cone. At this point, the reader might note the structural similaritybetween the expansions of the one-photon [Eq. (15.125)] and two-photon [Eq. (33.20)] wavefunctions.


33.2 Two-photon Schrodinger equation in direct space

In the previous section we introduced a two-photon two-times wave function concept inspace-time, viz., Φ2(r, r

′, t, t′). If t and t′ are identical the dyadic quantity

Φ2(r, r′, t, t) ≡ Φ(r, r′, t) (33.21)

is what one in usual terminology would call a two-particle wave function. Below, we shallrealize that the two-photon wave function Φ(r, r′, t) satisfies a Schrodinger-like quantummechanical wave equation. We start with some considerations which refer to the two-timeswave function, however.

Letting the transverse global photon Hamilton operator,

Hph =∑

m

~ωma†mam, (33.22)

act on the general two-photon state [Eq. (33.4)] one has

Hph|Φ2〉 =1

2L−3

∑

m,i,j

~ωmφij a†mama

†i aj |0〉. (33.23)

Since

a†mama†i a

†j |0〉 =

(

δmj a†ma

†i + δima

†ma

†j

)

|0〉, (33.24)

as the reader may verify moving the annihilation operator am to the right by means of thecommutator relation in Eq. (15.28), and using the ground state condition in Eq. (15.64),Eq. (33.23) can be reduced to

Hph|Φ2〉 =1

2L−3

∑

i,j

~φij

(

ωj a†j a

†i + ωia

†i a

†j

)

|0〉, (33.25)

an expression which is identical to

Hph|Φ2〉 = L−3∑

i,j

~ωiφij a†i aj |0〉, (33.26)

because φij = φji. The representation of the state space quantity Hph|Φ2〉 in configuration

space is given by the scalar product of Hph|Φ2〉 with the two-photon mean position state|R2〉(r, r′, t, t′). By combining Eqs. (33.12) [in bra form] and (33.26), and utilizing Eq. (33.6)it is easy to show that

〈R2|Hph|Φ2〉(r, r′, t, t′) =L−6∑

i,j

(qiqj)− 1

2 eiej exp [i (qi · r− ωit+ qj · r′ − ωjt′)]

+ ejei exp [i (qj · r− ωjt+ qi · r′ − ωit′)]~ωiφij . (33.27)

By an interchange of summation indices (i↔ j) in the part containing the dyadic productejei, and elimination of φij in favor of Φij [Eq. (33.18)], one finally obtains

〈R2|Hph|Φ2〉(r, r′, t, t′) = L−6∑

i,j

~ (ωi + ωj)Φij

qiqjeieje

i(qi·r+qj·r′−ωit−ωjt′). (33.28)


For t′ = t, one has the expansion

〈R2|Hph|Φ2〉(r, r′, t) = L−6∑

i,j

~cqi + qjqiqj

Φijeiejei(qi·r+qj ·r′)e−ic(qi+qj)t. (33.29)

The two-photon wave function in configuration space, 〈R2|Φ2〉(r, r′, t), is obtained from Eq.(33.17) setting t′ = t. Hence,

〈R2|Φ2〉(r, r′, t) = L−6∑

i,j

(qiqj)−1 Φijeieje

i(qi·r+qj·r′)e−ic(qi+qj)t (33.30)

in view of Eq. (33.18). A comparison of Eqs. (33.29) and (33.30) leads to the conclusionthat the two-photon wave function in configuration space satisfies the dynamical equation

i~∂

∂t〈R2|Φ2〉(r, r′, t) = 〈R2|Hph|Φ2〉(r, r′, t). (33.31)

This equation is the quantum mechanical Schrodinger-like wave equation for the two-photonwave function. The corresponding Schrodinger equation for the one-photon wave functionwas given in Eq. (15.133).

33.3 Two-photon wave packet correlations

33.3.1 First-order correlation matrix

It appears from the analysis in Sec. 16.5 that the first-order two-photon correlation matrix,

G(1)2 (x1, x2) =

2ǫ0c

~〈Φ2|AT (x1)AT (x2)|Φ2〉, (33.32)

under the rotating-wave approximation, in which energy nonconserving terms in the atom-field interaction are neglected, is given by

G(1)2 (x1, x2) =

2ǫ0c

~〈Φ2|A(−)

T (x1)A(+)T (x2)|Φ2〉. (33.33)

The subscript “2” on the correlation matrix is meant to indicate that we are dealing with atwo-photon state, and the superscript “(1)” refers as before to first-order correlation. Space-time coordinates are abbreviated in the usual manner, i.e., x1 = (ct1, r1) and x2 = (ct2, r2).To investigate the first-order correlation matrix for a general two-photon state

|Φ2〉 =1

2L−3

∑

j,k

φjk a†j(0)a

†k(0)|0〉, (33.34)

we expand the positive and negative-frequency parts of the transverse vector potentialoperator in the helicity basis for the polarization (other choices of basis can of course bemade). Thus,

A(+)T (x) =

[

A(−)T (x)

]†= L− 3

2

∑

i

(

~

2ǫ0ωi

)1

2

ai(0)eieiqix, (33.35)


with the relativistic four-component notation

qix = qi · r− ωit. (33.36)

Since

aia†j a

†k|0〉 =

(

δkia†j + δij a

†k

)

|0〉, (33.37)

one obtains

A(+)T (x2)|Φ2〉 =

1

2L− 9

2

∑

i,j,k

(

~

2ǫ0ωi

)1

2

φjkeieiqix2

(

δkia†j + δij a

†k

)

|0〉

=1

2L− 9

2

∑

i

(

~

2ǫ0ωi

)1

2

eieiqix2

∑

j

φjia†j |0〉+

∑

k

φik a†k|0〉

. (33.38)

If the dummy summation index k is renamed j, Eq. (33.38) may be written as

A(+)T (x2)|Φ2〉 =

1

2L− 9

2

∑

i,j

(

~

2ǫ0ωi

)1

2

eieiqix2 (φji + φij) a

†j |0〉

= L− 3

2

(

~

2ǫ0c

)1

2 ∑

j

[

L−3∑

i

φij

q1

2

i

eiei(qi·r2−ωit2)a†j

]

|0〉, (33.39)

remembering now that qi = ωi/c, and φji = φij . By introducing the definitions

Φ(j)i ≡ Φij ≡ q

1

2

i φij ≡ q1

2

i φ(j)i , (33.40)

we have

A(+)T (x2)|Φ2〉 = L− 3

2

(

~

2ǫ0c

)1

2 ∑

j

Φ(j)1 (x2)a

†j(0)|0〉, (33.41)

where

Φ(j)1 (x2) = L−3

∑

i

Φ(j)i

qieie

i(qi·r2−ωit2). (33.42)

In the continuum limit the expression in Eq. (33.42) has the form given in Eq. (15.125).

Therefore, we can conclude that Φ(j)1 (x2) is a one-photon wave function (subscript 1). To

each j in φij = q−1/2i Φ

(j)i , one has a photon wave function Φ

(j)1 in q-space for the helicity

species s, remembering i ≡ (q, s).Since the adjoint of Eq. (33.41), yet with x2 replaced by x1 is given by

〈Φ2|A(−)T (x1) = L− 3

2

(

~

2ǫ0c

)1

2 ∑

i

[

Φ(i)1 (x1)

]∗〈0|ai(0), (33.43)

the first-order correlation matrix in Eq. (33.33) becomes

G(1)2 (x1, x2) = L−3

∑

i,j

[

Φ(i)1 (x1)

]∗Φ

(j)1 (x2)〈0|aia†j |0〉. (33.44)


Since 〈0|aia†j |0〉 = δij , one finally has

G(1)2 (x1, x2) = L−3

∑

j

[

Φ(j)1 (x1)

]∗Φ

(j)1 (x2). (33.45)

The first-order correlation matrix for a polychromatic two-photon state hence can be ex-

pressed in terms of one-photon wave functions constructed on the basis of the φij ≡ φ(j)i

coefficients. The correlation matrix (multiplied by L3) consists of a sum over j of tensor

products. The jth term is an outer product of the vectorial one-photon wave function Φ(j)1

and its complex conjugate (Φ(j)1 )∗ taken at the space-time points (ct2, r2) and (ct1, r1), re-

spectively. It is perhaps not surprising that a first-order correlation matrix for a two-photonstate can be expressed in terms of effective one-photon wave functions constructed from aset of two-photon coefficients. If the space-time points coincide, x1 = x2 = x, the trace ofthe correlation matrix becomes

Tr

G(1)2 (x, x)

= L−3∑

j

[

Φ(j)1 (x)

]∗·Φ(j)

1 (x). (33.46)

The probability density for observing our polychromatic two-photon object in the space-point r at time t hence is proportional to a sum of effective single-photon probabilitydensities.

In the continuum limit, where

Φ(i)1 (x) ⇒ Φ

(s)1 (q;x), (33.47)

the first-order two-photon correlation function is given by the integral expression

G(1)2 (x1, x2) =

∑

s

∫ ∞

−∞

[

Φ(s)1 (q;x1)

]∗Φ

(s)1 (q;x2)

d3q

(2π)3. (33.48)

33.3.2 Second-order correlation matrix

In analogy with the form given in Eq. (33.33) for the first-order correlation matrix of atransverse two-photon state, a second-order [superscript (2)] correlation matrix related to|Φ2〉 is defined by

G(2)2 (x1, x2, x3, x4) =

(

2ǫ0c

~

)2

〈Φ2|A(−)T (x1)A

(−)T (x2)A

(+)T (x3)A

(+)T (x4)|Φ2〉. (33.49)

As we shall realize below, it is possible to express G(2)2 (x1, x2, x3, x4) as a single tensor

product of two two-photon two-times wave functions (one complex conjugate, yet).

By repeated use of the commutator relation [aα(0), a†β(0)] = δαβ [Eq. (15.28) for t = 0],

and the ground state condition aγ(0)|0〉 = 0, the reader may verify that

ai(0)aj(0)a†k(0)a

†l (0)|0〉 = (δikδjl + δilδjk) |0〉. (33.50)

This implies that A(+)T (x3)A

(+)T (x4)|0〉 is a Hilbert vector in the same ray as the ground

state, |0〉. In turn, this implies that if one inserts a complete set of transverse photon states

between A(−)T (x2) and A

(+)T (x3) in Eq. (33.49), only a single operator, |0〉〈0|, contributes a

nonvanishing result. Hence, one may write the second-order correlation matrix as follows:

G(2)2 (x1, x2, x3, x4) =

(

2ǫ0c

~

)2

〈Φ2|A(−)T (x1)A

(−)T (x2)|0〉〈0|A(+)

T (x3)A(+)T (x4)|Φ2〉,

(33.51)


or equivalently in terms of the two-photon mean position state [Eq. (33.12)]

G(2)2 (x1, x2, x3, x4) = 〈Φ2|R2(x1, x2)〉〈R2(x3, x4)|Φ2〉. (33.52)

The definition in Eq. (33.14) finally gives one the sought for relation between the second-order correlation matrix and the transverse two-photon two-times wave function, viz.,

G(2)2 (x1, x2, x3, x4) = Φ∗

2(x1, x2)Φ2(x3, x4). (33.53)

In studies of the joint probability density that photodetection is registered at the space-timepoints x1 and x2 the second-order correlation matrix

G(2)2 (x1, x2, x2, x1) = Φ∗

2(x1, x2)Φ2(x2, x1) (33.54)

is of central importance [155, 214].The second-order correlation matrix for a single-photon state, |Φ1〉[≡ |Φ〉, Eq. (15.80)],

i.e.,

G(2)1 (x1, x2, x3, x4) =

(

2ǫ0c

~

)2

〈Φ1|A(−)T (x1)A

(−)T (x2)A

(+)T (x3)A

(+)T (x4)|Φ1〉 (33.55)

always vanishes because aiaj a†k|0〉 = 0 for all i, j and k. Physically, the reason that

G(2)1 (x1, x2, x3, x4) = 0 (33.56)

relates to the fact that a single photon cannot be detected (destroyed) at two differentspace-time points.

34

Spontaneous one- and two-photon emissions

This chapter culminates with a discussion of the photon wave mechanical picture of thecorrelated spontaneous emissions from a three-level atom. The photon spin entanglementof the two-photon cascade J = 0 → J = 1 → J = 0 is briefly analyzed in Sec. 32.5, payingattention to the correlations in the spin state space. Measurements are always carried out indirect space, and for this reason alone it is of interest to study the space-time correlations ofthe cascade process. The existing theories focus attention on the far-field correlations [214].In the far field the spatial extension of the rim zone of the atom is unimportant. However,if one imagines that the two detectors needed to measure the second-order correlationmatrix at two different space-time points are moved into the rim zone of the atom, thelack of spatial localization of the transverse photons mixes with the quantum mechanicalentanglement phenomenon. The standard theory of two-photon cascade entanglement isextended to the domain of near-field electrodynamics by relating the final two-photon statein Hilbert space to photon wave mechanics in direct space.

34.1 Two-level atom: Weisskopf–Wigner theory of spontaneousemission

As preparation for a study of the two-photon cascade process, it is useful to begin withan investigation of the spontaneous single-photon emission between two isolated atomiclevels. The description, resulting in an approximate expression for the wave function of thetransverse photon (embryo), and the related first-order correlation matrix, will help us tounderstand fundamental aspects of the role of the rim zone in spontaneous emission.

34.1.1 Atom-field Hamiltonian in the electric-dipole approximation.RWA-model

In order to describe the spontaneous emission of a T-photon from a two-level atom, westart by setting up an expression for the relevant nonrelativistic one-electron atom-fieldHamiltonian, H . As usual, we divide the Hamiltonian operator into three pieces, i.e.,

H = HA + HF + HI . (34.1)

The part HF is associated to the free transverse radiation field (and denoted by HT in Sec.15.2). In terms of the mode (q, s) creation and destruction operators one has

HF =∑

q,s

~ωq

(

a†qsaqs +1

2

)

. (34.2)

401


If we denote the normalized upper and lower energy eigenstates of the atom by |a〉 and |b〉,respectively, the associated energies by Ea and Eb, the transition frequency by

Ω =Ea − Eb

~, (34.3)

and choose the ground-state energy to be zero (Eb = 0), the atomic Hamilton operator isgiven by

HA = ~Ω|a〉〈a|. (34.4)

We can also express HA in terms of the flip (lowering) operator

b ≡ |b〉〈a|, (34.5)

and its adjoint (raising) operator

b† = |a〉〈b|. (34.6)

It follows from the orthonormalization conditions

〈a|a〉 = 〈b|b〉 = 1, (34.7)

〈a|b〉 = 0, (34.8)

that

b|a〉 = |b〉, b|b〉 = 0, (34.9)

b†|b〉 = |a〉, b†|a〉 = 0, (34.10)

and

HA = ~Ωb†b. (34.11)

In order to obtain an expression for the interaction Hamiltonian in the electric-dipoleapproximation we start from the classical Hamiltonian

H =1

2m

(

p+ eAextT (r, t)

)2 − eφ, (34.12)

describing (approximately) the interaction between an electron of mass m and charge −e(e > 0) and an external (ext) field given by the transverse vector potential Aext

T (r, t).With no loss of generality it may be assumed that the external scalar potential is zero,φext(r, t) = 0, [127]. In Eq. (34.12), φ hence is the Coulomb potential of the electron inour two-level atom. In writing down the expression for H the retarded part of the nucleus-electron coupling is also neglected (a good approximation, here). It is assumed next thatthe spatial variation of Aext

T (r, t) across the atom is negligible. With the atom placed atr = r0 we thus put

AextT (r, t) ≃ Aext

T (r0, t) (34.13)

in Eq. (34.12). This gives us the Hamiltonian in the so-called electric-dipole (ED) approxi-mation, viz.,

HED =1

2m

(

p+ eAextT (r0, t)

)2 − eφ. (34.14)

Spontaneous one- and two-photon emissions 403

A gauge transformation [Eqs. (10.5) and (10.6)] referring to the external potential turnsHED into

(HED)′ =1

2m

[

p+ e(

AextT (r0, t) +∇χ(r, t)

)]2 − eφ+ e∂

∂tχ(r, t). (34.15)

With the choice

χ(r, t) = − (r− r0) ·AextT (r0, t) (34.16)

for the gauge function, one obtains

∇χ(r, t) = −AextT (r0, t), (34.17)

and

∂

∂tχ(r, t) = − (r− r0) ·

∂

∂tAext

T (r0, t)

= (r− r0) · EextT (r0, t). (34.18)

The gauge function in Eq. (34.16) hence transforms the ED Hamiltonian in Eq. (34.14) into

(HED)′ =p · p2m

− eφ+ e (r− r0) · EextT (r0, t). (34.19)

The part p ·p/(2m)− eφ is the atomic Hamiltonian in the new gauge, and when quantizedthis gives us in the framework of the two-level model the atomic Hamilton operator in Eq.(34.11).

The last part, e(r − r0) · EextT (r0, t), upon quantization, is the interaction Hamiltonian

in the electric-dipole approximation. Leaving out for notational simplicity the superscript“ext,” we have

HI = −d · ET (r0, t), (34.20)

where

d = −e (r− r0) ≡ −eR (34.21)

is the electric dipole operator in the r-representation. In the Interaction Picture the plane-wave expansion of the transverse electric field operator gives at the site of the atom

ET (r0, t) =∑

q,s

[

i

(

~ωq

2ǫ0L3

)1

2

aq,s(0)εqsei(q·r0−ωqt) + h.c.

]

, (34.22)

cf. the analysis in Sec. 16.4.For what follows it is useful to write d in so-called second-quantized form. Since the

identity operator (1) for our two-level atom is given by

1 = |a〉〈a|+ |b〉〈b|, (34.23)

one may write d = −eR in the form

d = 1d1 =∑

i,j=a,b

|i〉dij〈j|, (34.24)


where

dij = 〈i| − eR|j〉 (34.25)

is the matrix element of the electric dipole operator between the states i and j. Since thesingle-electron particle Hamiltonian is invariant under spatial inversion (r ⇒ −r), the parityoperator commutes with HA. The eigenstates of HA may therefore be assumed to have adefinite parity. In turn this implies that the diagonal elements of the electric dipole operatorvanish, i.e.,

〈i| − eR|i〉 = 0, i = a, b. (34.26)

In second-quantized form the ED-operator hence is given by

d =dab|a〉〈b|+ dba|b〉〈a|=dabb

† + dbab. (34.27)

By combining Eqs. (34.20), (34.22), and (34.27) [in the Schrodinger Picture] one finallyobtains the following expression for the interaction Hamiltonian in the Schrodinger Picture:

HI =∑

q,s

~(

gqsb†(0)− g∗qsb(0)

)

(

aqs(0)eiq·r0 − a†qs(0)e

−iq·r0) , (34.28)

where

gqs =1

i~

(

~ωq

2ǫ0L3

)1

2

dab · εqs (34.29)

is the so-called coupling constant. Later on, it will turn out to be fruitful to rewrite gqsin terms of the transition current density from the ground state to the excited state, Jab.From the general relation [127, 53]

Jij = −iωqdij , (34.30)

one hence has the connection

gqs =(

2ǫ0~ωqL3)− 1

2 Jab · εqs. (34.31)

The interaction Hamiltonian in Eq. (34.28) consists of four types of terms. The term con-

taining b†aqs describes a process where the atom is excited from the lower to the higher

state and a photon is annihilated in the mode (q, s). The term proportional to ba†qs de-scribes the opposite process, where the atom is deexcited under the emission of a photoninto the (qs)-mode. The energy is conserved in both these processes. The term containing

baqs describes a process where the atom is deexcited while annihilating a photon in mode

(qs). In such a process energy is lost. In the process described via b†a†qs, the atom is excitedsimultaneously with a creation of photon in (qs). In this process energy is gained. Closeto resonance the energy nonconserving processes are less important, and in the so-calledrotating-wave approximation (RWA) [155, 214], which we shall use in what follows, onlythe energy conserving terms are kept. In the RWA-model and Schrodinger Picture the totalparticle-field Hamiltonian thus is given by

H =~Ωb†(0)b(0) +∑

q,s

~ωqa†qs(0)aqs(0)

+∑

q,s

~(

gqsb†(0)aqs(0)e

iq·r0 + g∗qsb(0)a†qs(0)e

−iq·r0)

, (34.32)


dropping the zero-point energy, which in the present context plays no role. In the followingit is convenient to work in the Interaction Picture. In this picture, the Hamilton operatoris given by (Sec. 16.3)

ˆH = exp

(

i

~H0t

)

H exp

(

− i

~H0t

)

, (34.33)

where

H0 = HA + HF (34.34)

is the Hamiltonian describing the uncoupled field and particle dynamics. It is obvious thatˆH0 = H0, and to determine the interaction Hamiltonian in the Interaction Picture terms ofthe type

exp

(

i

~H0t

)

b†(0)aqs(0) exp

(

− i

~H0t

)

=exp(

iΩtb†(0)b(0))

b†(0) exp(

−iΩtb†(0)b(0))

× exp(

iωqta†qs(0)aqs(0)

)

aqs(0) exp(

−iωqta†qs(0)aqs(0)

)

, (34.35)

and their Hermitian conjugates, have to be calculated. Using the operator expansion theo-rem [155, 151]

exp(

αA)

B exp(

−αA)

= B + α[

A, B]

+α2

2!

[

A,[

A, B]]

+ · · · , (34.36)

which the reader may prove by expanding the exponentials exp(±αA) in Taylor series in

α around α = 0, the Interaction Picture operatorsˆb(t) and âqs(t), and their adjoints, can

readily be obtained. Thus, with the help of the commutator relation

[

b†b, b]

= −b (34.37)

one gets

ˆb(t) = eiΩtb†bbe−iΩtb† b

= b(0)

∞∑

n=0

(−iΩt)nn!

= b(0)e−iΩt, (34.38)

and then

ˆb†(t) = b†(0)eiΩt. (34.39)

For the electromagnetic field, the commutator relation

[

a†qsaqs, aqs]

= −aqs (34.40)

leads to

âqs(t) = aqs(0)e−iωqt, (34.41)

â†qs(t) = a†qs(0)eiωqt. (34.42)


Gathering the information above, it appears that the RWA interaction Hamiltonian in theInteraction Picture is given by

ˆHI(t) = ~∑

q,s

[

gqsb†(0)aqs(0)e

iq·r0ei∆qt + g∗qsb(0)a†qs(0)e

−iq·r0e−i∆qt]

, (34.43)

where

∆q = Ω− ωq (34.44)

is the frequency detuning of the q-mode frequency from the atomic transition frequency(Bohr frequency). In the following we shall drop the tilde (∼) indicating that given operatorsand state vectors are represented in the Interaction Picture.

34.1.2 Weisskopf–Wigner state vector

It appears from the analysis in the previous subsection that the state vector of the coupledatom-field system in the rotating-wave approximation will have the form

|ψ(t)〉 = ca(t)|a〉 ⊗ |0〉+∑

q,s

cb,qs(t)|b〉 ⊗ |1qs〉. (34.45)

In the Interaction Picture the basis state vectors |a〉 ⊗ |0〉 and |b〉 ⊗ |1qs〉 are time in-dependent. Under the assumption that the atom at time t = 0 is in the excited state, thecoefficients satisfy

ca(0) = 1, cb,qs = 0 (∀(q, s)) . (34.46)

The dynamical development of the state vector |ψ(t)〉 is determined by the Schrodingerequation, which in the Interaction Picture reads

i~d

dt|ψ(t)〉 = HI(t)|ψ(t)〉, (34.47)

where HI(t) is given by Eq. (34.43). To determine the yet unknown coefficients, ca(t) andcb,qs(t) [∀(q, s)], one takes the inner product of Eq. (34.47) with the time independent basevectors |a〉 ⊗ |0〉 = |a, 0〉 and |b〉 ⊗ |1qs〉 = |b, 1qs〉. The inner product with |a, 0〉 gives forthe left- and right-hand sides of the Schrodinger equation

i~d

dt〈a, 0|ψ(t)〉 = i~ca(t), (34.48)

and

〈a, 0|HI(t)|ψ(t)〉 = ~∑

qs

gqseiq·r0ei∆qtcb,qs(t). (34.49)

The inner product with |b, 1qs〉 gives for the chosen (q, s) the following results:

i~d

dt〈b, 1qs|ψ(t)〉 = i~cb,qs(t), (34.50)

〈b, 1qs|HI(t)|ψ(t)〉 = ~g∗qse−iq·r0e−i∆qtca(t). (34.51)


Hence, from the Schrodinger equation in the Interaction Picture we have obtained thefollowing set of linear first-order differential equations among the probability amplitudes

ca(t) = −i∑

q,s

gqs(r0)ei∆qtcb,qs(t), (34.52)

cb,qs(t) = −ig∗qs(r0)e−i∆qtca(t), (34.53)

where

gqs(r0) = gqs(0)eiq·r0 (34.54)

is a coupling constant which includes the dependence on the atomic position, r0.By integration of Eq. (34.53) one obtains in view of the initial conditions in Eq. (34.46)

cb,qs(t) = −ig∗qs(r0)∫ t

0

e−i∆qt′

ca(t′)dt′, (34.55)

and upon substitution of this expression into Eq. (34.52), one gets the following linearintegro-differential equation for ca(t):

ca(t) = −∑

q,s

|gqs(r0)|2∫ t

0

ei∆q(t−t′)ca(t′)dt′. (34.56)

In the frameworks of the ED and RW approximations, this equation is still an exact equationderived from the Schrodinger equation in the Interaction Picture, Eq. (34.47). To solve Eq.(34.56), it is assumed that one can replace ca(t

′) by its value at t′ = t. This approximationis the basis for the Weisskopf–Wigner theory [214, 151, 245] With ca(t

′) ≃ ca(t), the modelbecomes memory-less, i.e., local in time (Markow approximation). The integral over timenow becomes

∫ t

0

ca(t′)ei∆q(t−t′)dt′ ≃ 2πca(t)

[

1

2π

∫ t

0

ei∆q(t−t′)dt′]

= ca(t)ei∆qt − 1

i∆q. (34.57)

It appears from Eq. (34.57) that the integral over time has an appreciable magnitude onlyin the vicinity of the resonance frequency, i.e., for ∆q ≈ 0. This fact may be underlinedstudying the limit t→ ∞. Since

1

2π

∫ ∞

0

ei∆qτdτ = δ(+)(∆q), (34.58)

where δ(+)(∆q) is the positive-frequency part of the Dirac delta function [155], one has inthe Markow approximation

limt→∞

∫ t

0

ca(t′)ei∆q(t−t′)dt′ ≃ 2πca(t)δ

(+)(∆q) = πca(t)

[

δ(∆q)−i

πP

1

∆q

]

, (34.59)

where P denotes the Cauchy principal value. The part related by the P (1/∆q) is needed foran understanding and a calculation of the Lamb shift [155, 242, 139, 14], but is unimportantfor our discussion of the spontaneous emission process. In the limit t → ∞, one yields thefollowing differential equation for ca(t):

ca(t) = −Γaca(t), (34.60)


with

Γa = π∑

q,s

|gqs(r0)|2δ(Ω− ωq). (34.61)

The solution to Eq. (34.60) with ca(0) = 1, viz.,

ca(t) = exp (−Γat) (34.62)

shows that the two-level atom initially in the excited state |a〉 in vacuum decays expo-nentially in time in the Weisskopf–Wigner theory. Since |ca(t)|2 = exp(−2Γat), the decayconstant for the probability density is 2Γa, corresponding to a life time τa = 1/(2Γa) in thespontaneous photon emission process.

Let us now derive an explicit expression for Γa carrying out the summation over thepolarization states, and replacing the summation over the q-modes by an integration overq-space. It appears from Eqs. (34.29) and (34.61) that the polarization enters the expressionfor Γa via the unit vectors εqs (s = 1, 2). Since

2∑

s=1

ε∗qsεqs = U− κκ, (34.63)

with κ = q/q, as usual, one obtains in the continuum limit

Γa =π

2ǫ0~d∗ab · I · dab, (34.64)

where

I =

∫ ∞

−∞ωqδ(Ω− ωq) (U− κκ) d3q

(2π)3. (34.65)

It is obvious that I must be just the unit tensor multiplied by a certain scalar. The integra-tion in Eq. (34.65) is carried out easily using spherical coordinates, and one gets

I =1

(2π)24

3

(

Ω

c

)3

U, (34.66)

as the reader may show. By combining Eqs. (34.64) and (34.66) one obtains

Γa =1

4πǫ0

2Ω3

3~c3|dab|2. (34.67)

The quantity 2Γa is identical to the famous Einstein A-coefficient [155, 150, 67]. Havingdetermined ca(t) [Eq. (34.62)], the probability amplitude cb,qs(t) is readily obtained. Thus,from Eq. (34.53) one has

cb,qs(t) = −ig∗qs(r0)∫ t

0

e−i∆qt′

e−Γat′

dt′, (34.68)

with the initial condition cb,qs(0) = 0 [Eq. (34.46)]. Upon integration we find

cb,qs(t) = g∗qs(r0)1− exp [i (ωq − Ω)− Γa] t

ωq − Ω+ iΓa. (34.69)

In the Weisskopf–Wigner theory the global state vector of the coupled atom-field systemtherefore is given by

|ψ(t)〉 = e−Γat|a〉 ⊗ |0〉+ |b〉 ⊗∑

q,s

g∗qs(0)e−iq·r0

[

1− ei(ωq−Ω)te−Γat

ωq − Ω + iΓa

]

|1qs〉, (34.70)

remembering that the atomic transition must be electric-dipole allowed.


34.2 Two-level atom: Wave function of spontaneouslyemitted photon

34.2.1 Photon wave function in q-space

In Sec. 15.4 the photon embryo concept was introduced, see Eq. (15.81). The embryo statefor a transverse photon relates to the situation where field-matter coupling is present. Forthe spontaneous emission process discussed in Sec. 34.1, it appears from Eq. (34.70) thatthe state vector of the photon embryo, |Ψ(t)〉, is given by

|Ψ(t)〉 =∑

q,s

g∗qs(0)e−iq·r0

[

1− ei(ωq−Ω)te−Γat

ωq − Ω+ iΓa

]

a†qs(0)|0〉. (34.71)

In spontaneous emission the photon is only released from the atom in the limit t→ ∞, i.e.,effectively after a time which is much larger than the spontaneous lifetime (t ≫ Γ−1

a ). Thestate vector of the photon emitted from our two-level atom placed at r = r0, |Φ〉, hence hasthe (qs)-mode expansion

|Φ〉 = limt→∞

|Ψ(t)〉 =∑

q,s

g∗qs(0)e−iq·r0

ωq − Ω+ iΓaa†qs(0)|0〉. (34.72)

If one inserts here the expression given in Eq. (34.29) for gqs(0), |Φ〉 takes the form

|Φ〉 = L− 3

2

∑

q,s

(2ǫ0~ωq)− 1

2 Jba · ε∗qse−iq·r0

ωq − Ω + iΓaa†qs(0)|0〉, (34.73)

where Jba = J∗ab is the transition current density from |a〉 to |b〉. From the analysis in Sec.

15.4, we know that when the single-photon state vector is written in the form given inEq. (15.80), the scalar photon wave function in wave-vector space must be identified withthe quantity Φqs = q1/2φqs, see Eq. (15.92). Thus, we can conclude that the scalar wavefunction of the spontaneously emitted photon with polarization index s is given by

Φqs = (2ǫ0~c)− 1

2 Jba · ε∗qse−iq·r0

ωq − Ω+ iΓa, (34.74)

when the two-level atom is located at the position r0. Resolved in the helicity basis, thevectorial photon wave function of the helicity species s = + or − has the following form inthe continuum limit

Φs(q) = (2ǫ0~c)− 1

2e−iq·r0

cq − Ω+ iΓaJba · e∗s(q)es(q), (34.75)

cf. Eq. (15.126).The squared norm of the photon wave function becomes

〈Φ|Φ〉 =∑

q,s

|gqs(0)|2[

(ωq − Ω)2+ Γ2

a

]−1

, (34.76)

as the reader readily may show starting from Eq. (34.72). It appears from the analysis inSec. 34.1 that the Weisskopf–Wigner theory relates to an approximation of the form

1

(ωq − Ω)2 + Γ2a

≃ π

Γaδ(ωq − Ω). (34.77)


[The integrals of the right- and left-hand sides of Eq. (34.77) over all frequencies (−∞ <ω < ∞) are identical]. If the approximation in Eq. (34.77) is inserted in Eq. (34.76) oneobtains

〈Φ|Φ〉 ≃ π

Γa

∑

q,s

|gqs|2δ(ωq − Ω) = 1, (34.78)

The last member of Eq. (34.78) follows from the expression given for the decay constant inEq. (34.61). As one might have anticipated from the initial conditions in Eq. (34.46), thespontaneously emitted single-photon state is normalized.

34.2.2 The general photon wave function in r-space

On the basis of the mean position state introduced in Eq. (15.121), the vectorial wavefunction of a photon in state |Φ〉 is given by

Φ(r, t) =

(

2ǫ0c

~

)1

2

〈0|A(+)T (r, t)|Φ〉 (34.79)

in space-time. By inserting the explicit expression for |Φ〉 [Eq. (34.72)] and the (q, s)-mode

expansion of A(+)T (r, t) [positive-frequency part of Eq. (15.30)] into Eq. (34.75) one has

Φ(r, t) =L− 3

2

∑

q′,s′

∑

q,s

(q′)− 1

2 εq′s′g∗qs(0)

× ei(q′·r−ωq′ t) e−iq·r0

ωq − Ω + iΓa〈0|aq′s′(0)a

†qs(0)|0〉. (34.80)

Since

〈0|aq′s′(0)a†qs(0)|0〉 = δqq′δss′ , (34.81)

Eq. (34.80) reduces to

Φ(R, t) =L− 3

2

∑

q,s

q−1

2 g∗qs(0)εqsei(q·R−ωqt)

ωq − Ω + iΓa, (34.82)

where R = r− r0 is the vectorial distance from the atom to the “point of observation.” Bymeans of the expression given for gqs(0) in Eq. (34.31), Eq. (34.82) goes into

Φ(R, t) =

[

L−3∑

q,s

(2ǫ0~c)− 1

2 q−1 ei(q·R−ωqt)

ωq − Ω + iΓaε∗qsεqs

]

· J∗ab. (34.83)

The summation over the polarization states gives U−κκ [Eq. (2.95)], and in the continuumlimit the general photon wave function hence is given by

Φ(R, t) =

[

(2ǫ0~c)− 1

2

∫ ∞

−∞(U− κκ) q

−1ei(q·R−ωqt)

ωq − Ω+ iΓa

d3q

(2π)3

]

· Jba. (34.84)

Physically, the plane-wave superposition in Eq. (34.84) consists of two parts, viz., a partΦIN (R, t) describing a photon field converging on the atom (incoming field), and a partΦOUT (R, t) referring to a field diverging from the atom (outgoing field). In our study ofthe spontaneous emission process we will neglect ΦIN (R, t). The division of Φ(R, t) into


its incoming and outgoing parts is achieved by referring the integration over q-space tospherical coordinates, (q, θ, φ). The polar axis is chosen along the R-direction, and thepolar angle is denoted by θ. Thus,

Φ(R, t) =

[

(2ǫ0~c)− 1

2

∫ ∞

0

I(R; q)qe−icqt

cq − Ω + iΓa

dq

(2π)2

]

· Jba, (34.85)

where

I(R; q) =1

2π

∫ π

0

∫ 2π

0

(U− κκ) eiqR cos θ sin θdφdθ (34.86)

is the relevant angular integral. The integrals in Eq. (34.86) easily can be done, rememberingthat κ = (sin θ cosφ, sin θ sinφ, cos θ). I urge the reader to prove that

I(R; q) =4π

iq(G(R; q) +G(R;−q)) , (34.87)

where

G(R; q) =iq

4π

U− eReRiqR

−[

1

(iqR)2− 1

(iqR)3

]

(U− 3eReR)

eiqR (34.88)

is the standard outgoing dyadic Green function [127], and G(R,−q) the correspondingingoing one. The quantity eR is a unit vector in the R-direction. It appears from thedivision in Eq. (34.87) that the outgoing photon wave function formally (see Sec. 34.2.4) isgiven by the integral form

ΦOUT (R, t) =

[

(2ǫ0~c)− 1

2

πi

∫ ∞

0

G(R; q)e−icqt

cq − Ω + iΓadq

]

· Jba. (34.89)

Although the expression for ΦOUT (R, t) has been reduced to a single integral over themagnitude of the q-vectors, the physical interpretation of the result in Eq. (34.89) is quitecomplicated, as we shall see now.

34.2.3 Genuine transverse photon wave function

The outgoing wave function always has near-, mid-, and far-field parts. This is not sur-prising because the “tail” of ΦOUT (R, t) necessarily must be in contact with the atom ina spontaneous process where the decay toward the ground state is exponential. In a stim-ulated emission process where the atom with certainty is found in the lower state after afinite time, the released photon wave function will be located entirely in the far field aftersome time.

Let us consider first the far-field (FF) contribution to ΦOUT (R, t), viz.,

ΦFFOUT (R, t) =

(2ǫ0~c)− 1

2

4π2iR

[∫ ∞

0

eiq(R−ct)

cq − Ω + iΓadq

]

(U− eReR) · Jba. (34.90)

Since the Weisskopf–Wigner theory of spontaneous emission for a two-level atom essen-tially deals with electromagnetic mode frequencies in the vicinity of the atomic transitionfrequency (RWA model), we extend the lower limit of the integration in Eq. (34.90) to−∞. With this approximation the integral can be evaluated by contour integration alonga semicircle (of infinite radius). The first-order pole of the integrand is located in the lower


halfplane of the complex q-plane at (Ω− iΓa)/c. For R > ct, the integral vanishes becausethe contour must lie in the upper halfplane. For R < ct, the contour encircles the pole andthe integral equals 2πiRes[(Ω− iΓa)/c]. Hence,

∫ ∞

0

eiq(R−ct)

cq − Ω+ iΓadq ≃

∫ ∞

−∞

eiq(R−ct)

cq − Ω + iΓadq

=2π

icθ

(

t− R

c

)

exp

[

−i (Ω− iΓa)

(

t− R

c

)]

, (34.91)

where θ is the Heaviside unit step function. With the approximation in Eq. (34.91), thefar-field photon wave function becomes

ΦFFT (R, t) =

(2ǫ0~c)− 1

2

2πcRθ

(

t− R

c

)

exp

[

−i (Ω− iΓa)

(

t− R

c

)]

(eReR −U) · Jba.

(34.92)

For a reason which will become clear when we discuss the near- and mid-field parts of theoutgoing photon wave function, the subscript OUT has been replaced by T (for divergence-free), ΦFF

OUT ≡ ΦFFT . The trace of the first-order correlation matrix, generally given by

Eq. (16.95), equals the probability density for detecting a transverse photon in a givenspace-time point. In the far field one has on the basis of Eq. (34.92) the result

TrG(1)(x, x) =[

ΦFFT (R, t)

]†·[

ΦFFT (R, t)

]

=1

8π2~ǫ0c31

R2θ

(

t− R

c

)

exp

[

−2Γa

(

t− R

c

)]

Jab · (U− eReR) · Jba.

(34.93)

The presence of the step function in Eq. (34.93) implies that the connection between thephoton emission starting at time t = 0 from the atom located at r0, and the detection atthe space point r is time-like for far-field separations |R| = |r− r0|. It must be emphasizedhere that this conclusion is based on a calculation in which (i) nonresonant terms areneglected, and (ii) the relevant integral [Eq. (34.91)] is extended to (all) negative frequencies.In general one cannot conclude that the correlation between quantum events is time-like,even if these events are far-field separated. The correlation issue has been studied extensivelyin relation to theoretical calculations of the energy transfer between a pair of atoms (theso-called Fermi problem [72]). It was first pointed out by Shirokov [217, 218] that withoutthe approximations (i) and (ii) one obtains a noncausal result in the Einsteinian sense forthe probability of energy transfer. Later on the same conclusion was reached by others [208,235, 84]; see also [95, 195, 196, 197, 198]. It has turned out that the causality/noncausalityconclusion depends on the so-called state specification in the energy transfer process. Agood introduction to the Fermi problem is given in the textbook by Louisell [151]. Thepresence of space-like couplings does not imply that the signals can move faster than thevacuum speed of light, however [123, 127].

Although the outgoing photon wave function satisfies the criterion

∇ ·ΦOUT (R, t) = 0, R 6= 0, (34.94)

one cannot conclude that ΦOUT (R, t) is a genuine divergence-free (transverse) vector field.To qualify as such a field the condition ∇ ·ΦOUT = 0 must be satisfied in the entire space,that is also at R = 0. The whole question of genuine transversality is hidden in the structureof the dyadic Green function, cf. Eq. (34.89). In the rim (near-field) zone of the atom, the


wave function ΦOUT (R, t) thus contains a rotational-free (longitudinal) part, relating tothe nonretarded Coulomb field associated to the atomic dipole oscillations. It is knownon general ground that this part can be subtracted from ΦOUT (R, t) replacing the dyadicGreen function G(R; q) by the retarded transverse (T) dyadic propagator [127]

GT (R; q) =iq

4π

eiqR

iqR(U− eReR)−

[

eiqR

(iqR)2 − eiqR − 1

(iqR)3

]

(U− 3eReR)

. (34.95)

The outgoing transverse photon wave function, ΦT (R, t), hence is given by the integralexpression

ΦT (R, t) =

[

(2ǫ0~c)− 1

2

πi

∫ ∞

0

GT (R; q)e−icqt

cq − Ω+ iΓadq

]

· Jba, (34.96)

and the one-photon field indeed is a genuine transverse vector field because

∇ ·ΦT (R, t) = 0, ∀R. (34.97)

It is obvious from a comparison of Eqs. (34.88) and (34.95) that G and GT have identicalfar-field parts. In turn this implies that ΦT (R, t) approaches Φ

FFT (R, t) as R → ∞. This is

the reason that ΦFFOUT was renamed ΦFF

T above.

34.2.4 Spontaneous photon emission in the atomic rim zone

In the framework of the Weiskopf–Wigner theory for spontaneous emission from a two-levelatom with transition frequency Ω, it is obvious that only the interaction of the atom withelectromagnetic modes of frequencies close to Ω is well described. In consequence only thecontributions to the integral from the vicinity of Ω/cmake sense, physically. Mathematicallyone faces an additional problem in Eq. (34.89): The q-integral is divergent, because of the q−3

dependence of the Green function near q = 0. Thus, some renormalization of the integrandis needed for q → 0. Notwithstanding the physical limitations of the Weisskopf–Wignermodel, the q-integral related to the transverse photon wave function, ΦT (R, t) [given inEq. (34.96)], does not diverge. The physical reason for this stems from the fact that thetransverse propagator, describing the propagation of the field in space-time, must existindependent of the structure of the source emitting the field [127]. Let us examine this inthe context of spontaneous emission.

We start by defining a vectorial source field in the space-frequency domain, S(+)(r;ωq),by

S(+)(r;ωq) ≡ Kδ(r− r0)θ(ωq)

ωq − Ω+ iΓa, (34.98)

where

K =

(

2

ǫ0~c

)1

2 Jba

ic. (34.99)

The presence of the delta function, δ(r−r0), is due to the circumstance that the source atomgenerating the photon wave function is treated as a point source, located at r = r0. Thesuperscript (+) on S, shall remind us that a polychromatic photon is composed of positivefrequencies only, in agreement with the fact that the integral in Eq. (34.96) extends overthe interval 0 ≤ q ≤ ∞. The effective source field for the spontaneous emission therefore


is a superposition of positive-frequency components. The unit step function, θ(ωq), in Eq.(34.98) formally allows one to extend the q-integral to negative q’s [−∞ < q < ∞]. Thisextension is convenient when we use the folding theorem below.

A combination of Eqs. (34.96) and (34.98) allows one to write the transverse wavefunction [ΦT (R, t) ≡ ΦT (r, t)] as a double integral over r′-space and ωq, viz.,

ΦT (r, t) =1

2π

∫ ∞

−∞GT (r− r′;ωq) · S(+)(r′;ωq)e

−iωqtdωqd3r′. (34.100)

The frequency integral is identical to a folding integral of GT and S(+) in the time domain.Thus,

ΦT (r, t) =

∫ ∞

−∞GT (r− r′, t− t′) · S(+)(r′, t′)dt′d3r′. (34.101)

where, with r− r′ = R and t− t′ = τ ,

GT (R, τ) =1

2π

∫ ∞

−∞GT (R;ωq)e

−iωqτdωq, (34.102)

and

S(+)(r′, t′) = Kδ(r′ − r0)

∫ ∞

0

e−iωqt′

ωq − Ω+ iΓa

dωq

2π. (34.103)

The space-time form of the transverse propagator can be obtained by inserting Eq.(34.95) into (34.102). The integration over frequency can be carried out, and after a some-what lengthy calculation one obtains [123, 127, 119]

GT (R, τ) =1

4πRδ

(

R

c− τ

)

(U− eReR)

− c2τ

4πR3θ(τ)θ

(

R

c− τ

)

(U− 3eReR) , (34.104)

a result already cited in Eq. (22.11). The far-field part of the propagator (∼ R−1) onlyis nonvanishing on the outgoing light cone [R = cτ ]. When this part is inserted into Eq.(34.101), one regains the result given in Eq. (34.90) for the far-field part of the transversephoton wave function, ΦFF

OUT = ΦFFT . The near-field part of GT (∼ R−3) is causal in the

sense that it is nonvanishing only for t′ < t [due to the step function θ(τ)], but space-like,i.e., nonvanishing only for R > c(t − t′) [due to the step function θ(R/c − τ)]. The factthat the near-field part of GT (R, τ) is different from zero in front of the light cone doesnot imply that the Einsteinian causality is broken in the rim zone of the atom. As brieflytouched upon in Sec. 22.1.1, and discussed in detail in my book on the quantum theoryof near-field electrodynamics [127], the extension of the spatial localization of a transversephoton emitted from a delta-function source [∼ δ(r− r0)U] is given by the extension of thetransverse delta function, δT (r−r0). The source region of a T-photon therefore includes therim zone of the atom. Once generated somewhere in the rim zone, the T-photon propagateswith the speed of light. The near-field part of the transverse photon wave function hencecontains a fingerprint of our inability to localize a T-photon completely in space-time. Ifone inserts the near-field part of GT into Eq. (34.101), and makes use of the expressiongiven for S(+)(r′, t′) in Eq. (34.103), one obtains the following expression for the near-field


part of the T-photon wave function:

ΦNFT (r, t) =

c2

4πR3(3eReR −U) ·K

×∫ t

t−Rc

(t− t′)

[

∫ ∞

0

e−iωqt′

ωq − Ω + iΓa

dωq

2π

]

dt′, R = r− r0, (34.105)

where, as indicated, R is the vectorial distance between the atom and the r-coordinateentering the wave function. One may change the order of the integrations in Eq. (34.105),and since

∫ t

t−Rc

(t− t′)e−iωqt′

dt′ =1

ωq

(

1

ωq− iR

c

)

exp

[

iωq

(

R

c− t

)]

− 1

ω2q

exp (−iωqt) , (34.106)

the expression for the near-field part of the T-photon wave function reduces to a singleintegral over frequency, namely,

ΦNFT (r, t) =

c2

8π2R3(3eReR −U) ·K

×∫ ∞

0

[

1

ωq

(

1

ωq− iR

c

)

eiωq(Rc−t) − 1

ω2q

e−iωqt

]

dωq

ωq − Ω+ iΓa. (34.107)

The outgoing photon wave function, ΦOUT , given by the q-integral in Eq. (34.89) issingular at ωq = cq = 0, due to the singularity of the Green function GT (R; q) at q = 0;see Eq. (34.88). The transverse propagator GT (R; q) has no singularity at q = 0, and asthe reader may show, inserting Taylor series expansion of exp(iqR) around q = 0 in Eq.(34.95), one has

GT (R; q = 0) =U+ eReR

8πR. (34.108)

The result in Eq. (34.108) shows that the R−3-singularity in G(R; q) at R = 0 is replacedby a R−1-singularity in GT (R; q). Physically, this is a very satisfactory result becausethe integration of the product GT and a given source density distribution (with finitespatial extension) over the source domain is absolutely convergent [123, 133]. A similarintegration using G(R; q) only will be conditionally convergent due to the presence of theR−3-singularity.

The general remarks given above manifest themselves in the case of spontaneous emissionin that

limωq→0

∫ t

t−Rc

(t− t′) e−iωqt′

dt′ =1

2

(

R

c

)2

+O(ω1q). (34.109)

In consequence, the integrand entering the expression for ΦNFT (r, t) in Eq. (34.107) has no

singularity at ωq = 0. Near ωq = 0, the R-dependence of the integrand is ∼ R−3R2 = R−1,as expected.

In the framework of the Weisskopf–Wigner theory of spontaneous emission for the two-level atom we may extend the lower limit of integration in Eq. (34.107) to−∞, and thereafterevaluate the integral by contour integration, as described in the text between Eqs. (34.90)and (34.91). Since the integral only has one first-order pole at ωq = Ω− iΓa (as before) one


evanescent contribution

r = ct

<

space-like contribution

r = c

<

t

FIGURE 34.1Schematic illustration of the space-time structure of the transverse propagator, GT (r, τ)[upper two figures] and the T-photon wave function, ΦT (r, t) [lower two figures] in the rimzone of an ED point source (atom) located at r = 0. The propagator consists of two pieces,viz., a part [left] which is nonvanishing solely on the outgoing light cone, δ(r−cτ), and a part[right] which is space-like, θ(r − cτ). The light-cone piece, which is proportional to r−1, isthe only contribution to GT (r, τ) in the far field of the source. The space-like piece, which isproportional to τr−3 and obeys microcausality [θ(τ)], originates in the spatial delocalizationof the photon embryo. In the framework of the Weisskopf–Wigner theory the T-photon wavefunction consists of a time-like part, θ(t − r/c), containing far-field(r−1) [left], mid(r−2)-and near(r−3)-field pieces [right], and an evanescent near-field part, r−3θ(t). The extensionof the rim zone is given in light gray tone in the figures to the right.

obtains the following result for the near-field part of the transverse photon wave function:

ΦNFT (r, t) =

c2

4πiR3(3ereR −U) ·K

×[

1

Ω− iΓa

(

1

Ω− iΓa− iR

c

)

θ

(

t− R

c

)

e−iΩ(t−Rc )e−Γa(t−R

c )

− 1

(Ω− iΓa)2 θ(t)e

−iΩte−Γat

]

, (34.110)


or with insertion of the expression for K [Eq. (34.99)]

ΦNFT (r, t) =

(

2c

ǫ0~

)1

2 (U− 3eReR) · Jba

4πR3

×[

1

Ω− iΓa

(

1

Ω− iΓa− iR

c

)

θ

(

t− R

c

)

e−iΩ(t−Rc )e−Γa(t−R

c )

− 1

(Ω− iΓa)2 θ(t)e

−iΩte−Γat

]

. (34.111)

It appears from Eq. (34.111) that what we have called the near-field part of the T-photonwave function has near-field ∼ R−3 as well as mid-field (∼ R−2) parts, a result one mighthave expected for a field quantity. The mid-field part always is time-like, cf. the presenceof the step function θ(t − R/c). The near-field part of ΦNF

T (r, t), however, contains an“evanescent” (evan) part

ΦNFT,evan(r, t) =

(

2c

ǫ0~

)1

2 (3eReR −U) · Jba

4πR3

θ(t)

(Ω− iΓa)2 e

−iΩte−Γat, (34.112)

which is non-propagating, decays away from the atom as R3, and exhibits exponentiallydamped [exp(−Γat)] harmonic oscillations [exp(−iΩt)] in time. The evanescent componentof the T-photon wave function obeys microcausality [presence of the step function θ(t) inEq. (34.112)], and the component relates to the lack of perfect spatial photon localizationin the emission process. Although the near-field part of the transverse propagator is space-like, the propagating part of the T-photon wave function is time-like; see Fig. 34.1. TheEinsteinian causality hence is obeyed in the Weisskopf–Wigner theory for the spontaneousemission process. As the reader may prove to herself

∇×ΦNFT,evan(r, t) = 0, (34.113)

and it is the substraction of such a divergence-free and rotational-free part, ΦNFT,evan(r, t),

from the total outgoing photon wave function, ΦOUT (r, t) [Eq. (34.89)], here taken in thepole approximation, which in the general treatment [without the pole approximation] re-places the divergent integral for ΦOUT (r, t) [Eq. (34.89)] by the non-divergent integral forΦT (R, t) [Eq. (34.96)].

34.3 Three-level atom: Spontaneous cascade emission

34.3.1 Two-photon state vector

Let us consider now the spontaneous cascade emission of light from a three-level atom inthe framework of the Weisskopf–Wigner theory. The level scheme for the atomic decay isshown in Fig. 34.2. The atom, initially in the upper state |a〉, emits a photon of wave vectorq, polarization s, and (angular) frequency ωq = cq. Thereby, it decays to the intermediatestate |b〉. The Bohr transition frequency is denoted by Ωα = (Ea − Eb)/~. Subsequently,the atom decays to the ground state |c〉 emitting a photon of wave vector k, polarizationt, and frequency ωk = ck. The associated Bohr transition frequency is denoted by Ωβ =(Eb − Ec)/~. The flip operators for the |a〉 → |b〉, and |b〉 → |c〉 transitions we denote by

bα = |b〉〈a|, (34.114)


and

bβ = |c〉〈b|, (34.115)

respectively. It is assumed that the two transitions are electric-dipole allowed. The appropri-ate ED coupling constants for the |a〉 → |b〉 and |b〉 → |c〉 transitions are denoted by gαqs and

gβkt, cf. Eq. (34.29) [or Eq. (34.31)]. In the Interaction Picture, the interaction Hamiltonianfor the three-level system thus is given by

HI(t) =~∑

q,s

[

gαqs(r0)b†α(0)aqs(0)e

i(Ωα−ωq)t + h.c.]

+ ~∑

k,t

[

gβkt(r0)b†β(0)akt(0)e

i(Ωβ−ωk)t + h.c.]

(34.116)

in the rotating-wave approximation. As before, the atomic position, r0, enters the formalismvia phase factors on the coupling constants, cf. Eq. (34.54).

|c

a

b

W

W

G

G

,

,

FIGURE 34.2Spontaneous cascade decay [|a〉 → |b〉 → |c〉] of a three-level atom with Bohr transitionfrequencies Ωα [|a〉 → |b〉] and Ωβ [|b〉 → |c〉] and amplitude decay constants Γa (|a〉-level)and Γb (|b〉-level). In general, the emitted light is in an entangled two-photon state. In thelimit Γa ≫ Γb the two photon wave packets are emitted independently, but in differentquantum states. The non-interacting photons are in a correlated two-boson state.

The state vector of the coupled atom-field system has the general form

|ψ(t)〉 =ca(t)|a, 0〉+∑

q,s

cb,qs(t)|b, 1qs〉

+∑

q,s

∑

k,t

cc,qs,kt(t)|c, 1qs, 1kt〉, (34.117)

and the task now is to determine the time dependence of the yet unknown probabilityamplitudes, ca(t), cb,qs(t), and cc,qs,kt(t), under assumption that the atom at time t = 0 isin the upper excited state:

ca(t) = 1, cb,qs(0) = 0, cc,qs,kt(0) = 0. (34.118)

The last two initial conditions in Eq. (34.118) of course hold for all (q, s) and (k, t). Thedynamical time evolution of the state vector |ψ(t)〉 is determined by inserting Eqs. (34.116)


and (34.117) into the Schrodinger equation in the Interaction Picture [Eq. (34.47)]. Utilizingthe orthonormality of the time independent base vectors |a, 0〉, |b, 1qs〉 and |c, 1qs, 1kt〉, andtaking the appropriate inner products, one finds that the probability amplitudes satisfy thefollowing set of coupled first-order differential equations:

ca(t) = −i∑

q,s

gαqs(r0)ei(Ωα−ωq)tcb,qs(t), (34.119)

cb,qs(t) = −i[

gαqs(r0)]∗e−i(Ωα−ωq)tca(t)− i

∑

k,t

gβkt(r0)ei(Ωβ−ωk)tcc,qs,kt(t), (34.120)

cc,qs,kt(t) = −i[

gβkt(r0)]∗e−i(Ωβ−ωk)tcb,qs(t). (34.121)

I urge the reader to verify the correctness of Eqs. (34.119)-(34.121). In the spirit of theWeisskopf–Wigner approximation we now assume that the spontaneous decays |a〉 → |b〉and |b〉 → |c〉 are well described by the exponential decay rates Γa and Γb. Mathematically,the decay constant Γa is obtained by combining Eqs. (34.119) and (34.120), and assumingthat the terms containing the cc,qs,kt(t)’s are so small that they can be neglected in a firstapproximation. Following the analysis in Subsec. 34.1.2, the two decay constants are givenby

Γa = π∑

q,s

|gαqs(r0)|2δ(Ωα − ωq), (34.122)

Γb = π∑

k,t

|gβkt(r0)|2δ(Ωβ − ωk). (34.123)

Again, one may express Γa and Γb in terms of the transition dipole moments associatedwith the |a〉 → |b〉 and |b〉 → |c〉 transitions. Hence,

Γa =1

4πǫ0

2Ω3α

3~c3|dab|2, (34.124)

Γb =1

4πǫ0

2Ω3β

3~c3|dbc|2, (34.125)

cf. Eq. (34.67). The general set of coupled equations for the probability amplitudes, we thusreplace by

ca(t) = −Γaca(t), (34.126)

cb,qs(t) = −i[

gαqs(r0)]∗e−i(Ωα−ωq)tca(t)− Γbcb,qs(t), (34.127)

cc,qs,kt(t) = −i[

gβkt(r0)]∗e−i(Ωβ−ωk)tcb,qs(t), (34.128)

in what essentially amounts to the Weisskopf–Wigner approximation. The differential equa-tions above now can be solved one by one. By inserting the solution of Eq. (34.126), viz.ca(t) = exp(−Γat), into Eq. (34.127), the first-order inhomogeneous differential equationfor cb,qs(t) is readily solved. With the initial condition cb,qs(0) = 0 one gets

cb,qs(t) = −i[

gαqs(r0)]∗e−Γbt

∫ t

o

e−i(Ωα−ωq)t′

e−(Γa−Γb)t′

dt′

= i[

gαqs(r0)]∗ e−i(Ωα−ωq)te−Γat − e−Γbt

i (Ωα − ωq) + Γa − Γb. (34.129)


In the long-time limit (t→ ∞), one has

cb,qs(∞) = 0, (34.130)

in agreement with our physical expectation. By inserting Eq. (34.129) into Eq. (34.128), oneobtains by simple integration, and use of the initial condition cc,qs,kt(0) = 0 the followingexplicit result for the two-photon probability amplitude:

cc,qs,kt(t) =[

gαqs(r0)]∗ [

gβkt(r0)]∗ 1

i (ωq − Ωα)− (Γa − Γb)

×[

e−i(Ωβ−ωk)te−i(Ωα−ωq)te−Γat − 1

i (Ωβ − ωk) + i (Ωα − ωq) + Γa− e−i(Ωβ−ωk)te−Γbt − 1

i (Ωβ − ωk) + Γb

]

. (34.131)

In the long-time limit, Eq. (34.131) is reduced to

cc,qs,kt(∞) =−[

gαqs(0)gβkt(0)

]∗e−i(q+k)·r0

[i (ωq + ωk − Ω)− Γa] [i (ωk − Ωβ)− Γb], (34.132)

where

Ω = Ωα +Ωβ =Ea − Ec

~, (34.133)

is the Bohr frequency for the transition |a〉 → |c〉.After sufficiently long time the atom has decayed to the ground state, and the electro-

magnetic field is in the two-photon state

|Φ2〉 = limt→∞

|Ψ(t)〉

=∑

q,s

∑

k,t

−[

gαqs(0)gβkt(0)

]∗e−i(q+k)·r0

[i (ωq + ωk − Ω)− Γa] [i (ωk − Ωβ)− Γb]|1qs, 1kt〉. (34.134)

In compact form one may write |Φ2〉 as follows:

|Φ2〉 =∑

q,s

∑

k,t

cc,qs,kt(∞)a†qs(0)a†kt(0)|0〉, (34.135)

where |0〉 is the global photon vacuum state.

34.3.2 Two-photon two-times wave function

The dyadic two-photon wave function associated to the state in Eq. (34.135) is obtainedvia the two-photon mean position state in Eq. (33.12). The tensorial object relating to thespace-time positions (r1, t1) and (r2, t2), viz.,

Φ2(r1, r2, t1, t2) =2ǫ0c

~〈0|A(+)

T (r1, t1)A(+)T (r2, t2)|Φ2〉 (34.136)

taken for t1 = t2, is what one would call the two-photon wave function. For |Φ2〉 given byEq. (34.135), and with the plane-mode expansions inserted for the two transverse vectorpotentials one gets

Φ2(r1, r2, t1, t2) =L−3∑

q′,s′

∑

k′,t′

∑

q,s

∑

k,t

(q′k′)− 1

2 cc,qs,kt(∞)

×[

εq′s′εk′t′ei(q′·r1−ωq′ t1)ei(k

′·r2−ωk′ t2)

× 〈0|aq′s′(0)ak′t′(0)a†qs(0)a

†kt(0)|0〉

]

. (34.137)


The summations over (q′, s′) and (k′, t′) readily can be carried out since the matrix elementobviously is given by

〈0|aq′s′(0)ak′t′(0)a†qs(0)a

†kt(0)|0〉 = δk′kδt′tδq′qδs′s + δk′qδt′sδq′kδs′t, (34.138)

where all the δ’s are Kronecker delta’s. Hence,

Φ2(r1, r2, t1, t2) =L−3∑

q,s

∑

k,t

(qk)− 1

2 cc,qs,kt(∞)

×[

εqsεktei(q·r1−ωqt1)ei(k·r2−ωkt2)

+ εktεqsei(k·r1−ωkt1)ei(q·r2−ωqt2)

]

. (34.139)

A further reduction of Eq. (34.139) can be obtained by carrying out the summations overthe polarization states. These states enter cc,qs,kt(∞) via the coupling constants gαqs(0) and

gβkt(0) [see Eq. (34.132)]. If one uses for the coupling constants the generic form in Eq.(34.31), the summations relating to the first term in the square bracket of Eq. (34.139) arecarried out as follows:

∑

s

∑

t

εqsεkt(

ε∗qs · J∗ab

)

(ε∗kt · J∗bc) = J∗

ab ·[

∑

s

∑

t

ε∗qsεqsεktε∗kt

]

· J∗bc

= J∗ab ·[(

∑

s

ε∗qsεqs

)(

∑

t

εktε∗kt

)]

· J∗bc

= J∗ab · (U− qq)

(

U− kk)

· J∗bc, (34.140)

where q = q/q and k = k/k. Using this result, and an analogous one relating to the secondterm in the square bracket of Eq. (34.139), and letting for simplicity the atom be locatedat r0 = 0, one obtains

Φ2(r1, r2, t1, t2) =− L−6 (2ǫ0~c)−1∑

q

∑

k

(qk)−1

2 F

×[

J∗ab · (U− qq)

(

U− kk)

· J∗bce

i(q·r1−ωqt1)ei(k·r2−ωkt2)

+ J∗bc ·(

U− kk)

(U− qq) · J∗abe

i(k·r1−ωkt1)ei(q·r2−ωqt2)]

. (34.141)


F = [i (ωq + ωk − Ω)− Γa]−1 [i (ωk − Ωβ)− Γb]

−1 . (34.142)

As in the one-photon case, we turn to the continuum limit and refer the integrations overthe q- and k-space to spherical coordinates. Denoting the relevant solid angle elements bydSq and dSk two angular double integrals appear, viz.,

J(r1, r2; q, k) = (2π)−2

∫

4π

∫

4π

(U− qq)(

U− kk)

eiq·r1eik·r2dSqdSk, (34.143)

and a similar one with q and k interchanged. The quantity J is a product of integrals overq and k, and each of these integrals has the form given in Eq. (34.86). Having in mind thatthe integral in Eq. (34.86) can be done and essentially expressed in terms of the in- andoutgoing dyadic Green function [Eqs. (34.87) and (34.88)], one obtains

J(r1, r2; q, k) =

(

4π

i

)2

(qk)−1 (G(r1; q) +G(r1;−q)) (G(r2; k) +G(r2;−k)) . (34.144)


The q- and k-photon fields converging on the atom are neglected in our description of thetwo-photon cascade process. Hence,

J(r1, r2; q, k) ⇒

JOUT (r1, r2; q, k) = − (4π)2

qkG(r1; q)G(r2; k). (34.145)

By now, we have reached the following expression for the outgoing two-photon object re-ferring to two times t1 and t2:

Φ2(r1, r2, t1, t2) ⇒Φ2,OUT (r1, r2, t1, t2) =

(

2π2ǫ0~c)−1

×

Jba ·[∫ ∞

0

G(r1; q)G(r2; k)e−ic(qt1+kt2)Fdqdk

]

· Jcb

+ Jcb ·[∫ ∞

0

G(r1; k)G(r2; q)e−ic(kt1+qt2)Fdqdk

]

· Jba

.

(34.146)

We now know from the discussion in Subsec. 34.2.3 that the fields emerging from the|a〉 → |b〉 and |b〉 → |c〉 transitions both contain a rotational-free part in the rim zoneof the atom when the description is based on the standard Green function, G. Genuinetransverse vector fields are obtained by replacing G by GT everywhere in Eq. (34.146). Thetwo-photon two-times wave function of particular interest in relation to the spontaneousatomic cascade emission hence is

Φ2,T (r1, r2, t1, t2) = Φ(1)2,T (r1, r2, t1, t2) +Φ

(2)2,T (r1, r2, t1, t2), (34.147)

where

Φ(1)2,T (r1, r2, t1, t2) =

(

2π2ǫ0~c)−1

× Jba ·[∫ ∞

0

GT (r1; q)GT (r2; k)e−ic(qt1+kt2)Fdqdk

]

· Jcb (34.148)

and

Φ(2)2,T (r1, r2, t1, t2) =

(

2π2ǫ0~c)−1

× Jcb ·[∫ ∞

0

GT (r1; k)GT (r2; q)e−ic(kt1+qt2)Fdqdk

]

· Jba. (34.149)

The superiority of Φ2,T (r1, r2, t1, t2) [Eq. (34.147)] to Φ2,OUT (r1, r2, t1, t2) [Eq. (34.146)]appears in the description of near-field cascade emission; cf. the discussion of the sponta-neous one-photon emission in the atomic rim zone (Subsec. 34.2.4).

34.3.3 The structure of Φ2,T (r1, r2, t1, t2)

We finish our analysis of the spontaneous cascade emission by a qualitative discussion ofthe structure of the transverse two-photon two-times wave function given by Eqs. (34.147)-(34.149).

The joint probability density for transverse photon detection at the space-time pointsx1 and x2 relates to the second-order correlation matrix

G(2)2 (x1, x2, x2, x1) = Φ∗

2,T (x1, x2)Φ2,T (x2, x1), (34.150)


see Eq. (33.54). The detection “amplitude” Φ2,T (x2, x1) is the sum of two parts [Eq.

(34.147)]. The part Φ(1)2,T (x2, x1) corresponds to a situation where the amplitude of the

photon released in the |a〉 → |b〉 transition interacts with the detector placed at r1 [prop-agator: GT (r1; q)], and the photon amplitude from the |b〉 → |c〉 transition interacts withthe detector at r2 [propagator: GT (r2; k)]. In agreement with this it is the scalar productsGT (r1; q) · Jba and GT (r2; k) · Jbc which enter the expression for Φ2,T (x2, x1). The part

Φ(2)2,T (x2, x1) relates to the situation where the amplitude of the first [|a〉 → |b〉] photon

interacts with the detector at r2, and the second [|b〉 → |c〉] photon amplitude couples tothe detector at r1. In a registered joint detection probability the two amplitude processesinterfere. The considerations above are illustrated in Fig. 34.3.

|c

>

<

<

>

1

FIGURE 34.3Schematic illustration of the registration of two correlated photons in detectors D1 and D2.The two entangled T-photon wave packets are emitted in a (spontaneous) cascade emission[|a〉 → |b〉 → |c〉] from a three-level atom. The two-photon vectorial detection amplitude

Φ2,T is the sum of two parts, Φ(1)2,T and Φ

(2)2,T . The part Φ

(1)2,T corresponds to detection

of the photon released in the |a〉 → |b〉 transition in D1 and the photon emitted in the

|b〉 → |c〉 transition in D2 [fully drawn lines]. The part Φ(2)2,T relates to the opposite situation

[indicated by the dashed lines]. In the second-order correlation matrix G2(x1, x2, x2, x1) =Φ∗

2,T (x1, x2)Φ2,T (x2, x1), referring to the space-time points x1 and x2, the two amplitudeprocesses interfere.

The integral over the (qk)-domain [Eqs. (34.148) and (34.149)] does not separate into aproduct of integrals over q and k because the resonance factor F(ωq, ωk) is not factorizablein general. In an approximate sense a factorization is possible if

Γa ≫ Γb. (34.151)

The condition in the inequality (34.151) means that we are considering a situation wherethe natural lifetime in the |a〉-level is much shorter than that in the |b〉-level. Essentially, theemission of the photon from the upper transition thus has been completed before the emis-sion process of the photon from the lower transition has evolved significantly. For Γb/Γa ≪ 1,the resonance factor may be approximated by

F ≃ [i (ωq − Ωα)− Γa + i (ωk − Ωβ)− Γb]−1

[i (ωk − Ωβ)− Γb]−1. (34.152)

We know from the previous analysis of the spontaneous one-photon process decay that theemission spectrum is centered on the atomic transition frequency. When Γb ≪ Γa the radia-tive broadening of the |b〉-level is much smaller than that of the |a〉-level. In consequence,


the resonance factor factorizes approximately, i.e.,

F ≃ [i (ωq − Ωα)− Γa]−1

[i (ωk − Ωβ)− Γb]−1. (34.153)

By inserting Eq. (34.153) into Eq. (34.148) one obtains

Φ(1)2,T (r1, r2, t1, t2) =

[

(2ǫ0~c)− 1

2

πi

∫ ∞

0

GT (r1; q)e−icqt1

cq − Ωα + iΓadq

]

· Jba

×[

(2ǫ0~c)− 1

2

πi

∫ ∞

0

GT (r2; k)e−ickt2

ck − Ωβ + iΓbdk

]

· Jcb. (34.154)

In view of Eq. (34.96) it appears that Φ(1)2,T is just a tensor product of the transverse single-

photon wave functions belonging to the upper [ΦabT ] and lower [Φbc

T ] transitions:

Φ(1)2,T (r1, r2, t1, t2) = Φab

T (r1, t1)⊗ΦbcT (r2, t2). (34.155)

It is obvious that also Φ(2)2,T becomes a tensor product of single-photon wave functions when

Γb ≪ Γa. Altogether, the transverse two-time two-photon wave function takes the form

Φ2,T (r1, r2, t1, t2) = ΦabT (r1, t1)⊗Φbc

T (r2, t2) +ΦbcT (r1, t1)⊗Φab

T (r2, t2), Γa ≫ Γb.(34.156)

When Γa ≫ Γb it is meaningful to consider the spontaneous cascade process as an emissionin which two photons well described by single-particle wave packets Φab

T and ΦbcT partic-

ipate. In this sense the two photons can be said to be independent quantities. Here, onemust however again remember the words of Bohr: “(i) No (elementary) phenomenon isa phenomenon until it is a registered (observed) phenomenon, and (ii) the quantum me-chanical formalism permits well-defined applications referring only to closed phenomena”[29, 30, 31]. In our case the atom-field interaction in the cascade process (the preparationprocess) is a part of the Bohr closed phenomenon which finally leads to the finite field state.In the cascade process the emitted photons are in an entangled state [Eq. (34.156)]. Thetwo-photon state Φ2,T (r1, r2, t1, t2) is not a factorizable (simple tensor product) state. Thetwo wave packet photons are non-interacting for Γa ≫ Γb. As particles they are identical,but they are in different quantum states. The result in Eq. (34.156) therefore agrees withthe general formula for non-interacting bosons [Eq. (31.5)].

Let us now return to the general case, and discuss the structure of Φ(1)2,T (r1, r2, t1, t2).

The structure of Φ(2)2,T (r1, r2, t1, t2) is the same, basically, and from Φ

(1)2,T one may determine

Φ(2)2,T by appropriate changes of variables and indices. A tensorial source field referring to

the two-photon emission is introduced in the space-frequency domain by

S(+)2 (r1, r2;ωq, ωk) = S

(+)2 (ωq, ωk)δ(r1)δ(r2), (34.157)

where

S(+)2 (ωq, ωk) = K2 [ωq + ωk − Ω+ iΓa]

−1 [ωk − Ωβ + iΓb]−1 θ(ωq)θ(ωk), (34.158)

with

K2 = − 2

ǫ0~c3Jcb ⊗ Jba. (34.159)


The presence of the delta functions in Eq. (34.157) is associated with the fact that the two-photon field is generated by an atom which is assumed to be point-like (and here locatedat r0 = 0). As explained previously, this is the electric-dipole approximation. The stepfunctions θ(ωq) and θ(ωk) in Eq. (34.158) in an explicit manner show that the two-photon

wave function is composed of positive frequencies only [cf., the superscript (+) on S(+)2 ]. The

quantity K2, given by Eq. (34.159), is a tensor product of the transition current densitiesrelated to the lower and upper transitions. A comparison of Eqs. (34.99) and (34.159),makes it obvious that the expression for K2 is the natural generalization from a one-photon

to a two-photon source. The space independent part of the source field, S(+)2 (ωq, ωk), is

transferred to the time domain by Fourier integral transformation. Hence,

S(+)2 (t1, t2) = K2

∫ ∞

−∞

θ(ωq)θ(ωk)ei(ωqt1+ωkt2)

(ωq + ωk − Ω + iΓa) (ωk − Ωβ + iΓb)

dωqdωk

(2π)2, (34.160)

at least in a formal sense. Utilizing the folding theorem (twice) on Eq. (34.148) [with

S(+)2 (ωq, ωk) introduced], the reader may show that

Φ(1)2,T (r1, r2, t1, t2) =

∫ ∞

−∞GT (r1, t1 − t′1)GT (r2, t2 − t′2) : S

(+)2 (t′1, t

′2)dt

′1dt

′2. (34.161)

In Eq. (34.157) a new symbol “:” has been used. Remembering that the tensorial structure

of S(+)1 is given by K2 ∝ Jcb ⊗ Jba, the symbol is defined by the operation

GT (1)GT (2) : Jcb ⊗ Jba = Jba ·GT (1)GT (2) · Jcb. (34.162)

For brevity, the relevant arguments of the transverse propagators are denoted just by 1and 2. With GT (r1, t1− t′1) and GT (r2, t2− t′2) given by Eq. (34.104) [with the appropriate

identification of the space and time arguments], and S(+)2 (t′1, t

′2) given by the double integral

in Eq. (34.160), it appears that the near- and far-field structures, and the combinations ofthese may be determined, using the same procedure as in the single-photon case. Withreference to the far-field (FF) and near-field (NF) parts of the transverse propagator [Eq.(34.104)], the product GT (1)GT (2) leads to combinations of the types FF-FF, FF-NF

(NF-FF), and NF-NF in Φ(1)2,T (r1, r2, t1, t2). In the Weisskopf–Wigner pole approximation,

time-like and evanescent contributions occur in various combinations.

34.3.4 Far-field part of Φ(1)2,T (r1, r2, t1, t2)

The far-field part of the two-photon two-times wave function, Φ2,T , is the sum of the far-

field parts of Φ(1)2,T and Φ

(2)2,T . Below, I determine and discuss the far-field part of Φ

(1)2 . It is

left as an exercise for the reader to carry out the corresponding calculation for Φ(2)2 .

We start from the far-field part of Eq. (34.161), viz.,

Φ(1)2,T (r1, r2, t1, t2|FF ) =

∫ ∞

−∞GFF

T (r1, t1 − t′1)GFFT (r2, t2 − t′2) : S

(+)2 (t′1, t

′2)dt

′1dt

′2,

(34.163)

where

GFFT (R, τ) =

1

4πRδ

(

R

c− τ

)

(

U− RR)

(34.164)


is the far-field part of the transverse propagator; see Eq. (34.104). Note that eR ≡ R = R/R.The far-field wave function is a product of a tensorial part T, and a scalar part, S:

Φ(1)2,T (r1, r2, t1, t2|FF ) = TS. (34.165)

It appears from Eqs. (34.159)-(34.162) and (34.164) that the tensor part is given by

T = (U− r1r1) (U− r2r2) : K2

= − 2

ǫ0~c3Jba · (U− r1r1) (U− r2r2) · Jcb. (34.166)

The presence of the two Dirac delta functions, δ[(r1/c) − t1 + t′1] and δ[(r2/c) − t2 + t′2],immediately allows one to carry out the integrations over t′1 and t′2 in Eq. (34.163). In view

of the explicit form of S(+)2 (t′1, t

′2) [Eq. (34.160)], it is seen that the scalar part of the far-field

wave function is given by the following integral over the double frequency domain:

S =

(

1

4π

)21

r1r2

∫ ∞

0

e−iωq(t1−r1/c)e−iωk(t2−r2/c)

(ωq + ωk − Ω+ iΓa) (ωk − Ωβ + iΓb)

dωq

2π

dωk

2π. (34.167)

In the spirit of the Weisskopf–Wigner theory, we extend the lower limits of the integrals to−∞. Then, the integrals in Eq. (34.167) can be done by residue calculations. The integrationover ωq is carried out first. A first-order pole is located in the lower halfplane of the complexωq-plane at ωq = Ω− ωk − iΓa. Using appropriate contours [214], one obtains

S =

(

1

4π

)2(1

2π

)2(−2πi)

r1r2θ(

t1 −r1c

)

e−(iΩ+Γa)(t1− r1c )

×∫ ∞

−∞

e−iωk( r1c−t1− r2

c+t2)

ωk − Ωβ + iΓbdωk. (34.168)

The last integrand has a first-order pole at ωk = Ωβ−iΓb, and by means of relevant contours[214], we get

S =−(

1

4π

)21

r1r2θ(

t1 −r1c

)

θ[(

t2 −r2c

)

−(

t1 −r1c

)]

× e−(iΩ+Γa)(t1− r1c )e−(iΩβ+Γb)[(t2− r2

c )−(t1−r1c )]. (34.169)

Altogether, we have obtained the following result for the transverse two-photon wave func-

tion part Φ(1)2,T in the far field and in the pole approximation:

Φ(1)2,T (r1, r2, t1, t2|FF )

=2

ǫ0~c3Jba · (U− r1r1) (U− r2r2) · Jcb

θ(

t1 − r1c

)

θ[(

t2 − r2c

)

−(

t1 − r1c

)]

(4π)2r1r2

× exp[

− (iΩ+ Γa)(

t1 −r1c

)]

exp

− (iΩβ + Γb)[(

t2 −r2c

)

−(

t1 −r1c

)]

.

(34.170)

The unit step function θ(t1−r1/c) shows the field emission from the upper transition [|a〉 →|b〉] satisfies the Einsteinian Causality, an expected result in the far field (remembering thatwe have used the pole and rotating wave approximations). From the product of the twostep functions we can conclude that the wave function is nonvanishing only if

t2 −r2c> t1 −

r1c> 0. (34.171)


The first inequality tells us that the field emission from the lower transition [|b〉 → |c〉]begins later than that from the upper transition, as it is necessary physically because theelectron initially is in the |a〉-state. The inequality t2− r2/c > 0, shows that also the secondfield emission obeys the Einsteinian Causality.

Finally, we consider the case where Γa ≫ Γb. Since, now

exp[

− (iΩ+ Γa)(

t1 −r1c

)]

exp[

− (iΩβ + Γb)(

t2 −r2c− t1 +

r1c

)]

= exp[

− (iΩ+ Γa − iΩβ − Γb)(

t1 −r1c

)]

exp[

− (iΩβ + Γb)(

t2 −r2c

)]

≃ exp[

− (iΩα + Γa)(

t1 −r1c

)]

exp[

− (iΩβ + Γb)(

t2 −r2c

)]

, (34.172)

one obtains

Φ(1)2,T (r1, r2, t1, t2|FF )

=

[

(

2

ǫ0~c

)1

2 1

4πr1cJba · (U− r1r1) θ

(

t1 −r1c

)

e−(iΩα+Γa)(t1− r1c )

]

⊗[

(

2

ǫ0~c

)1

2 1

4πr2cJbc · (U− r2r2) θ

[(

t2 −r2c

)

−(

t1 −r1c

)]

e−(iΩβ+Γb)(t2− r2c )

]

=ΦabT (r1, t1|FF )⊗Φbc

T (r2, t2|FF ). (34.173)

The last member of Eq. (34.173) follows from the expression given for the far-field partof the single-photon wave function in Eq. (34.92). Our calculation hence has reached con-tact to the first part of single-particle entangled two-photon wave function given in Eq.(34.156). It is possible to obtain also an explicit expression for the near-field part of thetransverse two-photon wave function, following the procedure used to determine the spon-taneous one-photon near-field emission in Subsec. 34.2.4. In the rim zone of the atom, aphysically interesting interplay takes place between the two-photon entanglement and thespatial localization problem for the photon as emphasized in the presentation [NFO-11,Beijing] partly summarized in [128].

Bibliography

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Index

AAbelian Higgs model, 369–371, 375Aharonov-Bohm effect, xvi, 101Analytical signals, 4, 10–12

Hilbert transform pair (conjugatepair), 4, 12

Landau-Peierls-Sudarshan (LPS)equation, 25, 164, 207

propagation, 24–26Angular momentum, 103, 143–146

conservation, 21, 170quantized transverse field, 170∑

operator, 143from spherical to Cartesian

representation, 146–150spin current density in weakly

relativistic Pauli theory, 305two-photon polarization states, 390

Angular spectrum representation ofmonochromatic fields, 216, 233

Annihilation operators, 168, 169, 184, 196,206, 217, 315, 324, 338, 345,348–349, 401

commutation relation, 337Heisenberg equations of motion for

near-field, 217, 232, 233, 234–235two-photon wave function, 396

Antineutrino, 101, 144Antiphoton, 11, 115, 151, 329

energy wave function, 151four-current density, 332photon propagating backward in time,

204Atom as point-like entity, 215Atomic 1s ⇔ 2s transition, See Hydrogen 1s

⇔ 2s transitionAtomic and field correlation matrices,

192–193, See also Correlationmatrices

Atomic rim zone, See Rim zoneAtoms in near-field contact, 278

BBase vector transformation, 14–15Beam splitter experiment, 100fBianchi identity, 75Biphoton, xvii, See also Two-photon

phenomenaBispinor wave functions, 115–116, 153–156,

303Black-body body radiation, 380–382Bogolubov transformation, 325, 347–349,

357, 360Bohr, Niels, xiii–xiv, 3, 63, 99–100, 178,

197, 323, 350, 382, 389, 424Bohr transition frequency, 383, 417Boosts, xvii, 22, 29, 103, 140, 323–325,

332–337Bose-Einstein statistics, 99, 379, 380Bosons, 379

gauge bosons, 102Goldstone model, 326, 368Higgs bosons, 371two-boson state, 379–380, 382vector boson field, 132–136

Breathing mode transitions, xvii, 279,285–286, 289–292

Rabi frequency, 295two-level dynamics, 292–295

CCartesian representation of angular

momentum operator (∑

), 146–150Cerenkov angle, 268Cerenkov shock waves, 220, 260–262

transverse and longitudinal momenta,269–272

transverse and longitudinal responsetheory in matter, 263–265

transverse phenomenon, 266–269Charge conservation, 93, 224, 257, 301, 307,

320Chiral couplings, 114

441

442 Index

Christoffel symbol, 42, 59, 60, 61, 70–73, 76,80

Coherent statesLPS operator and, 164near-field, 235single-mode, 206–207

Commutation relations, between free-fieldoperators, 208–211

Commutation relations, covariant matrices,314–316

Feynman photon propagator, 316–317Commutator relations and physical vacuum

state, 339–340Compton wave number, 28, 96, 128,

133–134, 367Feynman photon propagator and, 309for transverse photon, 132

Conductivity, transverse and longitudinalresponse functions, 264–265

Conductivity tensor, 130–131Conservation laws, 4, 21–22, 24

one-photon energy, 152for photons, 144

Constitutive relations for curvedspace-time, 44, 79, 85–87

permittivity and permeability for staticmetrics, 88–89

permittivity and permeability inexpanding universe, 89–90

Constitutive relations in Minkowskianspace, 87–88

Contravariant four-current density, 34, 44,84

evanescent fields and, 221–223, See alsoEvanescent fields

Contravariant vectors or tensors, 29–30,67–68

covariant derivative, 70, 82covariant divergence, 80–81four-divergence of, 30four-vectors and four-tensors, 31–33vacuum Maxwell equations in General

Relativity, 79Correlation matrices, xvi, 187, 192–196,

280, 317entanglement and two-photon matrix,

383photon detection probability density,

384, 422–423photon propagators and, xvii, 311

commutator and covariantquantization, 314–316

covariant correlation matrix, 313–317T-photon time-ordered correlation ofevents, 311–313

single-photon wave packets, 163–164supermatrix, 202–203transverse photon propagator and,203–204

two-photon wave packet correlations,397–400

Young-type interference experiment,198–201

Covariance, 5, 33–35, 323Dirac equation and relativistic spin 1/2

current density, 299–301longitudinal and scalar photons in, 3,

5, 102, 120–122, See alsoLongitudinal photons; Scalarphotons

potential in Minkowski space, 127Principle of General Covariance, 43,

44, 69, 95quantized electromagnetic field, 161

Covariant correlation matrix, 313–314commutator and covariant

quantization, 314–316Feynman photon propagator, 316–318Jordan-Pauli scalar propagator, 317

Covariant curl in Riemann space, 80–81Covariant derivative, 43–44, 69–70

extension of free Maxwell equations tocurved-space time, 82

gauge-covariant derivative, 44, 95–96generalization of Lorenz condition to

curved space-time, 92generalization of Maxwell-Lorentz

equations to curved space-time, 72metric compatibility, 76parallel transport, 70–71vacuum Maxwell equations in General

Relativity, 79Covariant divergence in Riemann space,

80–81Covariant potential, 332–337, 345Covariant vectors or tensors, 29–30, 36–37,

44, 68, 79, 83Creation operators, 168, 169, 173, 184, 197,

315, 345, 348, 391, 401commutation relations, 337

Current density, See also Four-current

Index 443

density; Four-potential photonwave mechanics; Transitioncurrent density; Transverse currentdensity

breathing mode, xviidiamagnetic (or diamagnetic

transition), xvii, 132, 279,284–285, 289, 290–292, 294

dynamical equations for E and A, 129electric dipole, 288field-spinless boson interaction,

133–135four-potential description for

evanescent fields, 221–223, Seealso Evanescent fields

free Maxwell equations in curvedspace-time, 84

microscopic ML equations in curvedspace-time, 84–85

Minkowski current density four-vector,44

paramagnetic (or paramagnetictransition), 279, 284–286, 293

parts of gauge independent currentdensity, 279

screened canonical particle momentum,272–273

sheet model, 216, 223, See also Sheetcurrent density

single-particle (electron) transitioncurrent density, 283–286

spin current density in weaklyrelativistic Pauli theory, xvii, 280,301, 303–306

T-photon source domain, xvi, 130–131transverse and longitudinal response

theory in matter, 264–265transverse vector potential source

domain, 257vectorial scattering problem, 237–238

Current density, spin 1/2 in relativisticquantum mechanics, 297

continuity equations for spin and spacefour-current densities, 306–307

covariant form of Dirac equation,299–301

Dirac matrices, 297–299Gordon decomposition of Dirac

four-current density, 280, 301–302weakly relativistic spin current density,

303—306

Curved space-time, 67, 79, See also Generalrelativity, optics of

constitutive relations, 44, 79, 85–87covariant curl and divergence, 80–81covariant derivative, 69–70gauge-covariant derivative, 95–96

dynamical field equations, 85eikonal theory and null geodesics,

91–94, See also Eikonal theoryEinstein field equations, 43, 74–75electrodynamics in potential

description, 91–94generalization of Maxwell-Lorentz

equations, xv, 43, 72, 127geodesic deviation of light rays, 76–78infinitesimal squared distance, 55, 68light rays in, 52–53massive test particle motion, 43Maxwell equations with E, B, D, and

H fields, 83–84Maxwell equations with normal

derivatives, 81–83Maxwell-Lorentz theory extension to,

xv, 43metric compatibility, 76microscopic Maxwell-Lorentz

equations, 84–85parallel transport, 70–71as passive medium, 49permittivity and permeability for static

metrics, 88–89permittivity and permeability in

expanding universe, 89–90potential formulation, 44, 91–94Riemann curvature tensor, 71–74tensor fields, 67–68time-like metric geodesics, 56–59vacuum Maxwell equations in General

Relativity, 79

Dd’Alembertian operator, 9, 33, 207de Broglie relations, 43, 381Delta function identity, 157Density matrix operator, 193Deviation vector, 77Diamagnetic (or diamagnetic transition)

current density, xvii, 132, 279, 284,289, 290–292, 294

Diamagnetic interaction, 102, 103, 130, 326,363

444 Index

Dielectric constitutive relation, 87Dielectric tensor, 86, 87–88Dielectric-vacuum system, optical tunneling

process and, 245–246Dirac conjugate wave function, 280, 300Dirac delta function, 26, 110, 152, 257, 330

commutation relations betweenfree-field operators, 209–210

expansion of sheet current density, 216,223–224

far-field part of two-photon two-timeswave function, 426

Huygens propagator in wavevector-frequency domain, 247

plane-mode decomposition of covariantpotential, 334

retarded scalar propagator, 247Dirac equation, 96, 101, 113–114, 115,

297–299covariant form of, 299–301Gordon decomposition of four-current

density, 301–302weakly relativistic spin current density,

303Dirac four-current density, 301

Gordon decomposition of, 301–302Dirac matrices, 297–299Divergence-free fields, 3, 4, 7, 8–9, 22, 30Doppler shift, 37, 325, 337Duality transformation, 38Dual vectors, 29Dyadic transverse photon propagator, 251,

256Dynamical field equations, 129, 180, 366

free Higgs field, 371generalized Landau-Peierls-Sudarshan

equations, 207

EEffective canonical particle momentum, 273Eikonal equation, 48, 94

Hamilton-Jacobi form of Newton’ssecond law, 42, 50

Eikonal theory, xiv–xv, 41–42, 44, 48–51,61, 94, See also Geometrical optics

Einstein, Albert, xivlight quanta hypothesis, 18, 50, 99

Einstein A-coefficient, 408Einstein causality, 88, 241, 242, 256Einstein field equations, 43, 74–75Einstein formula (E = mc2), 86

Einstein-Podolsky-Rosen (EPR) critique ofquantum mechanics, 389

Einstein tensor, 74–75Electodynamics in curved space-time, See

Curved space-timeElectric-dipole (ED) approximation,

402–403, 425Electric dipole current density, 288Electric field [E(r, t)], See also

Electromagnetic fieldbase vector transformation, 14–15classical electromagnetics (optics) in

free space, 4, 7divergence-free and rotational-free

vector field, 8–9, See alsoLongitudinal vector fields;Transverse vector field

dynamical equations for E and A, 129field equations in inhomogeneous

vacuum, 47free-field commutators, 209–210geometrical analysis of polarization

state, 15–18Lorentz transformation of, 5, 35–38near-field and gauge photons, 123no independent existence from B, 3, 7,

36quantization of vector potential, 161,

See also Quantizedelectromagnetic field

Electromagnetic energy density, 22Electromagnetic field, See also Electric

field; Field-matter interactions;Magnetic field; Quantizedelectromagnetic field

classical electromagnetics (optics) infree space

analytical signals, 10–12, 24–26, Seealso Analytical signals

base vectors transformation, 14–15conservation laws, 21–22, 24Maxwell equations and waveequations, 7–8

monochromatic plane-waveexpansion, 13–14

polarization states, 14–18, See alsoPolarization

Riemann-Silberstein (RS) formalism,22–24

transverse and longitudinal vectorfields, 8–10

Index 445

wave-packet modes, 18–21, See alsoWave-packet modes

classical electromagnetics in free space,4

differentiable everywhere insemiclassical electrodynamics, 8

divergence-free vector field, 3energy of transverse electromagnetic

field, 166equivalent mass (E = mc2), 86free field concepts, 3global vacuum concept and, xivlocal gauge invariance, 95monochromatic plane-wave expansion,

13–14, 171–173relativistic coupling of spin 1/2

particle, 279–280wave-packet modes, xiv, See also

Wave-packet modesElectromagnetic surface waves, 216Electronic transitions, 1s ⇔ 2s, 279,

286–292Electron neutrino, 117Electron velocity operator, 145Energy conservation equation, 21Energy flux density, 22Energy-momentum relation for plane-wave

photons, 13Energy-momentum relation for relativistic

particle, 128, 297Energy-momentum tensor, 43, 75, 113, 326Energy of single photon, 152–153Energy of transverse electromagnetic field,

166Energy operators, quantized transverse

field, 168–171Energy wave function, See Photon energy

wave functionEntanglement, xvii, 189, 382–384, See also

Two-photon phenomenaentangled (quantum) states, 382, 384,

390, 424entangled polarization states, 390–392measurement on two-part physical

system, 387–389spontaneous two-photon cascade

emission, 383Euler-Lagrange equations, 53, 57, 74–75Evanescent fields, 216–217, 227–228

field-quantized approach, 231–233

four-potential description in the Lorenzgauge, 221–223

four-potential photon wave mechanics,229–231

frustrated total internal reflection andphoton tunneling, 237

Heisenberg equations of motion fornear-field annihilation operators,217, 232, 233, 234–235

incident fields generating evanescenttunneling potentials, 243–246

incident single-photon wave function,248–249

near-field part of T-photon wavefunction, 417

sheet current density, 223–225T-photon wave function component,

383transverse, longitudinal, and scalar

potentials, 225–228Expanding universe, permittivity and

permeability in, 89–90Exponential decay rates, 419Exponential T-photon source confinement,

xvii, 279, 287, 292, 305

FFar-field photon detection schemes, 215Fermat’s principle, 42, 53, 56Fermi-Dirac statistics, 379Fermions, 379

two-fermion state, 380Fermi problem, 412Feynman meson propagator, 138Feynman photon propagator, xvii, 103, 280,

309–311, 316–318, 375longitudinal and scalar photon

propagators, 318massive photon propagator and, 138,

309near-field free-field commutators, 210

Field-matter interactionscovariant formalism, See Covarianceentangled states, 189, See also

Entanglementfar-field photon detection schemes, 215field commutation relations and, 209four-current density and, 102geometrical optics and the “light

particle,” 45

446 Index

Gupta-Bleuler-Lorenz condition andphoton vacuum state, 338, 341

longitudinal and scalar photons and, 3,102, 121–122

massive vector boson field, 132–136mean position states and, 178, See also

Mean position statesphoton embryo concept, 409polychromatic single photon state, 174possibility of transverse photon

effective mass, xv–xviquantal photon gas and, 382single-atom field detectors, 163single-photon emissions and time

development of, 215spatial resolution problem, 277vector and scalar potentials, 101

Field-quantized electromagnetic field, SeeQuantized electromagnetic field

Flat Robertson-Walker metric, 89, 90Fock states, 161, 173, 338, 345–346, 348,

354, 356, 393Foldy-Wouthuysen wave mechanics, 145Four-current density, 34, 102, 122, 133–135,

217, 237, 329–332, See alsoCurrent density; Four-potentialphoton wave mechanics

continuity equations for spin and spacefour-current densities, 306–307

field emission from classicalpoint-particle, 255–259

four-potential of point-particle inuniform motion in vacuum,260–262

Gordon decomposition, 280, 301–302relativistic spin 1/2 current density, 301T-photon and antiphoton, 332

Four-dimensional delta function, 84Four-dimensional Riemann space, 55–56Four-divergence of contravariant vector, 30Four-momentum, 71, 103

Feynman photon propagator, 309–311kinematic four-momentum operator, 95photon wave-packet, 153–155

Four-potential photon wave mechanics, 363,See also Four-current density

evanescent fields and, 229–233, 256field emission from classical

point-particle, 255–259free space wave equation, 102

gauge transformations, 332–333,341–343

longitudinal and scalar parts in freespace, 323, See also Longitudinalpart of electric field; Scalarpotential

Lorentz boost and, 323–324oscillatory character, 263outward propagation from given

electronic source, 280physical (T-photon) vacuum, 337–341plane-mode decomposition of covariant

potential, 333–337, 345point-particle in uniform motion in

vacuum, 260–262relativistic coupling of spin 1/2

particle, 279–280relativistic spin 1/2 current density,

299–301transverse and longitudinal response

theory in matter, 263–265vacuum state considerations, 346

Four-tensors, 31–33Four-vectors

covariant four-vector, 36–37Lorentz transformation, 31–33Minkowski current density, 44mode functions, 336–337

Four-wave vector (covariant four-vector),36–37

Free-falling inertial observers, xvii, 325, 350Free-field operators, 205

commutation relations, 208–211generalized Landau-Peierls-Sudarshan

equations, 207–208Maxwell operator equations, 205–207

Frustrated total internal reflection (FTIR),41, 217–219, 237–238, 241, 251–253

GGalileo group, 29Gauge bosons, 102Gauge-covariant derivative, 44, 95–96Gauge photon, 102, 122–125Gauge transformation, 92, 105–107,

124–125, 134–135, 323, 332–333,341–343

unitary gauge, 370–371Generalized polarization and magnetization

fields, 85General Relativity, optics of

Index 447

eikonal equation, 42, 48, 50, 94, Seealso Eikonal theory

equation of motion for masslessparticle, 62–63

free-falling inertial observers, xvii, , 350geodesics, 42–43, 52–53four-dimensional Riemann space,55–56

gravitational redshift, 62–65metric tensor, 55–56Newtonian limit and weak staticgravitational field, 59–61

null geodesics and “light particles,”61–62

time-like, 56–59light particle propagation, xv, 42–43light rays in curvilinear space-time,

52–53local inertial frame, 56Maxwell-Lorentz equations, 44, See

also Maxwell-Lorentz equationspotential formulation, 44vectors and tensors, 29–30

General Relativity, space-time structure of,67, See also Curved space-time

constitutive relations, 85–87covariant derivative, 69–70Einstein field equations, 43, 74–75energy-momentum tensors, 43, 75geodesic deviation of light rays, 76–78metric compatibility, 76microscopic ML equations in curved

space-time, 84–85parallel transport, 70–71Riemann curvature tensor, 71–74tensors, 67–68vacuum Maxwell equations, 79

Geodesic deviation of light rays, 76–78Geodesics, xv, 42–43, 52–53, 55–65, See also

Riemann curvature tensorcovariant derivative, 69–70four-momentum, 71Newtonian limit and weak static

gravitational field, 59–61null geodesics, xv, 42, 55, 61–62, 90, 94relative acceleration and, 77–78time-like, 43, 56–59

Geodetic line, 52–53Geometrical optics, xiv–xv, 41–51, 99, See

also Eikonal theoryeikonal equation, 42, 48, 50, 94

geometrical wave surfaces and rays,48–51

inhomogeneous vacuum concept, 46–47light rays in curvilinear space-time, 94parallel transport, 71polarization states, 45–46

Geometrical wave surfaces, 48Goldstone bosons, 326, 368, 374Goldstone model, 364–371, 375Gordon decomposition of Dirac four-current

density, 280, 301–302Gravitational fields, 44

covariance and, 95geodesics, 59–61

Gravitational redshift, xv, 43, 62–65Green function, 25, 218, 222, 244, 383,

411–413, 415, 421–422Ground state, xvii, 173, 363–365Gupta-Bleuler-Lorenz condition, 338, 341Gupta-Bleuler theory, 315–316Gyrotropic constitutive relation, 86

HHamilton-Jacobi form of Newton’s second

law, 42, 50Hamilton operators/Hamiltonians

canonical quantization of transverseelectromagnetic field, 168–169

dynamical equation for photon wavefunction in direct space, 180

electric-dipole approximation, 402–403interaction Hamiltoniansingle-photon state observation,190–192

spontaneous single-photon emission,402–405

spontaneous two-photon emission,418

Maxwell operator equations and, 206photon vacuum and, 337polychromatic single photon state, 175propagation of analytical signal, 25quantum mechanical wave equation for

transverse photon helicity species,112

spontaneous single-photon emission,401–406

time evolution operator forsingle-photon wave-packetcorrelations, 187–188

Harmonic oscillator, 166–167

448 Index

Heisenberg equations of motion fornear-field annihilation operators,217, 232, 233

Heisenberg Picture, 174, 191, 202, 217, 235Heisenberg uncertainty relations, 381Helicity basis, 172, See also Polarization

antiphoton and, 115Cartesian representation of angular


), 149four-momentum of photon

wave-packet, 154monochromatic plane-waves, 4neutrino wave mechanics, 101, 114–115new T-photon mean position states,

179photon conservation law, 144photon wave mechanics, 101polychromatic single photon state,

176–177transverse photon as spin-1 particle,

110–112two-photon two-times wave function,

395two-photon wave packet correlations,

397Helicity basis vectors, 110Helmholtz’s theorem, 9, 21Hermitian operators

canonical quantization of transverseelectromagnetic field, 167–168

new T-photon mean position states,178

observable defined, 386photon velocity, 144

Higgs bosons, 371Higgs field, 102, 133, 326, 367–369, 375

Abelian Higgs model, 369–371, 375free form approximation, 373U(1) Higgs model, 368–371

Higgs particle model, 326Hilbert space

entangled photon polarization states,390

new T-photon mean position states,177–179

polychromatic single-photon theory,xvi, 162

tensor product space and quantummeasurement theory, 385–386

Hilbert transform pair (conjugate pair), 4,12

Homogeneous Lorentz transformation, 29Huygens scalar propagator, xvii, 122, 215,

221, 239–240, 246–247, 255, 309longitudinal and transverse

propagators and, 319massless Feynman photon propagator

and, 309–311retarded scalar propagator, 221–222,

239, 246–247in wave vector-frequency domain, 247,

260Huygens’ wave theory of light, 42Hydrogen 1s ⇔ 2s transition, xvii, 279,

286–289two-level breathing mode dynamics,

292–295

IInertial observers, xvii, 5, 262, 323–326, See

also Observer and observationconsiderations

four-vector mode functions, 336–337free-falling, xvii, 325, 350Lorentz and Lorenz-gauge

transformations of four-potential,332–333

Minkowskian space and, xvii, 323, 325,350

physical vacuum and, 337–343plane-mode decomposition of covariant

potential, 333–337Infinitesimal squared distance in curved

space, 55, 68Inhomogeneous vacuum, 42, 46–47, 49,

52–53, 99, 219evanescent, 222, See also Evanescent

fieldsphoton tunneling-coupled sheets,

250–251Interaction Hamiltonian, 190–192, 402–405,

418Interaction LaGrange, 367–368Interaction Picture, 187–192, 202, 403,

405–407, 418, 419Interference effects, xiv

field correlations in photon meanposition state, 201–204

single-photon states, 164, 197–198between transition amplitudes, 201two-photon interference, xvii, See also

Two-photon phenomena

Index 449

wave phenomena in classical optics, 41Young-type double-source experiment,

164, 198–201, 323Inverse coordinate transformation, 67–68

JJordan-Pauli scalar propagator, 317

KKinematic four-momentum operator, 95Kinetic and potential energy of light

particle, 43, 64–65Klein-Gordon equation, 102, 128, 133, 135,

297, 367–368Kramers-Kroenig relations, 245

LLaGrange density, 366–371, 374–375Lamb shift, 323, 407Landau-Peierls-Sudarshan (LPS) equation,

25, 164, 207–208Landau shock waves, 220, 272Leptonic electron, 117Leptonic interactions, 114Levi-Civita symbol, 34, 35, 110, 113Lewis, G. N., 45Lienard-Wiechert formula, 259Light particle, See Particle-like photonLight quanta hypothesis (Einstein), 18, 50,

99Light rays, 48–51

in curvilinear space-time, 52–53, 94eikonal theory and, See Eikonal theoryFermat’s principle, 53, 56geodesic deviation of, 76–78as particle trajectories, xv

Light speed, See Speed of lightLinear and space-time nonlocal microscopic

constitutive equation, 130Linear connection, 69–73, 76Linear response theory, 102, 252Local-field effects and dielectric tensors,

87–88Local gauge invariance, 44, 95Local inertial frame, 56, 63Localization of the photon, See Spatial

localization of the photonLongitudinal conductivity response

functions, 264–265Longitudinal delta function, 240–241

Longitudinal field momentum density,270–273


Longitudinal part of electric field, 36–37,323

evanescent fields and, 224–225,227–228, 231

Longitudinal photon propagator, xvii, 141,318–320

Longitudinal photons, 3, 5, 36, 101–102boosts and, 336covariant correlation matrix and, 314covariant field-matter interaction, 3,

102, 120–122Goldstone bosons, 374neutralization in free space, 120–122observation considerations, 323–324vacuum state considerations, 338–339wave equations, 119–120

Longitudinal polarization vector, 140, 318Longitudinal transition current density, 287Longitudinal vector fields, 8–9, 256Lorentz boost, xvii, 103, 140, 323–325,

333–337Lorentz invariant, xv, 36, 103, 155–157, 162,

175, 179, 246, 323, 333–334, 395,See also Lorentz transformation

Lorentz transformation, 5, 27–29of covariant and gauge potentials,

332–333duality transformation, 38form invariance of Maxwell-Lorentz

equations, 5, 33, See alsoCovariance

four-potential of point-particle inuniform motion in vacuum, 262

four-vectors and four-tensors, 31–33homogeneous, 29massive photon, xvi, 140Poincare (inhomogeneous) group, 29relativistic scalar product, 155–157of Riemann-Silberstein vectors, 38tensors, 29–30of (transverse) electric and magnetic

fields, 35–38Lorenz condition, 363

evanescent fields and, 221generalized to curved space-time, 92–93L- and S-photon neutralization in free

space, 120

450 Index

massive photon propagator and, 139,374

near-field and gauge photons, 123Proca equation and, 102, 128, See also

Proca equationLorenz gauge, 72, 92–93, 102, 119–120,

124–125, 128, 129, 256transformations of four-potential,

332–333, 341–343

MMagnetic field [B(r, t)], See also

Electromagnetic fieldclassical electromagnetics (optics) in

free space, 4, 7divergence-free and rotational-free

vector field, 8–9, See alsoLongitudinal vector fields;Transverse vector field

field equations in inhomogeneousvacuum, 47

free-field commutators, 209–210Lorentz transformation, 5, 35–38no independent existence from E, 3, 7,

36quantization of vector potential, 161,

See also Quantizedelectromagnetic field

Zeeman effect and spin current densityin weakly relativistic regime,305–306

Majorana particle, 115Many-body conductivity tensor, 130–131Markow approximation, 383, 407Massive photon field, 127, 132–136, See also

Mass of photondiamagnetic interaction, 130dynamical field equations, 129Proca equation, 128, 136

Massive photon propagator, 103, 136–141,309, 373–375, See also Photonpropagators

Mass of electromagnetic field (E = mc2), 86Mass of photon, See also Massive photon

fieldcovariant four-photon context, 128Goldstone bosons, 368, 374possibility without destroying gauge

invariance, 363Proca equation and massive photon

propagator, 103, 136–137, 138,373–375

’t Hooft gauge and propagator, 375transverse photon effective mass,

xv–xvi, 102–103, 116, 130, 326, 363U(1) Higgs model, 368–371vacuum state, xvii, 102, 326Goldstone and Higgs models,364–371, 375

spontaneous symmetry breaking,363–365, 368, 372

’t Hooft gauge and propagator, 375vacuum screening current, 372–373

Maxwell equations, xvclassical electromagnetics (optics) in

free space, 4, 7–8positive-frequency wave packetsolutions, 10–12, See alsoWave-packet modes

Riemann-Silberstein vectors, 22–24extension to curved space-time

(General Relativity), 79E, B, D, and H fields, 83–84normal derivatives, 81–83vector potential description andeikonal theory, 91–94

form invariance for Lorentztransformation, 5, 33–35, See alsoCovariance

Maxwell-Lorentz equations in GeneralRelativity, 44

Maxwell operator equations and,205–207

monochromatic plane-wave expansion,13–14

Riemann-Silberstein (RS) vectors, 4,22–24

Maxwell-Lorentz equations, 5charge conservation condition, 93covariant derivative, 43–44dynamical equations for E and A, 129extension to curved space-time, xv, 43,

72, 84–85, 127field-spinless boson interaction,

133–135form invariance for Lorentz

transformation, 5, 33, See alsoCovariance

“light particles” in classical physics, 41,45–47, See also Geometrical optics

Index 451

microscopic equations in curvedspace-time, 84–85

particles as point-like entities, 7propagation of point-particle through

matter, 263–265Maxwell operator equations, 205–207Mean position states, xvi, 164, 177–179,

201–204, 248, 311, 331, 382, 394,420

Measurement and measurement theory, 209,325–326, 360, See also Observerand observation considerations;Photon detection; Spatiallocalization of the photon

Bohr and, xiii–xiv, 99–100entangled states and, 382, 390–392gravitational redshift and, 63near-field interactions and photon

tunneling considerations, 237observable defined, 386physical photon vacuum state

description, 350, See also Photonvacuum

single-atom field detectors, 163, 190spatial resolution limit problem,

277–278tensor product space, 385–389, 391–392test particle concept, 56, 208–209two-part physical system, 387–390, See

also Entanglementvectorial scattering problem, 237–238wave-packet reduction, 387

Meissner effect, 103Metric compatibility, 73Metric geodesic equation, 56

parallel transport, 71time-like metric geodesics, 56–59

Metric tensors, 33, 44constitutive relations for curved

space-time, 86covariant and contravariant field tensor

relations and, 83–84covariant derivative, 69–70covariant vector definition, 68extension to four-dimensional Riemann

space, 55–56Riemann curvature tensor, 71–74

geodetic line and, 42Goldstone model, 366gravitational potential, 59

light rays in curvilinear space-time,52–53

massive photon propagator and, 137massless Feynman photon propagator

and, 309Minkowski tensor, 33, 55–56, 161static space-time system, 88–89time-like metric geodesics, 56–59

Microscopic susceptibility, 46Minimal substitution rule, 72, 93Minkowski current density four-vector, 44Minkowski space, 28, 55, 350, 352–361

constitutive relations in, 87infinitesimal squared distance, 55–56metric tensor and four-dimensional

Riemann space, 55observers and, xvii, 323, 325, 350Unruh effect, 352–361

potential and covariant field equationsin, 127

Rindler coordinatization, 352–354single-photon quantum optics, xvi, See

also Single-photon wave-packetstates

Minkowski tensor, 33, 55–56, 161, See alsoMetric tensors

Moment of energy conservation, 21Momentum, See also Angular momentum;

Four-momentumCerenkov field, 220conservation equation, 21energy-momentum (or stress) tensor,

43, 75, 113, 326energy-momentum relation for

plane-wave photons, 13energy-momentum relation for

relativistic particle, 128, 297geometrical optics and the “light

particle,” 42Heisenberg uncertainty relations, 381kinematic four-momentum operator, 95monochromatic photon

four-momentum, 71operators and quantized transverse

field, 170polychromatic single photon state, 176screened canonical particle momentum,

272–273transverse and longitudinal parts of

Cerenkov field, 269–272Monochromatic photon four-momentum, 71

452 Index

Monochromatic plane waves, xvi, 4eikonal theory, 51expansion of electromagnetic field,

13–14, 171–173expansion of transverse vector potential

operator, 168, 205, 330–331free-space wave equation, 205geometrical analysis of polarization

state, 15–18geometrical optics and the “light

particle,” 42incident fields generating evanescent

tunneling potentials, 243–246wave-packet basis for one-photon

states, 183–184wave-packet modes, 18–21

Muons, 117

NNear-field and far-field parts of transverse

vector potential, 257–259, See alsoNear-field photon emission in 3D

Near-field annihilation operator, SeeAnnihilation operators

Near-field free-field commutators, 210Near-field part of transverse photon wave

function, 414–417Near-field photon emission in 3D, 219, 427,

See also Photon emissionfield emission from classical

point-particle, 255–259four-potential of point-particle in

uniform motion in vacuum,260–262


transverse and longitudinal parts ofCerenkov field, 269–272

transverse and longitudinal responsetheory in matter, 263–265

transverse Cerenkov phenomenon,266–269

Near-field photon phenomena, xv, 3, 102,122–123, See also Photontunneling; Rim zone

atoms in near-field contact, 278coherent states, 235current density sheet electrodynamics,

251, See also Sheet current densityevanescent fields, 216–217

fingerprint for spatial localizationproblem, 414–415

Heisenberg equations of motion forannihilation operators, 234–235

non-propagating evanescent part, 383photon and atomic electron

distribution source domains, 215photon emission in 3D, 219, See also

Near-field photon emission in 3Dphoton tunneling and apparent

superluminality, 218, 241–243scattering and photon tunneling,

237–243spatial resolution problem, 277two-photon entanglement and spatial

localization problem, 426Neutrino velocity operators, 144Neutrino wave mechanics, 101, 113–118Newton, Isaac, 41Newtonian gravitational field, geodesics,

59–61Newton’s second law, Hamilton-Jacobi form

of, 42, 50Nirvana state, xviiNull geodesics, xv, 42, 55, 61–62, 71, 90, 94Null hypersurfaces, 94

OObservable, 386, 387Observer and observation considerations,

xvii, 323–327, 332–333, See alsoInertial observers; Measurementand measurement theory

Bohr on registered phenomena, xiii, 63,101, 162, 178, 197, 350

non-unique vacuum, 348–351physical photon vacuum state, See

Photon vacuumPlanck (thermal) spectrum, 325, 352,

360–361Rindler particles in Minkowski vacuum,

354–361Rindler space and, 352–354single-photon states, 189–193transverse photon manifestation, 3Unruh effect, 352–361

Oppenheimer photon wave equations, 103,143–146

Optical length of a curve, 49, 52, 53Optical tunneling, See Photon tunneling

Index 453

PParallel transport of a tensor, 70–71Paramagnetic (or paramagnetic transition)

current density, 279, 284–286, 293Parity violations in weak interactions, 114Parseval-Plancherel identity, 153, 269, 294Particle-like photon (“light particle”), xiv,

45–47, 99, See also Geodesicsbeam splitter experiment, 100fclassical physics and, xiv, 41–44,

99–100, See also Eikonal theory;Geometrical optics

eikonal theory, 48–51Einstein’s light quanta, 18, 50, 99field emission from classical

point-particle, 255–259inhomogeneous vacuum concept,

46–47, 99, See also Inhomogeneousvacuum

kinetic and potential energy of, 43,64–65

tunneling and, 217, See also Photontunneling

vectorial quantum mechanical waveequation, 101, See also Photonwave functions; Wave mechanics

wave functions and, 100–103Particle motion in classical physics, 41–42

eikonal theory, 49–50time-like geodesic equation, 43

Particle motion in curved space-time, Seealso General relativity, optics of;Geodesics

Newtonian limit and weak staticgravitational field, 59–61

parallel transport, 71time-like metric geodesics, 56–59

Particlesimaginary mass, 367monochromatic plane-wave expansion,

172point-like entities in microscopic

Maxwell-Lorentz theory, 7test particle concept, 56, 208–209

Pauli, Wolfgang, 114Pauli exclusion principle, 379Pauli spin matrices, 113–114, 151Pauli spin vector (σ) operator, 103, 143Permittivity and permeability

in expanding universe, 89–90for static metrics, 88–89

Phenomenon, according to Bohr, xiii, 3, 63,101, 162, 178, 197, 350, 424

Photon, as particle, See Particle-like photonPhoton-atom interaction, 190Photon beam splitter experiment, 100fPhoton conservation law, 144Photon correlation matrices, See

Correlation matricesPhoton detection, 323, 360, See also

Measurement and measurementtheory; Observer and observationconsiderations

far-field schemes, 215free-falling observers, xvii, 325, 350joint probability density, 383–384,

422–423single-electron atom system, 190

Photon embryo, xvi, 174, 241, 409Photon emission

breathing mode transitions, xvii, 279,285–286, 289–295

entanglement and two-photon cascadeemission in three-level atom, 383

evanescent fields, 216–217Fermi problem, 412field emission from classical

point-particle, 255–259near-and far-field contributions,

257–259, 427, See also Near-fieldphoton emission in 3D

photon embryo concept, 409, See alsoPhoton embryo

polychromatic photon, xvirotating wave approximation, 404–406screened canonical particle momentum,

272–273spontaneous single-photon emission,

401genuine transverse photon wavefunction, 411–413

Hamiltonian in the electric dipoleapproximation, 401–406

rim zone, 413–417wave function in q-space, 409–410wave function in r-space, 410–411Weisskopf-Wigner theory, 406–408,413

time development of field-atominteraction, 215

transverse photon propagator, 310, See

454 Index

also Photon propagators;Transverse photon propagator

two-photon spontaneous emissions,xviii

Photon energy wave function, xvi, 4, 23,103, 150

angular momentum operator (∑

),143–150

antiphoton energy wave function, 151bispinor wave functions, 115–116,

150–153four-momentum of photon

wave-packet, 153–155one-photon energy, 152Oppenheimer light quantum theory,

103, 143–146relativistic scalar product, 103, 155–157spin of photon wave-packet, 153–155

Photon field propagation, xviPhoton gas, 379–384Photon in free fall, 62–65Photon mass, See Mass of photonPhoton probability density, 151Photon propagators, xvii, See also Feynman

photon propagator; Huygens scalarpropagator; Transverse photonpropagator


correlation matrices and, xvii, 203–204covariant correlation matrix, 313–314covariant quantization of

electromagnetic field, 314–316dyadic transverse propagator and

photon tunneling-coupled sheets,251

Jordan-Pauli scalar propagator, 317longitudinal and scalar propagators,

318–320massive photon, 103, 136–141photon tunneling and, 239–240retarded propagator, 221–222, 239,

246–247, 309, 413spontaneous single-photon emissions,

413–417’t Hooft propagator, 326, 375T-photon time-ordered correlation of

events, 311–313Photons, See also Longitudinal photons;

Scalar photons; Transversephotons

angular momentum, 143–146antiphoton and, 11, 115, 151, 204, 329classical versus quantum, 161free photon as abstraction, xiv, 63, 99,

101gauge, 102, 122–123near-field, 122–123, See also Near-field

photon phenomenaone-photon energy, 152photon phenomenon, xiii–xivphysical (or real), 3virtual, xv, 3, 101–102

Photon source domains, xvi, 277–278, Seealso Super-confined T-photonsources

1s ⇔ 2s transition, 279, 286–289breathing mode transitions, xvii, 279,

285–286exponentially confined, xvii, 279, 287,

292fields in classical electrodynamics, 8photons and atomic electron

distribution, 215rim zone as, 9, 277–278, See also Rim

zonesuper-confined T-photon, xvii, 278–280transverse current density as, xvi–xvii,

130, 255, 257, 279, 283–286, 310Photon spatial localization, See Spatial

localization of the photonPhoton tunneling, 217–219

apparent superluminality, 218, 241–243dielectric-vacuum system and, 245frustrated total internal reflection

(FTIR), 217–219, 237–238,241–242, 251–253

incident fields generating evanescenttunneling potentials, 243–246

incident polychromatic single-photonstate, 247–249

near-field interaction and scatteringproblem, 237–238

scattering from single current-densitysheet, 218, 238–243

sheet current density and, 217, 223photon tunneling-coupled sheets,250–253

Photon vacuum, xvi, xvii, 113, 169,337–341, See also Vacuum state

Bogolubov transformation, 325,347–349

Index 455

coherent states, 207commutator relations, 339–340Gupta-Bleuler-Lorenz condition, 338,

341photon mass and, xvii, 102, 326photon observation considerations,

323–325physical phenomena in, 323quantized transverse field, 169spontaneous symmetry breaking,

363–365Photon velocity operators, 144–145Photon wave functions, 100–103, 107–109,

See also Photon energy wavefunction; Wave mechanics

bispinor wave functions, 115–116,153–156, 303

Dirac conjugate wave function, 280, 300dynamical field equations, 129, 180evanescent component, 383evanescent fields and, 221, See also

Evanescent fieldsfingerprint for spatial localization

problem, 414–415four-momentum, 153–155gauge transformation, 124–125incident polychromatic single-photon

state, 247–249Landau-Peierls-Sudarshan equations,

25, 164, 208longitudinal and scalar photon wave

equations, 119–120near-field and gauge photons, 122–125one-photon energy, 152–153photon and atomic electron

distribution source domains, 215polychromatic single photon state,

179–180p-polarization, 248–249quantized electromagnetic field, 162single-photon correlation matrix,

194–196single-photon wave-packet mode

interference, 197spin of photon wave-packet, 155spontaneously emitted photon, 383general transverse photon wavefunction, 411–413

wave function in q-space, 409–410wave function in r-space, 410–411

tentative function and equations forT-photons, 107–109

total internal reflection (TIR), 219T-photon mean position state,

177–179, 248, 331, See also Meanposition states

T-photon source domain and, 278transverse and longitudinal response

theory in matter, 264–265transverse scalar wave functions, 162two-photon correlations, 397–400two-photon Schrodinger equation in

direct space, 396–397two-photon two-times wave function,

382–384, 393–395, 400, 420–427Photon wave mechanics, See Wave

mechanicsPhoton wave-packets, 4, 162–164, 218

atomic and field correlation matrices,192–193

basis for one-photon states, 183–184extension to multi-photon wavepackets, 184–186

four-momentum, 153–155polychromatic single photon, 173–177reduction of the wave packet, 387Riemann-Silberstein vectors, 24scattering from single current-density

sheet, 218, 238–243time evolution operator, 186–189total spin, 155wave-packet mode interference,

197–198Physical photon, 3, See also Transverse

photonsPhysical photon vacuum state, See Photon

vacuumPlanck’s constant (h), xiv, xv, 161Planck spectrum, 325, 352, 360–361Plane-mode decomposition of

four-potential, 345Poincar group, 22, 29Polaritons, 86Polarization (or polarization vectors), See

also Helicity basisbase vector transformation, 14–15Cartesian representation of angular


), 149commutator and covariant

quantization, 315

456 Index

entangled photon polarization states,390–392

field-quantized approach to evanescentfields, 232

four-vector mode functions, 336–337generalized polarization in General

Relativity, 85geometrical analysis, 15–18geometrical optics and microscopic

Maxwell-Lorentz equations, 45–46longitudinal and scalar photon

propagators, 318longitudinal polarization vector, 140,

318massive photon propagator and, 103,

139–140massless Feynman photon propagator

and, 309monochromatic plane-waves, 4new T-photon mean position states,

179photon wave mechanics, 101plane-mode decomposition of covariant

potential, 335–336p-polarization of vectorial wave

function in photon tunnelingcontext, 248–249

Riemann-Silberstein vectors, 23spontaneous single-photon emissions,

408transverse Cerenkov phenomenon, 267transverse photon as spin-1 particle,

110–112two-photon two-times wave function,

395two-photon wave packet correlations,

397Polarization entanglement, 382Polychromatic single photons, xvi, 162

monochromatic plane-wave expansion,171–173

new T-photon mean position states,177–179

non-orthogonality of T-photon positionstates, 181

photon tunneling and incident state,247–249

T-photon wave function and dynamicalequation, 179–180

wave-packet modes, 173–177, See alsoSingle-photon wave-packet states

basis for one-photon states, 183–184extension to multi-photon wavepackets, 184–186

Polychromatic two-photon state, 393–394,399, See also Two-photontwo-times wave function

Potential description of classicalelectrodynamics, 44, 91–94

Poynting vector, 22, 49Principle of Equivalence, 43, 61, 62Principle of General Covariance, 43, 44, 69,

95Proca equation, xv, 326, 372–373

massive photon field, 128, 136massive photon propagator and, 103,

136–137, 138, 373–375photon effective mass and, 363

Projection operator, 116, 154, 387, 388Propagation of analytical signal, 24–26Proper time invariance, 27–28

QQuantal photon gas, 379–384, See also

EntanglementQuantized electromagnetic field, xvi,

161–164, 217angular momentum operator, 170canonical quantization of transverse

electromagnetic field, 165–168commutator and covariant correlation

matrix, 314–316energy, momentum, and spin operators,

168–171expansion of transverse vector

potential operator, 168, 205field-quantized approach to evanescent

fields, 231–233Hamilton operators, 168–169Landau-Peierls-Sudarshan (LPS)

equation, 164Maxwell operator equations and, 206monochromatic plane-wave expansion,

13–14, 171–173multi-photon states and, 171new T-photon mean position states,

177–179physical phenomena in photon empty

space, 323Rindler coordinatization of Minkowski

space, 352–354Quantum measurement theory, See

Index 457

Measurement and measurementtheory

Quantum mechanics, 103, 161, See alsoRelativistic quantum theory


),143–150

Bose-Einstein statistics, 99, 379, 380differentiable fields in semiclassical

electrodynamics, 8Einstein-Podolsky-Rosen (EPR)

critique, 389explaining frustrated total reflection, 41Fermi-Dirac statistics, 379gravitational redshift, 43, 63single-particle (electron) transition

current density, 283–286spin 1/2 current density, 297–298, See

also Spin 1/2 particleQuantum number, 146–150Quantum photon, 161Quantum states, 390

eigenstate for near-field annihilationoperator, 217, 235

emitted one-photon wave-packet field,163

entangled states, 382, 384, 390, 424,See also Entanglement

generalizing nonrelativistic scalarproduct of, 155

Maxwell operator equations and, 205photon gas, 379single-photon states and, 177, 189, 193,

199, 210transition current density and, 285

Quasi-classical state, 206–207

RRabi frequency, 295Rayleigh criterion, 277Real photon, 3, See also Transverse photonsReduction of the wave packet, 387Reflection, 41, See also Frustrated total

internal reflectionRefraction, 41Refractive index, 49, 51, 52, 219Relativistic Doppler shift, 37, 325, 337Relativistic quantum theory

operators associated with conservationlaws, 22

spin 1/2 current density, 297continuity equations for spin and

space four-current densities,306–307


Dirac matrices, 297–299Gordon decomposition of Diracfour-current density, 280, 301–302

weakly relativistic spin currentdensity, 303—306

Relativistic scalar product, 103, 155–157,179

Relativistic spinless boson particleinteractions, xvi

Relativity theory, optics in, See Generalrelativity, optics of; SpecialRelativity, optics in

Retarded propagator, 221–222, 239,246–247, 309, 413

Ricci tensor, 44, 74, 75, 92Riemann curvature tensor, 44, 71–73

algebraic properties of, 73–74relative acceleration and, 77–78Ricci tensor, 44, 74, 92

Riemann-Silberstein (RS) vectors, 4, 22–24,103, 150–151

Lorentz transformation, 38Riemann space, 70

covariant curl and divergence, 80–81Rim zone, xvi, xviii, 3, 9, 216, 338, See also

Near-field photon phenomena;Photon tunneling

implications of absence of, 278light propagation speed, 218Lorentz transformation, 3near-field free-field commutators, 210spatial photon localization for different

inertial observers, 5spontaneous single-photon emissions,

413–417T-photon spatial source domain, 9,

277–278transition current density and 1s ⇔ 2s

transition, 288two-photon entanglement and spatial

localization problem, 426Rindler particles in Minkowski vacuum,

354–361Rindler space, 352–354Robertson-Walker metric, 44, 90Rotating wave approximation (RWA), 163,

195, 404–406, 418

458 Index

Rotational-free fields, 8–9

SScalar photon propagator, xvii, 318–320Scalar photons, 3, 5, 36, 101–102

boosts and, 336covariant correlation matrix and, 314covariant field-matter interaction, 3,

102, 120–122neutralization in free space, 120–122observation considerations, 323–324vacuum state considerations, 338–339wave equations, 119–120

Scalar potential, 101, 323evanescent fields and, 228, 229–230

Scalar product, 332–333relativistic, 103, 155–157tensor product space and quantum

measurement theory, 385–386Scalar propagator, See Huygens scalar

propagatorScalar tensors, 29Scattering, 200f, See also Interference effects

near-field interactions and photontunneling, 237–243

one-photon process, 174wave-packet band, 218, 238–243weak quantum-mechanical nonlinearity

and, 380Schrodinger (or Schrodinger-like) equations,

xv, 101, 112, 119, 161, 180,283–285, 293, 301, 383, 396–397,406–407, 419

Schrodinger Picture, 174, 187–188, 190,202–203, 404

Screened canonical particle momentum,272–273

Sheet current density, 216–218, 223dielectric-vacuum system and photon

tunneling, 245field-quantized approach to evanescent

fields, 233frustrated total internal reflection and

photon tunneling, 237NF-photon field operators and, 235photon tunneling-coupled sheets,

250–253scattering from single current-density

sheet, 218, 238–243transverse, longitudinal, and scalar

potentials, 224–228

Sigma (σ) parameter and geodetic line,52–53

Single atom field detectors, 163Single-electron atom photon detector, 190Single-mode coherent state, 206–207Single-particle (electron) transition current

density, 283–286Single-photon correlation matrices, xvi,

163–164, 194–196, 202–204, Seealso Correlation matrices

Single-photon emission, See Photonemission

Single-photon energy, 152–153Single-photon interference phenomena, 164,

197–204, See also Interferenceeffects

Single-photon quantum optics inMinkowskian space, xvi

Single-photon wave-packet states, 173–177,350, See also Photon wave-packets

atomic and field correlation matrices,192–193

basis for one-photon states, 183–184extension to multi-photon wave

packets, 184–186observation considerations, 189–193photon tunneling and incident state,

247–249time evolution operator, 186–189wave-packet mode interference,

197–198Space-like separation, 28, 55Spatial localization of the photon, 21, 145

fingerprint in near-field transversephoton wave function, 414–415

Heisenberg uncertainty relations, 381mean position states, xvi, 164,

177–179, 201–204, 248, 311, 331,382, 394, 420

non-orthogonality of T-photon positionstates, 181

photon energy wave function concept,150

photon probability density, 151photon tunneling and, 237, 242rim zone and different inertial

observers, 5spacelike separation concept, 28spatial resolution limit problem, 277

Spatial resolution limit, 277–278Special Relativity, optics in, xiv, 5, 27

Index 459

covariance of free Maxwell equations,33–35

four-vectors and four-tensors, 31–33Lorentz transformations, 27–29, 31of electric and magnetic fields, 35–38proper time invariance, 27–28of Riemann-Silberstein vectors, 38

Principle of General Covariance, 43, 69privileged status of inertial observers,

xviitensors, 30–31

Speed of light (c), 9global vacuum concept and, xivphoton tunneling and apparent

superluminality, 218, 241–243Spherical to Cartesian representation for


),146–150

Spin 1/2 particle, 110–112current density in relativistic quantum

mechanics, 297continuity equations for spin andspace four-current densities,306–307


Dirac matrices, 297–299Gordon decomposition of Diracfour-current density, 280, 301–302

weakly relativistic spin currentdensity, 303—306

Dirac equation, 96, 101neutrino wave mechanics, 113relativistic coupling to classical

electromagnetic field, 279–280Spin-1 particles, 101, 110–112

Proca equation for, 128spherical representation for angular


), 147Spin correlation states for two photons,

390–392Spin current density in weakly relativistic

limit, xvii, 280, 301, 303–306continuity equations for spin and space

four-current densities, 306–307Spin of photon wave-packet, 155Spin of polychromatic single photon state,

176Spin operators, quantized transverse field,

170–171

Spontaneous photon emission, 383–384,401, See also Photon emission

polarization states, 408rotating wave approximation, 404–406three-level atom (two-photon emission)two-photon state vector, 417–420two-photon two-times wave function,420–427

transverse propagator and, 413–417two-level atom (single-photon

emission), 401genuine transverse photon wavefunction, 411–413

Hamiltonian in the electric dipoleapproximation, 401–406

rim zone, 413–417wave function in q-space, 409–410wave function in r-space, 410–411Weisskopf-Wigner theory, 406–408,413

wave functions for, 383, 409–411,420–427

Spontaneous symmetry breaking, 363–365,368, 372, 373

Static space-time, 88–89Stress (energy-momentum) tensor, 43Superconductivity, 102–103, 132, 363,

373–374Super-confined T-photon sources, xvii,

277–280

TTau neutrinos, 117Tensor fields and curvilinear space-time,

67–68Tensor product space, 385–389, 391–392Tensors, 29–30

contravariant four-current density, 34contravariant vectors, 29–30, 67–68covariant derivative, 43–44, 69–70, 95covariant vectors, 36–37, 68dielectric, 86, 87–88Einstein, 74–75energy-momentum (or stress), 43, 75,

113four-tensors and Lorentz

transformation, 31–33free-field commutators, 209, 210Lorentz transformation of transverse

electric and magnetic fields, 35–38many-body conductivity, 130–131

460 Index

Maxwell theory extension to GeneralRelativity, 79

metric compatibility and, 76microscopic susceptibility, 46Minkowski tensor, 33, 55–56, 161, See

also Metric tensorsparallel transport, 70–71Ricci (contracted Riemann curvature

tensor), 44, 74, 92Riemann curvature tensor, 71–74torsion tensor, 70transverse photon propagator, 203

Test particles, 56, 208–209Thermal spectrum, 325, 352, 360–361’t Hooft propagator, 326, 375Time evolution operator, single-photon

wave-packet correlations, 186–189Time-like geodesics, 43, 56–59Time-like separation, 28, 55t-matrix, 185Torsion tensor, 70Total internal reflection (TIR), 218–219Transition amplitudes, interference

between, 201Transition current density, 285–286,

329–3321s ⇔ 2s transition and breathing mode

transition, 286–295exponentially confined, 287, 292spontaneous single-photon emissions,

404Transverse and longitudinal response theory

in matter, 263–265Transverse conductivity response functions,

264–265Transverse current density, xvi, 36–37, 131,

See also Current density;Transverse vector potential

H atomic 1s ⇔ 2s transition, 286–289photon source domain and, xvi–xvii,

130, 255, 257, 279, 283–286, 310sheet current density, 224–225single-particle (electron) transition

current density, 283–286two-level breathing mode dynamics,

292–295Transverse delta function, 20–21, 25, 131,

211, 215–216, 218, 240–241Transverse field momentum density,

270–272Transverse part of vector potential, 101, 102

Transverse photon correlation matrix, 280,314–316, See also Correlationmatrices


covariant correlation matrix, 313–314Transverse photon four-current density, See

Four-current densityTransverse photon propagator, xvii, 141,

203–204, 310–311, See also Photonpropagators

T-photon correlation matrix, 313commutator and covariantquantization, 314–316

covariant correlation matrix, 313–314Transverse photons, 3, 172, See also

PhotonsCompton wave number, 132effective mass, xv–xvi, 102–103, 116,

130, 326, 363photon embryo, xvirim zone source domain, 9, 277–278,

See also Photon source domains;Rim zone

source domain, 130spatial localization, See Mean position

states; Spatial localization of thephoton

spin-1 particles, 101, 110–112transverse part of electric field, 36

Transverse photon wave functions, SeePhoton wave functions

Transverse polarization vectors, 140Transverse transition current density,

285–289Transverse vector field, 4, 8–9

wave-packet modes, 18–21Transverse vector potential, xv–xvi,

111–112, 130, 205, 255, 330, Seealso Transverse current density

evanescent fields and, 225–228, 231field emission from classical

point-particle, 255–259free fields and Heisenberg Picture, 191generalized Landau-Peierls-Sudarshan

equations, 207–208incident fields generating evanescent

tunneling potentials, 243–244Lienard-Wiechert formula, 259Lorentz and Lorenz-gauge

Index 461

transformations of four-potential,333

massless Feynman photon propagatorand, 310

near- and far-field parts, 257–259photon tunneling-coupled sheets,

250–253plane-mode decomposition of covariant

potential, 335–336quantization, 161, 165–168, See also

Quantized electromagnetic fieldsingle-particle (electron) transition

current density, 283–286single-photon correlation matrix, 194T-photon effective mass and, 363, See

also Mass of photonT-photon time-ordered correlation of

events, 311transverse Cerenkov phenomenon,

266–269Tunneling, See Photon tunnelingTwo-atom system, 278Two-boson state, 379–380, 382Two-fermion state, 380Two-particle wave function, 396–397Two-photon correlation matrix, 383Two-photon phenomena, xvii, 382–384, See

also Entanglemententangled polarization states, 390–392general polychromatic two-photon

state, 393–394Schrodinger equation in direct space,

396–397transverse two-photon mean position

state, 394wave mechanics formalism, xviiiwave packet correlations, 397–400

Two-photon spontaneous emission, xviiitwo-photon state vector, 417–420two-photon two-times wave function,

420–427Two-photon two-times wave function,

382–384, 393–395, 400, 420–427Two-point resolution problem, 277, 278

UU(1) Higgs model, 368–371Unitary gauge, 370–371Universe expansion, permittivity and

permeability and, 89–90Unruh effect, 352–361

VVacuum Maxwell equations in General

Relativity, 79Vacuum screening currents, xvii, 326,

372–373Vacuum state, 49, 102, 219, 345–348, 363,

See also Inhomogeneous vacuumanalogous to ground state, xvii, 173,

363–365Bogolubov transformation, 325,

347–349, 357, 360degeneracy, xvii, 102, 363–364, 368, 373evanescent, 222, See also Evanescent

fieldsHiggs free form approximation, 373inertial observers and, 324–325, See

also Inertial observersL- and S-photon neutralization in free

space, 120–122longitudinal and scalar photons in,

338–339massive photon, xviinear-field domain, See Rim zoneperspectives for different observers,

348–351photon mass and, xvii, 102, 326, 372Goldstone and Higgs models,364–371, 375

spontaneous symmetry breaking,363–365, 368, 372, 373

’t Hooft propagator, 375vacuum screening current, 372–373

physical photon-free vacuum concept,See Photon vacuum

quantized transverse field, 169Rindler particles in Minkowski vacuum,

354–361spontaneous symmetry breaking,

363–365, 368, 372, 373Vector boson field, 132–136Vector potentials, See Transverse vector

potentialVelocity operators, 144–145Virtual photons, xv, 3, 101–102, 374, See

also Longitudinal photons; Scalarphotons

WWave mechanics, xiv, xv–xvi, 130, See also

Four-potential photon wave

462 Index

mechanics; Photon wave functions;Photon wave-packets

bispinor wave functions, 115–116,150–153

classical electromagnetics (optics) infree space, 4–5, 7–26, See alsoElectromagnetic field

complex analytical signals, 10–12Dirac conjugate wave function, 280, 300dynamical field equations, 129, 180, 366Foldy-Wouthuysen formalism, 145four-potential description and

evanescent fields, 229–233gauge transformation, 92, 105–107generalized Landau-Peierls-Sudarshan

equations, 208helicity and, 101, 172, See also Helicity

basislongitudinal and scalar photon wave

equations, 119–120, See alsoLongitudinal photons; Scalarphotons

neutrino, 101, 113–118Oppenheimer photon wave equations,

103, 143–146Pauli spin matrices, 113–114photon energy wave, 4, 23, 103Proca equation and, 102, See also

Proca equationquantum field theory and, 103, 217single-photon wave packets, See Photon

wave-packetsspin 1/2 current density, See Spin 1/2

particletentative function and equations for

T-photons, 107–109transverse photon as spin-1 particle,

110–112transverse vector potential, xv–xvi, See

also Transverse vector potentialtwo-photon formalism, xviiitwo-photon two-times wave function,

382–384, 393–395, 400, 420–427vectorial scattering problem, 237–238wave function choice based on

T-photon mean position state, xviWave-packet modes, xiv, 4, See also Photon

wave-packetsclassical free field expansion, 18–21particle and antiparticle solutions, 10

Wave phenomenon of light in classical

optics, 41–44, See alsoGeometrical optics

Weak interaction parity violations, 114Weakly relativistic Pauli theory, xvii, 280,

301, 303–306Weisskopf-Wigner theory, xviii, 406–408,

411, 419Weyl equation, 101, 103, 114, 116Wigner, Eugene Paul, 114

XYoung-type double-source interference

experiment, 164, 198–201, 323

YZeeman effect, 305–306

Documents

Light - The Physics of the Photon