Light and Vacuum: The Wave Particle Nature of the Light and the Quantum Vacuum. Electromagnetic Theory and Quantum Electrodynamics Beyond the Standard Model

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LIGHT AND VACUUMThe Wave-Particle Nature of the Light and the Quantum Vacuum. Electromagnetic Theory and Quantum Electrodynamics Beyond the Standard ModelSecond Edition

system now known or to be invented, without written permission from the publisher.

To my wife Saliha,for her endless patience and warm support


“The infinite vacuum is the real essence of the cosmos, whateverexists has derived from the intrinsic action of the infinity resultingto positive and negative entities”

(Anaximander, 611-546 BC).


Acknowledgments

I’m sincerely very grateful to Professor Christos Flytzanis for the veryfruitful discussions and comments. I’m also particularly indebted toProfessor Pierre Richard Dahoo for reading the book and for hisuseful suggestions.

Constantin Meis

ix


Prologue

Throughout this book, we give the main principles of the electromag-netic theory and quantum electrodynamics (QED), both developedfor the understanding of light’s nature and for the explanation of theassociated phenomena when interacting with matter. Of course, it isnot in the scope of this manuscript to give a full and detailed presen-tation of these theories. Only selected theoretical topics have beenchosen, supported by experimental evidence, which are indispensablefor the understanding of the present status of the theories on thenature of light. The chapters of the book follow a well-defined schemein order to provide the reader the necessary knowledge for under-standing the wave and particle concepts of the light as well as theadvanced elaborations and the associated problematics at the end.

After the introduction and historical survey the Chapter 3resumes Maxwell’s electromagnetic wave theory while Chapter 4introduces the fundamentals of quantum mechanics employed for thedescription of the particle nature of light. In Chapter 5, we discuss themain difficulties encountered by both theories, to ensure a completeand coherent mathematical description of the simultaneous wave-particle nature of light put in evidence by experiments.

Finally, in Chapter 6, we enhance beyond the standard descriptionthe basic aspect of QED related to the quantization of the vectorpotential amplitude of the electromagnetic field to a single photonstate, and we advance further on its relationship with the classicalelectromagnetic wave theory and the vacuum.

The topics are all drawn from many works previously published,and given in the bibliography. The perspectives and elaborations

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on the photon vector potential and its relationship to the quantumvacuum are of my own, in the aim to raise questions and aspirefor further theoretical and experimental investigation, in order toimprove our knowledge and understanding on the real essence oflight and vacuum.

Note: In this book, the word “light” is not limited to the visiblefield, but concerns the whole electromagnetic spectrum ranging fromzero frequency to infinity.

Table of Contents

Acknowledgments ix

Prologue xi

1. Introduction 1

2. Historical Survey and Experimental Evidence 3

The concepts of light during the last 2500 years: corpuscles,ray optics, wave optics, electromagnetic wave theory andfinally quantum particle theory. . . . . . . . . . . . . . . . 3Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3. Basic Principles of the Electromagnetic Wave Theory 11

3.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . 113.2 Electromagnetic Wave Propagation . . . . . . . . . . 14

• Dispersion relation . . . . . . . . . . . . . . . . . . 14• Physical quantities involved in the

electromagnetic field . . . . . . . . . . . . . . . . . 16• Propagation equations of electromagnetic waves . . 18• Helmholtz equation . . . . . . . . . . . . . . . . . . 18• Energy flux of electromagnetic waves . . . . . . . . 19• TEM waves — Laplace equation . . . . . . . . . . 21

3.3 Scalar and Vector Potentials . . . . . . . . . . . . . . 23

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3.4 Vector Potential and Electromagnetic FieldPolarization . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Guided Propagation of Electromagnetic Waves . . . 28• Rectangular waveguide of cross sectiona, b(a > b) . . . . . . . . . . . . . . . . . . . . . . . 29

• Rectangular waveguide TEm,n modes . . . . . . . . 29• Rectangular waveguide TMm,n modes . . . . . . . 30• Circular waveguide of radius r0 . . . . . . . . . . . 31• Circular waveguide TEm,n modes . . . . . . . . . . 31• Circular waveguide TMm,n modes . . . . . . . . . . 32• Case of two parallel plates . . . . . . . . . . . . . . 33• Density of states . . . . . . . . . . . . . . . . . . . 34

3.6 Conclusion Remarks . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4. From Electromagnetic Waves to QuantumElectrodynamics 39

4.1 Elements of Quantum Mechanics . . . . . . . . . . . 39• Blackbody radiation and the ultraviolet catastrophe 39• Energy and momentum operators in quantum

mechanics . . . . . . . . . . . . . . . . . . . . . . . 41• Particle in a square potential well —

correspondence with the waveguides . . . . . . . . 434.2 Harmonic Oscillator in Quantum Mechanics . . . . . 44

• From the classical expressions to the quantummechanical ones . . . . . . . . . . . . . . . . . . . . 44

• Dirac representation, creation and annihilationoperators, a, a+ . . . . . . . . . . . . . . . . . . . . 47

• The Dirac equation . . . . . . . . . . . . . . . . . . 484.3 Quantum Electrodynamics (QED)

and the Photon Description . . . . . . . . . . . . . . 50• The photoelectric effect and the quantum

interpretation . . . . . . . . . . . . . . . . . . . . . 51• Compton scattering . . . . . . . . . . . . . . . . . 51• Low intensity Young’s double-slit interferences . . 53• Second Quantization . . . . . . . . . . . . . . . . . 54

Table of Contents xv

• Quantization process of theelectromagnetic field . . . . . . . . . . . . . . . . . 54

• The Photon Wave Function . . . . . . . . . . . . . 574.4 Interaction between Electromagnetic Waves

and Charged Particles, Reality of the VectorPotential . . . . . . . . . . . . . . . . . . . . . . . . . 60• Interaction Hamiltonian between an

electromagnetic wave and a charged particle . . . . 60• Reality of the vector potential, Ehrenberg–Siday

or Aharonov–Bohm effect. . . . . . . . . . . . . . . 614.5 Transition Rates and Vacuum Induced

Spontaneous Emission . . . . . . . . . . . . . . . . . 63• Photoelectric effect and the semi-classical

interpretation . . . . . . . . . . . . . . . . . . . . . 65• Spontaneous emission rate . . . . . . . . . . . . . . 66• Dipole approximation and spontaneous emission . 67• The photon spin . . . . . . . . . . . . . . . . . . . 69

4.6 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . 70• Nonrelativistic calculations: Bethe’s approach . . . 70


5. Theory, Experiments and Questions 77

5.1 Planck’s Constant and the Vacuum IntrinsicElectromagnetic Properties . . . . . . . . . . . . . . 77

5.2 Hamiltonian Issued from the Quantizationof the Electromagnetic Field . . . . . . . . . . . . . . 79

5.3 QED Singularities . . . . . . . . . . . . . . . . . . . 805.4 Electron-Vacuum Interactions and the Associated

Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 82• Spontaneous emission . . . . . . . . . . . . . . . . 83• Lamb shift . . . . . . . . . . . . . . . . . . . . . . . 84• Casimir effect . . . . . . . . . . . . . . . . . . . . . 85

5.5 Simultaneous Wave-particle Nature of the PhotonRevealed by the Experiments — Discussions . . . . . 88

xvi Light and Vacuum (Second Edition)

• Photoelectric effect, Compton’s Scatteringand Young’s double-slit experiment . . . . . . . . . 88

• Mizobuchi and Ohtake double-prismexperiment . . . . . . . . . . . . . . . . . . . . . . 89

• Grangier’s et al. experiments, photonindivisibility . . . . . . . . . . . . . . . . . . . . . . 90

• Hunter–Wadlinger experiments, the photonspatial expansion . . . . . . . . . . . . . . . . . . . 92


6. Analysis of the Electromagnetic Field QuantizationProcess and the Photon Vector Potential. Non-Local Photon Wave-Particle Representation and theQuantum Vacuum 99

6.1 Quantized Vector Potential Amplitude of a SinglePhoton State . . . . . . . . . . . . . . . . . . . . . . 99• Dimension analysis from Maxwell’s equations.

Vector potential amplitude proportionalto the frequency . . . . . . . . . . . . . . . . . . . 99

• Quantum equation for the photon vectorpotential. . . . . . . . . . . . . . . . . . . . . . . . 101

• Photon quantization volume. . . . . . . . . . . . . 106• Relation between the photon vector potential

and the electron charge. Definition of the vectorpotential amplitude quantization constant ξ. . . . . 111

6.2 Ground Level of the Electromagnetic Field.Quantum Vacuum Representation . . . . . . . . . . . 113• Analysis of the origin of the zero-point energy

in the QED formalism . . . . . . . . . . . . . . . . 113• Ground level of the electromagnetic field

and the vacuum . . . . . . . . . . . . . . . . . . . . 1176.3 The Quantum Vacuum Field Effects . . . . . . . . . 118

• Spontaneous emission . . . . . . . . . . . . . . . . 118• Lamb shift . . . . . . . . . . . . . . . . . . . . . . . 119

Table of Contents xvii

• Vacuum energy density and the cosmologicalconstant . . . . . . . . . . . . . . . . . . . . . . . . 122


7. Epilogue 131

Index 137

February 20, 2017 11:23 Light and Vacuum (Second Edition) 9in x 6in b2812-ch01 page 1

Chapter 1

Introduction

“Light is the means nature employs to observe itself”.

Man has always been fascinated by light. Since the rise of humanity,light is related to life and darkness to death. This concept seems tohave intrinsic roots in the human mind whatever the tribe or nation,on any continent and during any historic or pre-historic period.

The natural entities emitting light, helping man and animals tosee while eliminating the fears of darkness, like the sun, the moonand fire were venerated since the beginning and the absolute divinewas associated with them. Man quickly understood that life, as itis met on the surface of this planet, would be inexistent or at leastcompletely different without light. The forests are the main lags ofthe earth’s atmosphere contributing to the production and to therecycling of oxygen, a so crucial for the life element. In the absence ofthe sun’s light, plants, trees and forests are condemned to disappearand consequently animals would be unable to survive.

Within a biological point of view, light is the basic medium ofnature conferring the possibility to organisms possessing an appro-priate detection system, optical or not, to see or detect the shapesof the objects of the surrounding cosmos. By the same token, lightis the means nature has chosen to watch itself by way of live beings.The exchange of energy and information between atoms, moleculesand more complex systems occurs through light, permitting life toemerge in the cosmos. This notion, once understood, automaticallyattributes a unique character to light in the universe.

In the old historical times, light on earth was only emitted fromfire, volcanic lava or high temperature metallic objects. Ancient

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Greeks had such a profound respect for light that the only originthey could imagine was undoubtedly divine. In the Greek language,light is Phos (Φωs). This one-syllable word has quite a strong anddeep consonance that used to be attributed only to exceptional phys-ical entities, like Pyr (Πυρ) fire, or even the head of the Gods, Zeus(Zευs). Hence, according to the legend, Prometheus has stolen thefire from Gods in order to offer light and heat to mankind. Disregard-ing the fact if mankind was worthy of it, the value of the gift wasinvaluable. For his action, Prometheus was punished by the Gods tobe attached on the Caucasian mountains where an eagle was eatinghis liver in the day while it was regenerated during the night. Thecharacter of this endless torture without any chance of forgivenessor clemency gives witness to the extreme importance ancient Greeksattributed to the stolen entity from the Gods, fire. One simply hasto wonder what human civilization would be like without fire.

Not surprisingly, in most religions God is identified with light.However, light has an intrinsic nature and undergoes specific phys-

ical laws governing its behavior, and studied by science. Today, aquite considerable progress has been done in order to understandthe behavior laws but our mathematical representation of its wave-particle nature and its relationship to the vacuum is still incomplete.


Chapter 2

Historical Surveyand Experimental Evidence

The concepts of light during the last 2500 years: corpuscles, rayoptics, wave optics, electromagnetic wave theory and finally quan-tum particle theory.

As far as we know, the first attempts to study the nature of lightfrom the scientific point of view are due to ancient Greeks. Theybelieved light to be composed of corpuscles.

Thales of Miletus in the sixth century BC knew already that in agiven medium, light propagates in straight lines and that the light ofthe sun also obeys that property. Based on that knowledge and usinghis famous theorem, he measured the height of the Cheops pyramidin Egypt by comparing the length of its shadow to that of his stickpositioned vertically.

About two centuries later, Euclid published the book “Optica”in which, based on the rectilinear propagation of light, he developedthe laws of reflection by applying principles of geometry. Archimedes,developed the geometrical study of parabolas and ellipsoids and,according to the legend, he created big metallic mirrors for focusingthe sunlight in order to burn the Roman battle ships during the siegeof Syracuse. That was the first time in human history that light wasemployed as a weapon.

A couple of centuries later, in his book “Optics”, Ptolemy ofAlexandria made a full synthesis of all the previous knowledge on lightfollowing the concepts of Euclid, Archimedes and Heron of Alexandria,treating refraction, reflection and colors. Refraction of the moonlightand sunlight by the earth’s atmosphere was also analyzed.

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Euclid’s “Optica” and Ptolemy’s “Optics” are the first known sci-entific publications on light’s properties. Only a fraction of “Optics”has been saved, and this is thanks to Arab mathematicians of the7th to 10th centuries AC.

After the dominance of Christians during the third century AC,the first Byzantine emperors ordered the definite closure and destruc-tion of all Greek mathematical and philosophical academies that werespread from Greece to Egypt through the Middle East. Philosophersand scientists were persecuted and murdered preventing any possibil-ity of scientific development for more than thirteen centuries. The lastone of them was Hypatia of Alexandria who has been assassinatedby a crowd of fanatic christians because she was woman, scientistand philosopher.

Humanity had to wait until 1620 AC for Snell’s works on refrac-tion and Fermat’s principle, according to which light rays travel alongthe path of least time. It was at this period that many scientists like,Grimaldi, Boyle, Hook, Descartes and others began studying light’sproperties.

Half a century later, in the year of 1670, Newton had retaken theproposals of Pierre Gassendi, who had revived the ideas of ancientGreeks, and advanced the theory that light rays are composed of cor-puscles that travel rectilinearly. However, he went further beyond thisdescription by announcing that under specific physical conditions,the light corpuscles may give birth to waves in “aether”, without giv-ing it a precise definition. Surprisingly, in the first edition of his book“Opticks”, the light corpuscles and the generated “aether” waveswere replaced by particles submitted to a kind of “periodic relief”.

Parallel to Newton, in the year of 1680, Huygens developed aremarkable wave theory for light, deducing the laws of reflection andrefraction while demonstrating that wave propagation may not be incontradiction with the rectilinear propagation. Huygens’ wave theorywas a hard opponent to Newton’s particles concept.

In the beginning of the 19th century, Young’s experimentsdemonstrated that interference can be obtained by different waves,while Fresnel applied the wave theory to explain the diffractionpatterns observed experimentally. Furthermore, Young explained


Historical Survey and Experimental Evidence 5

some polarization observations by making the hypothesis that lightoscillations are perpendicular to the propagation axis. Nevertheless,not even a scientist could seriously consider the interference theoryand Young’s studies were berated by the journalists. Nearly twentyyears later, the diffraction theories of Young and Fresnel, though notidentical, were the only ones capable of predicting all the observeddiffraction patterns and that was really the crucial turnover point inthe 19th century following which the scientific community started togenerally accept the wave nature of light.

The wave theory was dominant until the beginning of 20th cen-tury. It is worth noting that until that time, nearly for more than2500 years, the main question for scientists was focused on the parti-cle or the wave nature of light, but nobody had advanced any specificquestions on the real nature of light, i.e., a corpuscle made of what?or a wave of what?

In 1865, James Clerk Maxwell published his remarkable workon the electromagnetic waves issued from . . . Maxwell’s equationsdescribing the relations between the electric and magnetic fields, andhas shown that the electromagnetic waves propagate in vacuum atthe same speed observed by astronomers for light. For the first timethe speed of light was related directly to the vacuum electric permit-tivity and magnetic permeability showing the natural relationshipbetween light and vacuum. Just a few years later, Hertz discoveredthe long wavelength electromagnetic radiation demonstrating that itpropagates at the speed of light confirming Maxwell’s theory.

It is quite amazing in the history of science that Maxwell’s andHertz’s works, carried out within roughly ten years, were absolutelyrevolutionary and decisive in our understanding about the nature oflight, which remained stationary for over 25 centuries.

At the same period, from 1880 to 1900, the works of Stefan, Wienand Rayleigh have shown for the first time the direct relationshipbetween the thermal radiation energy and the temperature of theemitting body, assimilated to a black body. However, the emittedradiation energy density of the black body as a function of the tem-perature calculated by Rayleigh, failed to describe the experimentalresults obtained at short wavelengths. Scientists had given the name



of “UV catastrophe” to this situation which revealed the necessity ofa new theoretical approach.

In the very beginning of the 20th century, Max Planck assumedthat the bodies are composed of “resonators”, a kind of oscillatorsthat have the particularity of emitting the electromagnetic energy inpackets of hν, thus proportional to the wave frequency ν times h, thatwas later called Planck’s constant. This hypothesis permitted Planckto establish the correct energy density expression for the radiationemitting from a black body with respect to temperature, which is inexcellent agreement with the experiment.

In 1902, Lenard pointed out that the photoelectric effect, discov-ered by Hertz fifteen years earlier, occurs beyond a threshold fre-quency of light and the kinetic energy of the emitted electrons doesnot depend on the incident light intensity.

The experiments carried out by Michelson and Morley demon-strated that the speed of light in vacuum is a universal physicalconstant. This fundamental discovery was the starting point for thedevelopment of the theory of special relativity, based on Lorentz’sset of equations followed by Poincare’s theoretical studies. Einsteingrouped all these works in an elaborated theory published in 1905.During the same year, based on the works of Planck and Lenard,Einstein published also an article stating that the electromagneticradiation is composed of quanta of energy hν and advanced a simpleparticle interpretation of the photoelectric effect. In a second articlein 1917, Einstein re-established Planck’s radiation density formulaand expressed that the light quanta have a momentum hν/c, wherec is the velocity of light in vacuum. He advanced that “the energyof a light ray when spreading from a point consists of a finite num-ber of energy quanta localized in points in space, which move with-out dividing and are only absorbed and emitted as a whole”. Thatwas a decisive step toward the particle theory of light, but Bohr,who was strongly opposed to the photon concept, announced in hisNobel lecture, “the light quanta hypothesis is not compatible with theinterference phenomena and consequently it cannot throw light in thenature of radiation”. The concept of light composed of quanta wasstill not generally accepted.



Compton published his works on X-ray scattering by electrons in1923 showing that the experimental results can only be interpretedbased on the light quanta model. On the other side, Wentzel in 1926and Beck in 1927 demonstrated that “the photoelectric effect may bequite well interpreted using Maxwell’s wave theory for radiation andquantum theory for the atomic energy levels without ever having tointroduce the light quanta” in their calculations. Although their arti-cles were an extremely strong argument in favor of the wave theory,it is quite surprising that exactly the same year, the light quantaconcept started to be universally accepted by many scientists andLewis introduced the word “Photon”, from the Greek word Phos(Φωs = light).

Obviously, just before 1930, the general picture on the nature oflight was extremely puzzling and confusing since both opposing sidesdefending the wave or the particle theory presented equally strongarguments. Not surprisingly, in 1928, Bohr inspired by De Broglie’sthesis on the simultaneous wave character of particles announcedthe “Complementarity Principle” according to which “light has bothwave and particle natures appearing mutually exclusive in each spe-cific experimental condition”.

From 1925 to 1930, new theories have shaken physics. Heisenbergdeveloped matrix-mechanics, while in parallel, Schrodinger inventedwave-mechanics which was revealed to be equivalent to matrix-mechanics. Dirac and Jordan developed the quantum theory of radi-ation. A few years later, enhancing further Dirac’s works, Pauli andHeisenberg developed the fundamentals of quantum electrodynamics(QED) theory which was further developed in the years of forties tosixties. According to this theory, photons are considered to be pointparticles and the energy of the electromagnetic field corresponds to,that of an ensemble of quantized harmonic oscillators. In the com-plete absence of photons, QED representation gives an infinite energyfor the vacuum state.

By the end of 20th century and the beginning of the 21st, astro-physical observations established the first estimations of the vac-uum energy density in the universe demonstrating that it is, roughly10−9 J/m3 putting in evidence that the universe is speeding up rather



than decelerating as it might be predicted by the general relativity inthe standard matter sources universe. In 1989, Weinberg had alreadyargued that when considering reasonable cut-offs in the UV spectrumfor the energy of high frequency photons, the vacuum energy densityobtained by QED is by far many orders of magnitude greater than themeasured one. This has been recently called the “quantum vacuumcatastrophe” in order to mark the analogy with the “UV catastro-phe” which, a century ago, entailed the necessity of developing a newtheory to account with.

Considering that the fundamentals of QED were established inthe years of 1930 to 1950, since then no new theories came to light.Nevertheless, the technological advances on microwaves during theSecond World War and the invention of masers and lasers in the yearsof the fifties and sixties, respectively, gave scientists an impulse foradvanced experiments on light’s nature. Some of them are brieflydescribed in this book.

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16. J.C. Garrison and R.Y. Chiao, Quantum optics, Oxford: Oxford UniversityPress. 2008.

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Chapter 3

Basic Principlesof the Electromagnetic

Wave Theory

3.1. Maxwell’s Equations

The works of Ampere and Faraday on the electric and magnetic fieldsduring the years of the 1860s were the starting point for establishingthe theory of electromagnetic waves.

Initially they were both investigating time varying electric andmagnetic fields. Faraday firstly understood that the curl of an electricfield E (r, t) (units: Volt m−1) at a position r and at an instant t, in amedium with magnetic permeability µ (units: Henry m−1) equals thetime variation of the magnetic field intensity H (r, t) (units: Amperem−1) times µ

∇× E (r, t) = −µ ∂∂tH (r, t) (3.1.1)

The last equation is still called Faraday’s law. A few years later,Ampere put in evidence that in a medium with electric permittivityε (units: Farad m−1), the curl of the magnetic field intensity H (r, t)equals the current density j (r, t) plus the time variation of the elec-tric field intensity E (r, t) times ε.

∇× H (r, t) = j (r, t) + ε∂

∂tE (r, t) (3.1.2)

where the current density j (r, t) (Coulomb m−2 s−1 or Ampere m−2)is related to the charge density ρ (r, t) (Coulomb m−3 ) through the

11



fundamental “continuity equation”

∇ ·j (r, t) = −∂ρ (r, t)∂t

(3.1.3)

expressing that the spatial variation of the current density equals thetime variation of the charge density implying simply charge conser-vation.

What one first notices is the natural fact that the electric andmagnetic fields cannot evaluate independently of the medium whichis expressed in this case through the electric permittivity and themagnetic permeability. We can thus introduce the notions of theelectric displacement flux density D (r, t) (units: Coulomb m−2) andmagnetic field flux density, that is the magnetic induction, B (r, t)(units: Webers m−2) which are related to the electric and magneticfields intensities E (r, t) and H (r, t) respectively through the tensorexpressions,

Dαβ(t) = εαβ · Eαβ(t); Bαβ(t) = µαβ · Hαβ(t) (3.1.4)

where the electric permittivity and magnetic permeability respec-tively are tensors εαβ and µαβ with α = x, y, z and β = x, y, z, char-acterizing the intrinsic electric and magnetic nature of the mediumin which the electric and magnetic field intensities subsist.

At that point it is important mentioning that the electric fieldE (r, t) and the magnetic induction B (r, t) are fundamental fieldswhile D (r, t) and H (r, t) are fields that include the response of themedium at macroscopic level.

In isotropic media, i.e., in media in which the electric and mag-netic properties are identical in all spatial directions, the electricfield is parallel to the electric displacement and the magnetic fieldflux intensity parallel to the magnetic field

D (r, t) = ε E (r, t) H (r, t) =1µB (r, t) (3.1.5)

Taking the divergence (∇·) of the Faraday’s and Ampere’s laws,(3.1.1) and (3.1.2), and using the continuity equation (3.1.3) one gets


Basic Principles of the Electromagnetic Wave Theory 13

directly Gauss’s laws

∇ · E (r, t) =ρ (r, t)ε

(3.1.6)

∇ · H (r, t) = 0 (3.1.7)

where (3.1.6) is also called Coulomb’s law.Physically speaking Gauss’s equations express the fact that indi-

vidual electric charges, positive as well as negative, subsist in natureand are expressed in the charge density ρ (r, t) in equation (3.1.6),while the absence of the magnetic monopoles is expressed in equation(3.1.7).

James Clerk Maxwell in 1873 unified equations (3.1.1), (3.1.2),(3.1.6) and (3.1.7) to a group called Maxwell’s equations which,joined to the continuity equation (3.1.3) express the relationshipbetween the electric and magnetic fields intensities in a medium witha given magnetic permeability µ and electric permittivity ε in thepresence of charges and current densities.

Frequently, Maxwell’s equations are written in such a way thatthey express the relation between the electric field intensity E (r, t)and the magnetic field flux intensity B (r, t), i.e., the fundamentalfields.

∇× E (r, t) = − ∂

∂tB (r, t) (3.1.8)

∇× B (r, t) = µj (r, t) + (εµ)∂

∂tE (r, t) (3.1.9)

∇ · E (r, t) =ρ (r, t)ε

(3.1.10)

∇ · B (r, t) = 0 (3.1.11)

The relations (3.1.8) to (3.1.11) together with the continuity equa-tion (3.1.3) are the fundamental equations governing the behavior ofthe electromagnetic fields. They are invariant under time reversal(t → −t), or space inversion (r → −r) or even both time reversaland space inversion. More precisely, the intrinsic space time trans-formation properties are such that the electric field E (r, t), andtherefore the electric displacement D (r, t), do not change sign under



time reversal but under space inversion. The opposite is valid forthe magnetic induction B (r, t) and the magnetic field H (r, t). Thisis also true for the response of a medium to an electric field and amagnetic induction consisting respectively by the electric polariza-tion P (r, t) (units: Coulomb m−2) and magnetization M (r, t) (units:Ampere m−1) expressed as following

P (r, t) = D (r, t) − ε0 E (r, t) (3.1.12)

M (r, t) =1µ0

B (r, t) − H (r, t) (3.1.13)

According to the set of equations (3.1.8) to (3.1.11), in spaceregions characterized by the absence of charges and current densities|j| = 0, ρ = 0, the physical vector entities E (r, t) and B (r, t) arerelated through the quantities ε and µ which appear to be of cru-cial importance in Ampere’s equation (3.1.9). Under these conditionsMaxwell’s equations get the simple form

∇× E (r, t) = − ∂

∂tB (r, t) (3.1.14)

∇× B (r, t) = (εµ)∂

∂tE (r, t) (3.1.15)

∇ · E (r, t) = 0 (3.1.16)

∇ · B (r, t) = 0 (3.1.17)

Starting from the last set of equations, Maxwell deduced the timevarying propagation of the electric and magnetic fields that gavebirth, for the first time, to a theoretical representation of the elec-tromagnetic waves.

3.2. Electromagnetic Wave Propagation

• Dispersion relation

Let us analyze how Maxwell’s equations lead naturally to theexpressions of propagation of the electromagnetic fields. We makethe hypothesis that the electric field intensity E and the magneticfield flux intensity B at any spatial position r vary in time with an



angular frequency ω = 2πν = 2π 1T , where ν and T are the frequency

and the period of the oscillation respectively, so as we can write[E (r, t)B (r, t)

]= Re

[E0 (r)B0 (r)

]eiωt

(3.2.1)

where Re denotes the real part of the equation.Then, Maxwell’s equations, more precisely Ampere’s and Fara-

day’s laws, are expressed through the angular frequency ω

∇× E (r, t) = iω B (r, t) (3.2.2)

∇× B (r, t) = −iω (εµ)E (r, t) (3.2.3)

∇ · E (r, t) = 0 (3.2.4)

∇ · B (r, t) = 0 (3.2.5)

The set of the above equation describes the behavior of the electricfield and magnetic field flux intensities of an electromagnetic har-monic wave oscillating with an angular frequency ω in an isotropicmedium with electric permittivity ε and magnetic permeability µ.

One should notice two important points:

a. the correspondence ∂∂t → −i ω, which as we shall see, will be of

crucial importance in quantum mechanics formalism,b. the curl inversion absence of symmetry between E and B which

is ensured through the factor −(εµ), and which is revealed to beof high importance in the electromagnetic field theory.

The first remark will be extensively discussed in the next chapter.However, the second reveals the role of the medium’s nature in thepropagation of the electromagnetic field.

Let us assume the media to be homogeneous. Therefore we can useplane wave solutions for E0 (r) and B0 (r) in (3.2.1) of the form ei

k·r

where k is a vector along the propagation axis called wave vector.Hence, from Maxwell’s equations we obtain

k × E (r, t) = ω B (r, t) (3.2.6)



k × B (r, t) = −ω (µε)E (r, t) (3.2.7)

k · E (r, t) = 0 (3.2.8)

k · B (r, t) = 0 (3.2.9)

The last set of equations entail that in isotropic media k is per-pendicular to E and B (while it is always perpendicular to D andB, even if the medium is anisotropic) and E is perpendicular to B.

Consequently, the unit vectors, e = E| E| , b = B

| B| , k = k

|k| constitute

a right-hand rectangular coordinate system.Considering equation (3.2.6) and taking the cross-product we get

k × k × E (r, t) = ωk × B (r, t) (3.2.10)

According to the vector property k×k× E (r, t) = (k · E (r, t))k−k2 E (r, t) = −k2 E (r, t) where we have also used equation (3.2.8), theequation (3.2.10) using (3.2.7) becomes

Ω2kE (r, t) = 0 (3.2.11)

with Ω2k = k2 − (µε)ω2. The solution to the last equation is Ω2

k = 0.Thus,

k2 = (µε)ω2 (3.2.12)

which is called the “dispersion relation” relating the wave vector kto the angular frequency ω through the medium intrinsic propertiesε and µ.

• Physical quantities involved in the electromagnetic field

Notice that 1εµ has the dimension of the square power of celerity.

Consequently, it is precisely the electric permittivity ε and magneticpermeability µ that fixes the velocity of the electromagnetic wave ina medium whose refractive index n is defined as

n =√

µε

µ0ε0(3.2.13)

with µ0 = 4π × 10−7Hm−1 and ε0 = 8.85 × 10−12Fm−1 ≈ 136π

10−9Fm−1, the magnetic permeability and electric permittivity ofthe vacuum.



This is already an amazing concept of the classical electromag-netic theory that confers electric and magnetic properties to the vac-uum state.

The characteristic impendence of a medium is given by

R =√µ

ε(3.2.14)

corresponding to the resistance (units: Ohms, Ω) of the medium whenan electric potential is applied.

Consequently, the vacuum itself is characterized by the impen-dence

Rvacuum =√µ0

ε0≈ 120π Ω (3.2.15)

which has the definite value of 376.7 Ω.It is extremely worthy noticing that the vacuum state in classical

electrodynamics is not an empty, fully inertial entity characterized bythe perfect “inexistence” but has its own physical, measurable impen-dence, Rvacuum, as well as electric, ε0, and magnetic, µ0, propertiesaffecting the propagation of the electromagnetic waves in vacuum.In fact, the speed of the electromagnetic waves in vacuum, c, resultsdirectly from the intrinsic properties of the vacuum itself

c =1√ε0µ0

≈ 2.9979 108 m s−1 (3.2.16)

Although the origin of the vacuum’s electromagnetic propertiesremains unknown, the last ones dictate the value of the highest pos-sible velocity in the universe, c, in the frame of course of the presentstate of knowledge.

The dispersion relation writes now

k2

(c2

n2

)= ω2 (3.2.17)

expressing a fundamental relationship between the wave vector k

and the angular frequency ω through the speed of the light c/n in amedium with refractive index n, where n = 1 for the vacuum.



• Propagation equations of electromagnetic waves

Taking the curl of equation (3.1.14) and using (3.1.15)

∇× ∇× E (r, t) = − ∂

∂t(∇× B(r, t)) = − ∂

∂t(µε)

∂

∂tE (r, t)

= −(µε)∂2

∂t2E (r, t) (3.2.18)

Considering the classical vector formalism

∇× ∇× E = ∇(∇ · E) − ∇2 E = −∇2 E (3.2.19)

So (3.2.18) writes

∇2 E (r, t) − (µε)∂2

∂t2E (r, t) = 0 (3.2.20)

which is the propagation equation for the electric field in a mediumwith refractive index n

∇2 E (r, t) − n2

c2∂2

∂t2E (r, t) = 0 (3.2.21)

An analogue equation can be obtained for the magnetic field fluxintensity B (r, t).

• Helmholtz equation

The vector relations of (3.2.6) to (3.2.9) show that k, E and B areperpendicular to each other and construct a right-hand coordinatesystem. Hence, the harmonic solutions of the propagation equationfor the electric field and magnetic field induction write

E (r, t) = E0 (r) e e−i(k·r−ωt) + c.c. (3.2.22)

B (r, t) = B0 (r)(k × e) e−i(k·r−ωt) + c.c. (3.2.23)

with c.c. denoting the complex conjugate part.From (3.2.12), (3.2.17), (3.2.21) and (3.2.22), we deduce the

Helmholtz equation for each component of the electric field(∂2

∂x2+

∂2

∂y2+

∂2

∂z2+ k2

)Eα = 0 α = (x, y, z) (3.2.24)



In other words, in the classical electromagnetic theory, the electricand magnetic fields oscillate perpendicularly along the propagationaxis k with an angular frequency ω at the speed c/n.

This is called simply a plane wave solution. In this representationthe wave vector in vacuum (n = 1) has the simple expression

k =ω

ck =

2π νck =

2πλk (3.2.25)

where λ is the wavelength, representing the spatial distance of acomplete oscillation of the electric and magnetic field vectors over aperiod T and k, the unit vector along the propagation axis.

For a given fixed value of the angular frequency ω, and the wave-length λ, the wave is called monochromatic (single color) and we willuse the notations Eω (r, t) and Hω (r, t) for the corresponding elec-tric and magnetic fields, which, according to the above considerationsobey the same propagation equation

∇2 Gω(r, t) − n2

c2∂2

∂t2Gω(r, t) = 0 (3.2.26)

where Gω(r, t) is either Eω (r, t) or Hω (r, t).More generally an electromagnetic wave is called Transverse

Electro-Magnetic (TEM) when both Eω(r, t) and Hω(r, t) lie in theplane that is transverse to the direction of propagation, with thesame token Transverse Electric (TE) corresponding to the configu-ration with all the components of the electric field, but not those ofthe magnetic field, in the transverse plane to the propagation axis.Conversely, a Transverse Magnetic (TM) electromagnetic wave hasall of the components of the magnetic field, but not those of theelectric one, in the plane transverse to the propagation direction.

• Energy flux of electromagnetic waves

The vector product

Sω = Eω × Hω (3.2.27)

represents the instantaneous magnitude and direction of the powerflow of the electromagnetic field and is called the Poynting vector.



The power density (units: Watt m−2) along the propagation axisis obtained by time averaging the Poynting vector over a period T

〈Sω〉T =1T

∫ T

0(Eω × Hω) dt =

12Re

Eω × Bω

µ

=12

( cn

)ε| Eω|2 k =

12

( cn

) 1µ| Bω|2k (3.2.28)

The quantity 〈Sω〉T is also called “Intensity” or “Irradiance” ofthe monochromatic electromagnetic wave.

The energy density (units: Joules m−3) of a transverse monochro-matic electromagnetic (TMEM) wave in vacuum oscillating at theangular frequency ω is obtained at the coordinate r and at an instantt by the expression:

Wω(r, t) =12

(ε0| Eω(r, t)|2 +

1µ0

| Bω(r, t)|2)

(3.2.29)

where ε0 and µ0 are the vacuum permittivity and permeabilityrelated to the velocity of light in vacuum c by the simple relation:

ε0µ0c2 = 1 (3.2.30)

implying once again that the vacuum in classical electrodynamicshas an electromagnetic essence.

Wω(r, t) has to be understood as a time varying scalar fieldentailing that the energy density has not fixed values at any spacecoordinate, due naturally to the periodic variation of the electric andmagnetic fields. Frequently, Wω(r, t) is expressed through the vacuumpermittivity only, getting for the total energy of a monochromaticelectromagnetic field in a volume V :

E(ω, ε0) =ε02

∫V

(| Eω(r, t)|2 + c2| Bω(r, t)|2)d3r (3.2.31)

It is quite important noticing that the energy of the electromag-netic waves depends directly on the vacuum intrinsic properties, theelectric permittivity ε0 and magnetic permeability µ0.

Furthermore, Eω depends on the integration volume V and it is atime varying function.



• TEM waves — Laplace equation

General solutions of Maxwell’s equations for TEM waves are with-out longitudinal field components.

From equations (3.1.5), (3.2.2) and (3.2.3), we can write the curlsof the electric and magnetic field as following

∇× E (r, t) = −iω µ H (r, t) (3.2.32)∇× H (r, t) = iω ε E (r, t) (3.2.33)

Using the plane wave expression (3.2.22) and (3.2.23) and con-sidering the propagation along the z-axis so that in Cartesian coor-dinates the components of the electric and magnetic field intensitiesof the transverse wave depend on (x, y), the equations (3.2.32) and(3.2.33) are expanded to the component expressions[

∂Ez

∂y+ iγEy

]+ iωµHx = 0 (3.2.34)

[∂Ez

∂x+ iγEx

]− iωµHy = 0 (3.2.35)

[∂Ey

∂x− ∂Ex

∂y

]+ iωµHz = 0 (3.2.36)

[∂Hz

∂y+ iγHy

]− iωεEx = 0 (3.2.37)

[∂Hz

∂x+ iγHx

]+ iωεEy = 0 (3.2.38)

[∂Hy

∂x− ∂Hx

∂y

]− iωεEz = 0 (3.2.39)

where we have introduced a general wave vector γ, called propagationfactor, in such a way that the difference between the square valuesof k (given by 3.2.12) and γ define a cut-off wave vector kc

k2c = k2 − γ2 (3.2.40)

According to the last set of six components equations, it is impor-tant noting that all the transverse field components can be calculated



with respect to the longitudinal components Ez and Hz

Ex = − i

k2c

[γ∂Ez

∂x+ ωµ

∂Hz

∂y

](3.2.41)

Ey = − i

k2c

[γ∂Ez

∂y− ωµ

∂Hz

∂x

](3.2.42)

Hx =i

k2c

[ωε∂Ez

∂y− γ

∂Hz

∂x

](3.2.43)

Hy = − i

k2c

[ωε∂Ez

∂x+ γ

∂Hz

∂y

](3.2.44)

Hence, the components of TE waves, for which Ez = 0, as wellas TM waves, for which Hz = 0, are immediately deduced fromequations (3.2.41) to (3.2.44).

Now, one may wonder what happens for a TEM wave, when bothEz and Hz are zero. Obviously, we cannot use equations (3.2.41) to(3.2.44). Hence, we have to apply again equations (3.2.34) to (3.2.38)from which, by eliminating Hx, we get simply

µεω2Ey = γ2Ey → γ = k = ω√µε (3.2.45)

entailing that the cut-off wave vector is zero for TEM waves

kc(TEM) = 0 (3.2.46)

From the Helmholtz equation, the (3.2.45) relation and the e−iγz

dependence of each component of the field we get(∂2

∂x2+

∂2

∂y2+

∂2

∂z2+ k2

)(Ex

Ey

)=(

00

)(3.2.47)

Because ∂2

∂z2

(Ex

Ey

)= −γ2

(Ex

Ey

)= −k2

(Ex

Ey

)equation (3.2.47) becomes

the Laplace equation for the transverse components of a TEM wave(∂2

∂x2+

∂2

∂y2

)(Ex

Ey

)=(

00

)(3.2.48)

It is easy to demonstrate that the magnetic field transverse com-ponents Hx, Hy also satisfy Laplace equation, entailing that the



transverse components of the TEM wave behave as static fieldsbetween conductors.

3.3. Scalar and Vector Potentials

The classical electromagnetic theory described briefly above repre-sents light as electromagnetic plane waves whose electric and mag-netic fields may oscillate at any frequency from “zero” to “infinity”.From mathematical point of view, Eω(r, t) and Bω(r, t) can be gen-erated from the coupling of scalar and vector potentials which arefrequently denoted by Φω(r, t) and Aω(r, t) respectively

Eω(r, t) = −∇Φω(r, t) − ∂

∂tAω(r, t) (3.3.1)

Bω(r, t) = ∇ × Aω(r, t) (3.3.2)

Obviously, the scalar and vector potentials are not preciselydefined by the last relations since equations (3.3.1) and (3.3.2) remainunaltered when considering any function σ(r, t) such as

Φω(r, t) → Φω(r, t) − ∂

∂tσ(r, t) (3.3.3)

and

Aω(r, t) → Aω(r, t) + ∇σ(r, t) (3.3.4)

In electromagnetic theory, this property is called Gauge Invari-ance which requires a supplementary condition for the definition ofthe scalar and vector potential Φω(r, t) and Aω(r, t).

In Lorentz Gauge, the supplementary condition is chosen to be

∇ · Aω(r, t) = −µε ∂∂t

Φω(r, t) (3.3.5)

From the propagation equation (3.2.20) and (3.3.1), in absence ofcharges and current densities |j| = 0, ρ = 0, and using the LorentzGauge condition, we get

∇2 Aω(r, t) − µε∂2

∂t2Aω(r, t) = 0 (3.3.6)



and

∇2Φω(r, t) − µε∂2

∂t2Φω(r, t) = 0 (3.3.7)

showing that all the electric and magnetic fields of a monochromaticelectromagnetic wave, Eω(r, t), Bω(r, t), Aω(r, t) and Φω(r, t) prop-agate in space satisfying, exactly the same propagation equation.

If we consider that the current and charge densities are not zero,it is easy to get from Maxwell’s equations (3.1.8) to (3.1.11) thecomplete propagation equations:

∇2 Aω(r, t) − µε∂2

∂t2Aω(r, t) = −µ J(r, t) (3.3.8)

and

∇2Φω(r, t) − µε∂2

∂t2Φω(r, t) = −ρ (r, t)

ε(3.3.9)

The general solutions of the last equations are:

A (r, t) =µ

4π

∫ J(r′, t− |r−r′|

c

)|r − r′| d3r (3.3.10)

and

Φ(r, t) =1

4πε

∫ ρ(r′, t− |r−r′|

c

)|r − r′| d3r (3.3.11)

in which the time delay appearing in the current and charge den-sities is conditioned by the speed of the electromagnetic waves invacuum, c.

At that level it is very important to note that the dimensionanalysis of (3.3.10) shows that the vector potential is inversely pro-portional to time, hence proportional to a frequency.

Now, let us consider a different Gauge condition which consider-ably simplifies the equations. In the so called Coulomb Gauge, thesupplementary condition is chosen as:

∇ · Aω(r, t) = 0 (3.3.12)



From (3.3.5) and (3.3.9), we get Poisson’s equation relating thescalar potential to the charge density:

∇2Φω(r, t) = −ρ (r, t)ε

(3.3.13)

whose solution is now independent of the time delay characterizingthe charge density:

Φ (r, t) =1

4πε

∫ρ (r ′, t′)|r − r′| d

3r′ (3.3.14)

Hence in the absence of charges Φ(r, t) = 0 and from (3.3.1),we get a quite important relation between the electric field and thevector potential in the Coulomb Gauge.

Eω(r, t) = − ∂

∂tAω(r, t) (3.3.15)

The last relation is principally used in classical and quantum elec-trodynamics to calculate the electric field starting from the vectorpotential.

3.4. Vector Potential and Electromagnetic FieldPolarization

Using the above relations, the propagation equation for the vectorpotential in vacuum writes

∇2 Aω(r, t) − 1c2∂2

∂t2Aω(r, t) = 0 (3.4.1)

In the case of a plane monochromatic electromagnetic field thevector potential is real (from mathematical point of view) and it isexpressed in the same way as the electric and magnetic fields

Aω (r, t) = A0(ω) (e−i(k·r−ωt+θ) + c.c.)

= 2 A0 (ω) cos(k · r − ω t+ θ) (3.4.2)

where θ is a phase parameter.For the Coulomb Gauge condition to be satisfied, the following

relation between the wave vector and the amplitude of the vectorpotential should hold:

k · A0(ω) = 0 (3.4.3)



entailing that the wave vector is perpendicular to the vector potentialso that the electromagnetic wave is called transverse.

Taking θ = 0 and using the relations (3.3.15) and (3.3.2), whileputting A0 → A0(ω) ε and k = ω

c k we get directly the electric fieldand magnetic induction expressed through the vector potential

Eω(r, t) = −2ωA0(ω) ε sin (k · r − ω t) (3.4.4)

Bω(r, t) = −1c2ωA0(ω)(k × ε) sin (k · r − ω t) (3.4.5)

The polarization of the radiation, i.e., the spatial orientation ofthe electric field, is also specified by the unit vector of the vectorpotential ε.

From the coupling of the equation (3.2.28) with the last expres-sions one gets in vacuum,

〈Sω〉T = 2ε0c ω2A20(ω) k (3.4.6)

expressing that the power density along the propagation axis(Poynting vector) is also expressed through the vector potential overa period T .

In the same way the energy density of the electromagnetic field isobtained by (3.2.29), (3.4.4) and (3.4.5):

Wω(r, t) = 4ε0ω2A20(ω) sin2(k · r − ωt) (3.4.7)

whose mean value over a period writes:

〈Wω〉T = 2ε0ω2A20(ω) =

1c|〈Sω〉T | (3.4.8)

Consequently, the only knowledge of the vector potential amplitudeA0(ω) is sufficient to deduce the principal physical properties of aTMEM wave of angular frequency ω in vacuum.

We can now have a close look to the vector orientation of theelectric field and that of the vector potential with respect to thepropagation axis. Following the equation (3.4.3), the unit vector ofthe vector potential and thus that of the electric field intensity, ε,lies in a plane which is perpendicular to the propagation vector k.



Considering two orthogonal unit vectors e1 and e2, defining the planeperpendicular to the wave vector k forming a right-handed system:

k = e1 × e2, e1 · e2 = 0 (3.4.9)

so as the unit vector ε of the vector potential can be decomposedsimply as

ε = α1e1 + α2e2 (3.4.10)

And since ε is a unit vector, the following relation between thecoefficients α1 and α2 holds

α21 + α2

2 = 1 (3.4.11)

Thus, we can define two complex orthogonal unit vectors L andR such as

L =1√2(e1 + ie2) and R =

1√2(e1 − ie2) (3.4.12)

appropriate for describing respectively, the Left-hand and Right-hand circularly polarized electromagnetic waves, corresponding totwo states of the radiation polarization.

In the first case, the electric field follows a circular left-hand pre-cession along the propagation axis with the angular frequency ω,while in the second one it follows a circular right-hand precession atthe same angular frequency.

From equations (3.4.2) and (3.4.12), one may easily deduce thecomponents of the vector potential of a circularly polarized planewave in Cartesian coordinates (e1 → x and e2 → y):

A(L)x (z, t) = A(R)

x (z, t) =√

2A0(ω) cos (kz − ω t)

A(L)y (z, t) = −A(R)

y (z, t) = −√

2A0(ω) sin (kz − ω t)

A(L)z = A(R)

z = 0 (3.4.13)

Using (3.3.15) and (3.4.13), the expressions of the corresponding elec-tric field components write:

E(L)x (z, t) = E(R)

x (z, t) = ωA(L)y = −ωA(R)

y



E(L)y (z, t) = −E(R)

y (z, t) = −ωA(L)x = −ωA(R)

x

E(L)z = E(R)

z = 0 (3.4.14)Consequently, an arbitrary state of polarization can be obtained

by the complex unit vector P consisting of a linear combination ofL and R:

P = βLL+ βRR

=1√2(βL(x+ iy) + βR(x− iy)) with |βL|2 + |βR|2 = 1

(3.4.15)For example, a linear polarization consists of the oscillation of the

electric and magnetic field in the orthogonal planes (x, z) and (y, z)along the propagation axis z.

Obviously, such a polarization is easily obtained by the combi-nation of two circularly polarized waves propagating in phase andhaving right and left polarization.

3.5. Guided Propagation of Electromagnetic Waves

In the previous chapter, we have seen that the electric and mag-netic fields of a monochromatic wave obey the propagation equation(3.2.26) in free space. It is of high interest to explore the way the elec-tromagnetic wave propagate in a waveguide of specific cross sectionalshape and dimensions. The problem consists of seeking solutions toMaxwell’s equations satisfying the particular boundary conditionsimposed by the waveguide shape.

Let us consider the simple case in which the inner walls of thewaveguide consists of a homogeneous and isotropic dielectric withfixed values of ε and µ. It is worth noting that a TEM mode canonly propagate in the presence of two or more guiding conductors.Consequently, a TEM mode cannot propagate in a hollow waveguidesince it is composed of a single conductor. Thus, we give below thecomponents of the electric and magnetic fields of transverse elec-tric (TE) and transverse magnetic (TM) modes satisfying Maxwell’sequations and the boundary conditions, in the case of rectangularand circular waveguides.



• Rectangular waveguide of cross section a, b (a > b)

We also assume here time-harmonic fields with an angular fre-quency ω, hence with an exponential dependence eiωt. The propa-gating fields must be solutions of Maxwell’s equations satisfying theboundary conditions, for the components of the electric and magneticfields on the waveguide walls.

• Rectangular waveguide TEm,n modes

Are characterized by the absence of the electric field componentalong the propagation axis Ez = 0. The magnetic field componentsHz has to satisfy the reduced (simplifies the time dependence) prop-agation equation [

∂2

∂x2+

∂2

∂y2+ k2

c

]Hz(x, y) = 0 (3.5.1)

The boundary conditions for the electric field tangential compo-nents are

ex(x, y) = 0 at (y = 0 and b) and

ey(x, y) = 0 at (x = 0 and a) (3.5.2)

The solutions in Cartesian coordinates (x, y, z) are given by

Hx = iH0z

γ

k2c

(mπa

)sin(mπx

a

)cos(nπy

b

)e−iγz (3.5.3)

Hy = iH0z

γ

k2c

(nπb

)cos

(mπxa

)sin(nπy

b

)e−iγz (3.5.4)

Hz = H0z cos(mπx

a

)cos(nπy

b

)e−iγz (3.5.5)

Ex = Hy

(ωµ

γ

); Ey = −Hx

(ωµ

γ

); Ez = 0 (3.5.6)

With m,n = 0, 1, 2, . . . and where γ is the propagation factor

γ =√k2 − k2

c =(µεω2 −

(mπa

)2 −(nπb

)2)1/2

(3.5.7)



with k2 = µεω2 and

k2c =

(2πλc

)2

=(mπa

)2+(nπb

)2(3.5.8)

is the cut-off wave vector and λc the cut-off wavelength correspondingto a cut-off angular frequency

ωc =kc√µε

and λc =2π√(

mπa

)2 +(

nπb

)2 (3.5.9)

Only modes with ω > ωc and λ < λc can propagate in this rectan-gular waveguide. The TE mode with the biggest wavelength (lowerfrequency) that can propagate in the rectangular wave guide witha > b corresponds to the values m = 1 and n = 0.

• Rectangular waveguide TMm,n modes

The TM m,n components are characterized by the absence of themagnetic field component along the propagation axis Hz = 0. Theelectric field components Ez has to satisfy the reduced propagationequation [

∂2

∂x2+

∂2

∂y2+ k2

c

]Ez(x, y) = 0 (3.5.10)

The boundary conditions for ex and ey are also satisfied by ez, i.e.,ez(x, y) = 0 at (x = 0 and a) and ez(x, y) = 0 at (y = 0 and b)

(3.5.11)The solutions in Cartesian coordinates (x, y, z) are given by

Ex = −iE0z

γ

k2c

(mπa

)cos(mπx

a

)sin(nπy

b

)e−iγz (3.5.12)

Ey = −iE0z

γ

k2c

(nπb

)sin(mπx

a

)cos(nπy

b

)e−iγz (3.5.13)

Ez = E0z sin(mπx

a

)sin(nπy

b

)e−iγz (3.5.14)

Hx = −Ey

(ωε

γ

); Hy = Ex

(ωε

γ

); Hz = 0 (3.5.15)

where n,m are integers (including 0). H0zand E0zare the amplitudesof the magnetic and the electric field respectively along the directionof propagation z.



As in the case of TEm,n modes in a rectangular waveguide, thecut-off wavelength λc for the TM m,n writes

λc =2π√(

mπa

)2 +(

nπb

)2 (3.5.16)

Hence, in a rectangular waveguide, only TEm,n and TM m,n modeswith a wavelength shorter than λc can propagate.

• Circular waveguide of radius r0

In a circular waveguide things are slightly different and it is morepractical to use the cylindrical coordinates (r, θ, z).

• Circular waveguide TEm,n modes

In the case of the rectangular waveguide, the electric field com-ponent along the propagation axis Ez = 0, while the magnetic fieldcomponent Hz is a solution to the equation[

∂2

∂ρ2+

1ρ

∂

∂ρ+

1ρ2

∂2

∂θ2+ k2

c

]Hz(ρ, θ) = 0 (3.5.17)

The solution for Hz(ρ, θ) has to be periodic in angular rotationso that Hz(ρ, θ) = Hz(ρ, θ + 2nπ), while the tangential componentof the electric field on the wall should vanish as Eθ(ρ = r0, θ) = 0.

Respecting these boundary conditions the field components havethe expressions

Hz = H0z cos (mθ)Jm

(R(J ′

m)n

ρ

r0

)e−iγz (3.5.18)

Hθ = H0z

γ

k2c

(m

ρ

)sin(mθ)Jm

(R(J ′

m)n

ρ

r0

)e−iγz (3.5.19)

Hρ = −iH0z

(γ

kc

)cos (mθ)J ′

m

(R(J ′

m)n

ρ

r0

)e−iγz (3.5.20)

Eθ = −Hρ

(ωµ

γ

); Eρ = Hθ

(ωµ

γ

); Ez = 0 (3.5.21)



where Jm is the Bessel function of the first kind of order m and R(J ′m)

n

is the nth root of J ′m, the derivative with respect to the argument.

The cut-off wavelength λc and the corresponding cut-off angular fre-quency ωc are also given as a function of the circular waveguide crosssection parameter r0.

λc =2π r0R

(J ′m)

n

(3.5.22)

ωc =R

(J ′m)

n

r0√µε

(3.5.23)

In a circular waveguide with radius r0 only the TEm,n modes witha wavelength shorter than λc can propagate.

• Circular waveguide TMm,n modes

The TM modes are obtained by solving for Ez, the propagationequation: [

∂2

∂ρ2+

1ρ

∂

∂ρ+

1ρ2

∂2

∂θ2+ k2

c

]Ez(ρ, θ) = 0 (3.5.24)

This is an identical equation to that of the Hz component in theTE case. Thus, the general solutions have the same form and areobtained with the boundary condition that the z component of theelectric field vanishes at the walls: Ez(ρ = r0, θ) = 0.

Ez = E0z cos(mθ)Jm

(R(Jm)

n

ρ

r0

)e−iγz (3.5.25)

Eθ = iE0z

γ

k2c

(m

ρ

)sin (mθ) Jm

(R(Jm)

n

ρ

r0

)e−iγz (3.5.26)

Eρ = −iE0z

(γ

kc

)cos(mθ)J ′

m

(R(Jm)

n

ρ

r0

)e−iγz (3.5.27)

Hθ = Eρ

(ωε

γ

); Hρ = −Eθ

(ωε

γ

); Hz = 0 (3.5.28)

The expression for the propagation factor is the same as in theTE modes while the cut-off wavelength and angular frequency cut-off



now write

λc =2π r0R

(Jm)n

(3.5.29)

and

ωc =R

(Jm)n

r0√µε

(3.5.30)

where R(Jm)n is the nth root of Jm, that is Jm(R(Jm)

n ) = 0.

• Case of two parallel plates

This is a particular configuration that accepts TE and TM wavesand can also support TEM waves since it is composed of two conduc-tors. As we have seen in 3.2, the TEM waves behave as the static fieldsbetween conductors and consequently they can be deduced from thescalar potential that satisfies Poisson’s equation (3.3.13) in the plane(x, y) and in the absence of charges.

∇2(x,y) Φ (x, y) = 0 (3.5.31)

Considering two parallel plates in the (x, z) plane separated bya distance y0, having a potential difference of U , and considering adefinite length along the x axis, x0, for 0 ≤ x ≤ x0 and 0 ≤ y ≤ y0,the boundary conditions for the scalar potential write

Φ(x, 0) = 0 and Φ (x, y0) = U (3.5.32)

The solution of (3.5.31) respecting the boundary conditions(3.5.32) is

Φ (x, y) =y

y0U (3.5.33)

The TE field is obtained immediately

E (x, y) = −∇(x,y)Φ(x, y) =1y0U ey (3.5.34)

and the complete expression of the electric field is

E (x, y, z) =U

y0e−ikz ey (3.5.35)

and the magnetic field becomes

H (x, y, z) =U

y0Re−ikz ex (3.5.36)



where R is the impendence of the medium between the parallel platesgiven by (3.2.14). Since Ez and Hz are zero here, the TEM fields inthis situation are analogue to plane waves in a homogeneous medium.

Now, one can easily calculate the potential V (y0) on the plate aty = y0 from the classical electrostatic formalism

V (y0) = −∫ y0

0

U

y0e−ikz dy = U e−ikz (3.5.37)

while the surface current density on the same plate is calculated asfollowing

I(y0) = −∫ x0

0(ey × H) · ez dx =

x0

y0

U

Re−ikz (3.5.38)

This is a quite interesting result showing that TEM waves canpropagate between two parallel plates with a potential difference U ,inducing a current density on the upper plate.

• Density of states

The guided propagation of the electromagnetic waves reveals thatwhatever the shape of the box the exponential part of the wave func-tion e−ik·r of (3.4.2) suffers the boundary conditions. This introducesnaturally the notion that the electromagnetic wave is composed of“states”, whose number is imposed by the shape and the dimensionsof a given volume and can be calculated.

Let us consider the simple case of a cube of a large side L andvolume V . Then the periodic boundary conditions, like in (3.5.8) forthe wave vector are

kx =2πLnx, ky =

2πLny, kz =

2πLnz (3.5.39)

with nx, ny, nz = 0,±1,±2, . . . .The number of modes in the range dkxdkydkz = d3k writes

dkxdkydkz =(

2πL

)3

dnxdnydnz (3.5.40)



Assuming that L is too large compared to the wavelength, nx, ny,nz can be considered as continuous variables

dnxdnydnz =(L

2π

)3

dkxdkydkz =V

8π34πk2dk (3.5.41)

Using the dispersion relation ω = kc and considering two direc-tions of polarization per state, the total number of states in theangular frequency interval dω writes

dn(ω) = Vω2

π2c3dω (3.5.42)

Notice that the same expressions will be used later in quantummechanics for the calculation of the number of photons in a volumeV , revealing an interesting relation between the notion of electro-magnetic field “state” and the concept of the “photon”.

Hence, the states density, corresponding to the number of statesper unit volume and unit frequency ν = ω/2π results readily from(3.5.42),

ρ(ν) =8πν2

c3(3.5.43)

Equation (3.5.41) is extremely useful for transforming the discretesummation over the wave vector k, characterizing each state to acontinuous summation over the angular frequencies

∑k

→ V

8π3

∫4πk2dk =

V

2π2c3

∫ω2dω (3.5.44)

This is the key relation used in all calculations in quantum elec-trodynamics for the description of a system with a large number ofphotons in order to obtain physical results independent of the vol-ume parameter V . It is of high importance to notice that the lastrelation has been obtained under the condition that all the photonswavelengths are much smaller than the characteristic dimensions ofthe volume V .

Finally, we have to remark that according to (3.5.42) the densityof states theory predicts that a single mode of the electromagnetic



field occupies a volume which is roughly

Vsinglemode ∝ 3π2c3

ω3(3.5.45)

As we have seen from the guided propagation descriptions a givenmode of the electromagnetic field can only stand within a vol-ume with characteristic dimensions equal or bigger than the cut-offwavelength. The last relation leads to the important conclusion thatthe minimum volume in which an electromagnetic field mode canstand is proportional to ω−3.

3.6. Conclusion Remarks

The principal conclusion that can be drawn out from the last chap-ter is that the electromagnetic waves can propagate in vacuum andthe propagation equations are immediately deduced from Maxwell’sequations.

However, in Maxwell’s theory the vacuum itself is not a com-pletely empty medium but has an electromagnetic nature, processinga magnetic permeability µ0 = 4π × 10−7Hm−1 and an electric per-mittivity ε0 = 8.85×10−12Fm−1 ≈ 1

36π 10−9Fm−1, and consequently

an impendence of a precise value, Rvacuum =√

µ0

ε0≈ 120πΩ.

The vacuum intrinsic electric and magnetic properties play a cru-cial role in Maxwell’s equations whatever the system of units used (SIor CGS) because they fix, the speed c of the electromagnetic wavesin vacuum c = 1√

ε0µ0≈ 2.9979 108 m s−1, which at the present status

of knowledge is considered to be the highest speed permitted in theuniverse.

Furthermore, it is important to note that the energy density andflux of the electromagnetic waves depend equally on the vacuumproperties ε0 and µ0.

It turns out that the electromagnetic waves, in other words light,appear to behave as a natural vibration of the vacuum, having intrin-sic electric and magnetic properties. The vector potential of the elec-tromagnetic field, issued from Maxwell’s equations is proportional tothe angular frequency.



On the other hand, it is of high interest to note the volume impor-tance for the existence of a “mode” of the electromagnetic field,issued from the guided propagation studies. All the cut-off wave-lengths calculated for the different shapes of waveguides entail thatlonger wavelengths beyond those values cannot stand within a volumeof a given shape and dimensions. Consequently, an electromagneticwave mode of a given wavelength, even the simplest one composedby a single state, cannot subsist in a volume whose dimensions corre-spond to a smaller cut-off wavelength. The characteristic volume cor-responding to a single mode of the electromagnetic field is inverselyproportional to the cube of the angular frequency.

Finally, the number of states per unit volume and unit frequencycalculated by ρ(ν) = 8πν2

c3is only valid in a volume whose dimensions

are considerably bigger than the wavelength of the electromagneticwaves.

Bibliography


2. M. Born and E. Wolf, Principles of optics, Cambridge: Harvard UniversityPress. 1999.

3. B.H. Bransden and C.J. Joachain, Physics of atoms and molecules, London:Longman Group Ltd. 1983.

4. S.L. Chuang, Physics of photonic devices, New Jersey: John Wiley & Sons.2009.


6. H. Haken, Light, Amsterdam–Oxford: North Holland Publishing. 1981.7. M.D. Pozar, Microwave engineering, 3rd Edition, New York: John Wiley &

Sons, Inc. 2005.8. F.H. Read, Electromagnetic radiation, New York: John Wiley & Sons. 1980.9. B.E.A. Saleh and M.C. Teich, Fundamentals of photonics, New York: John

Wiley & Sons. 2007.



Chapter 4

From Electromagnetic Wavesto Quantum Electrodynamics

4.1. Elements of Quantum Mechanics

• Blackbody radiation and the ultraviolet catastrophe

The study of the properties of radiation emitted by hot bodieswas the starting point for the development of the quantum theoryat the end of the 19th and the beginning of the 20th centuries. Theexperimental evidence has shown that the spectral distribution, inother words the distribution of the wavelengths of the radiation, aswell as the total energy radiated depend directly on the temperatureof the body.

The number of standing electromagnetic waves, i.e., the number ofmodes, per unit volume and unit frequency within a cavity is givenby the expression (3.5.43). It can be shown that for a sufficientlybig cavity this expression holds whatever its shape. Considering, theelectromagnetic waves to be in thermal equilibrium at temperatureT with the walls of the cavity, the probability p(E) that a modehas an energy E included between E + dE, is obtained in classicaltheory by

p (E) =e−E/kBT

kBT(4.1.1)

where kB is Boltzmann’s constant.

39



The mean energy of a mode is thus

E =∫ ∞

0Ep (E) dE =

∫ ∞

0Ee−E/kBT

kBTdE = kBT (4.1.2)

Consequently, all the modes have the same mean energy, indepen-dent of the frequency. Using (3.5.43), the energy per unit volume ofthe cavity at temperature T and per unit frequency writes

u (T, ν) =8πν2

c3kBT (4.1.3)

The last expression is known historically as the Rayleigh-Jeansenergy density which fails to describe the experimentally observedresults at high frequencies, often called the “ultraviolet catastrophe”,while giving an infinite total energy per unit volume of the cavitywhen summing up along all the frequencies

U(T ) =∫ ∞

0

8πν2

c3kBTdν → ∞ (4.1.4)

In order to resolve this problem, Max Planck introduced the orig-inal idea that the cavity walls are composed of local oscillators, eachone being capable of absorbing and emitting discrete quantities ofelectromagnetic energy nhν, where n is an integer and h is a uni-versal constant attributed in his honor to him, Planck’s constant.This absolutely revolutionary concept at the beginning of the 20thcentury, led in the following years to the development of the quantumtheory of the electronic energy levels in matter and in the quantiza-tion of the electromagnetic field.

Following Planck’s idea, the total energy of a mode is En = nhν

and the probability for a mode to be composed of n quantum ofelectromagnetic energy, called photons, is

pn = γ e−nhν/kBT (4.1.5)

where γ is a normalization factor so that the sum of all probabilitiesbe equal to 1.

Hence,

γ∑n

e−nhν/kBT = 1 ⇒ γ = 1 − e−hν/kBT (4.1.6)


From Electromagnetic Waves to Quantum Electrodynamics 41

Now, using β = 1kBT , the mean energy of each mode in thermal

equilibrium writes

ε =∑∞

0 nhν e−nhνβ∑∞0 e−nhνβ

= − d

dβlog

( ∞∑0

e−nhνβ

)

= − d

dβlog(

11 − e−hνβ

)=

hν

ehνβ − 1(4.1.7)

and the electromagnetic energy per unit volume and unit frequencynow becomes within Planck’s hypothesis

u (T, ν) =8πν2

c3hν

ehν/kBT − 1(4.1.8)

The last expression is Planck’s radiation law and interprets quitesatisfactorily all the experimental results at any frequency and tem-perature of a black body.

• Energy and momentum operators in quantum mechanics

In the first two decades of the 20th century, the introduction byMax Planck of the quantized energy levels of the atoms and moleculesresulting in quantized emissions and absorptions of the electromag-netic field, as well as many experiments demonstrating the wave prop-erties of electrons (Davisson and Germer), created the necessity toelaborate a new theory.

The birth of quantum mechanics came along with Niels Bohr’sand Erwin Schrodinger’s works in 1925, based on the ideas put for-ward by Louis de Broglie in 1924, following which the energy andthe momentum of particles are associated to a frequency and a wave-length, through the formalism already established previously for light

E = hν = ω P =h

λek = k (4.1.9)

A plane wave is associated to the particle

Ψ (r, t) = Ψ0 ei(k·r−ωt) (4.1.10)

From the last two equations, one can easily deduce

i∂

∂tΨ = EΨ − i∇Ψ = pΨ (4.1.11)



According to the last relations, a fundamental postulate of quan-tum mechanics was born according to which for a nonrelativisticparticle, either it is free or not, the energy and the momentum arerepresented by the operators

E → i∂

∂tp→ −i∇ (4.1.12)

At that point, we recall the correspondence already deduced fromMaxwell’s equations in (3.2) between the angular frequency and thetime derivative operator ω → i ∂

∂t getting from (4.1.12), E → ω,when considering plane wave solutions.

Taking into account the potential energy V of the particle in thetotal energy and using (4.1.11), we get the Schrodinger equationwhich consists the basis of quantum mechanics

i∂

∂tΨ =

(P 2

2m+ V

)Ψ = −

2

2m∇2Ψ + VΨ (4.1.13)

To complement the fundamentals of the quantum mechanical the-ory, we can now consider the action of the momentum operatorpα = −i ∂

∂α along a Cartesian coordinate α where α is x, y or zacting on a function φ(α). For that we apply the commutation oper-ator [α, pa] = αpα − pαα on a function φ(α)

[α, pa]φ(α) = −i(α∂φ(α)∂α

− ∂

∂ααφ(α)

)= iφ(α) (4.1.14)

The last result shows that the commutation between the coor-dinate α and the corresponding component of momentum opera-tor pα do not commute and can be generalized using the Cartesiancoordinates

[x, px] = [y, py] = [z, pz] = i (4.1.15)

All the other operators vanish, that is [α, pα′ ] = iδα,α′ where δα,α′

is Kronecker symbol such as δα,α′ = 1 when α = α′ and δα,α′ = 0 forα = α′.

The physical consequence of this is that for the non-commutingoperators a “simultaneous” measurement of the correspondingobservables is impossible.



• Particle in a square potential well — correspondence with thewaveguides

The simplest case of the application of Schrodinger equation isto consider a particle of mass m in a one-dimensional infinite squarepotential well, V (z) defined by

V (z) = 0 for |z| > z0V (z) → ∞ for z < −z0 and z > z0 (4.1.16)

Since the potential tends to infinity for z < −z0 and z > z0, theparticle has no possibility to be outside the borders of the poten-tial, consequently its wave function Ψ(z) must tend to zero in thatinterval.

For |z| < z0 where V (z) = 0, the Schrodinger equation is simplywritten as

− 2

2md2

dz2Ψ(z) = EΨ (z) (4.1.17)

Putting

k =

√2mE

(4.1.18)

the solution of (4.1.17) is a general trigonometric function of the form

Ψ(z) = c1 sin(kz) + c2 cos(kz) (4.1.19)

Taking into account that Ψ(z) must be zero at z = ±z0 we gettwo solutions:

Either

c1 = 0 and cos(kz0) = 0 (4.1.20)

so that the only permitted values of k are

kn =nπ

2z0(n = 1, 3, 5, 7, . . .) (4.1.21)

Or

c2 = 0 and sin (kz0) = 0 (4.1.22)

And the permitted values of k are now

kn =nπ

2z0(n = 2, 4, 6, 8, . . .) (4.1.23)



Consequently the energy is deduced from (4.1.18) and is found tobe quantified following the integer values of n

En =

2k2

2m=

2

8m

(nπ

z0

)2

(n = 1, 2, 3, 4, . . .) (4.1.24)

The wave function for both solutions can be normalized tounity by ∫ z0

−z0

|Ψn(z)|2dz = 1 (4.1.25)

getting for both solutions respectively

Ψn(z) =1

z1/20

cos(nπ

2z0z

)(n = 1, 3, 5, . . .) (4.1.26)

and

Ψn(z) =1

z1/20

sin(nπ

2z0z

)(n = 2, 4, 6, . . .) (4.1.27)

Comparing (4.1.21) and (4.1.23) with (3.5.8), we deduce thatwhen attributing a wave function to a particle entails automatically,it is submitted to boundary conditions obtaining for k equivalentexpressions with those of the guided electromagnetic waves.

4.2. Harmonic Oscillator in Quantum Mechanics

• From the classical expressions to the quantum mechanical ones

In order to have a precise understanding of the mathematicalprocess of the quantization of the electromagnetic field, with whichwe will deal in the next chapters, it is of outstanding importanceto know how the harmonic oscillator is described in the quantummechanical approach.

We may start with the simple case of one dimensional motionalong the x axis of a particle of mass m submitted to a central forceexpressed by the well-known Hooke’s law

F = −∂V (x)∂x

x

|x| = −Kx x (4.2.1)



where V (x) is the potential energy and Kx is the force constant.

V (x) =12Kx x

2 (4.2.2)

The kinetic energy at any coordinate x of the motion of the par-ticle is

Tx =12m

(∂x

∂t

)2

=P 2

x

2m(4.2.3)

where Px = m(∂x∂t ) is the momentum of the particle along the x axis.

The total energy of the system writes

E(x) = Tx + V (x) =P 2

x

2m+

12Kx x

2 (4.2.4)

Consequently, putting

ωx =(Kx

m

)1/2

(4.2.5)

the energy simply becomes

E(x) =P 2

x

2m+

12mω2

x x2 (4.2.6)

Notice that ωx has the dimensions of an angular frequency alongthe x axis.

As we have seen in the previous chapter, the general expressionof the three dimensions momentum operator in quantum mechanicsis

P = −i∇ (4.2.7)

which writes along the x axis

Px = −i ∂

∂x(4.2.8)

Considering ψ (x) to be the eigenfunction, the Schrodinger equa-tion for the linear harmonic oscillator derives immediately from(4.2.6),

− 2

2m∂2ψ(x)∂x2

+12mω2

xx2ψ(x) = E(x)ψ(x) (4.2.9)



Putting

ξ = x

(mKx

2

)1/4

(4.2.10)

and considering the dimensionless fraction

ηx =2E(x)

ωx(4.2.11)

we get the simplified equation

∂2ψ(ξ)∂ξ2

+ (ηx − ξ2)ψ(ξ) = 0 (4.2.12)

The asymptotic behavior of the above equation when |ξ| → ∞suggests looking for solutions of the form:

ψ(ξ) = e−ξ2/2H(ξ) (4.2.13)

where H(ξ) should be polynomial functions not affecting the asymp-totic behavior.

Solutions of (4.2.12) are found for

ηx = 2nx + 1 with ηx = 0, 1, 2, . . . (4.2.14)

called the Hermite polynomials Hn(ξ) whose general expression is

Hnx(ξ) = (−1)nxeξ2 dnxe−ξ2

dξnx(4.2.15)

Getting the complete eigenfunctions

ψnx(ξ) = Ce−ξ2/2Hnx(ξ) = C(−1)nxeξ2/2 d

nxe−ξ2

dξnx(4.2.16)

Where C is a constant which can be determined by imposing thesummation over the whole space of the square modulus of the wavefunction to be normalized to unity. And the eigenvalues, that is theenergy spectrum with E(x) → Enx(ωx) writes

Enx(ωx) = ωx

(nx +

12

)(4.2.17)

Hence, the energy of the linear one dimensional harmonic oscilla-tor in quantum mechanics can only get the discrete values of relation(4.2.17) with nx = 0, 1, 2, . . . .



This is easily generalized in a three dimensional harmonic oscil-lator for which the Schrodinger equation and the energy spectrumtake the expressions

− 2

2m∇2Ψ(r) +

12m[ω2

xx2 + ω2

yy2 + ω2

zz2]Ψ(r) = EΨ(r)

(4.2.18)

E = ωx

(nx +

12

)+ ωy

(ny +

12

)+ ωz

(nz +

12

)(4.2.19)

This is a quite amazing result showing that a zero energy statecannot exist, and even an one dimensional harmonic oscillator, incomplete absence of any external interaction, at zero temperature,has a minimum nonzero energy which equals to E0(ω) = 1

2ω depend-ing only on the potential and the mass of the particle accordingto (4.2.5).

• Dirac representation, creation and annihilation operators

For an one dimensional harmonic oscillator of mass m at a coor-dinate q with a momentum p = mdq/dt, employing the reducedexpressions for the canonical variables of position Q = |q|√m andmomentum P = |p|/√m, the total energy (4.2.6) becomes

E =12(P 2 + ω2Q2) (4.2.20)

The transition to the quantum mechanical Hamiltonian isobtained by imposing the following expressions for the momentumand position operators respectively

P = i

√ω

2(a+ − a) and Q =

√

2ω(a+ + a) (4.2.21)

where a, a+ are respectively the annihilation and creation operatorsof a quantum of energy of the harmonic oscillator, which can beexpressed by inversing equation (4.2.21)

a =

√1

2ω(ωQ+ iP ) and a+ =

√1

2ω(ωQ− iP ) (4.2.22)



Considering (4.1.15), the commutation relation between the posi-tion and momentum operators is

[Q,P ] = i (4.2.23)

and we get

[a, a+] = 1 (4.2.24)

Considering (4.2.21) and (4.2.20), we obtain directly the Hamil-tonian of the harmonic oscillator expressed with respect to the anni-hilation and creation operators

HHO =12(P 2 + ω2Q2) = ω

(a+a+

12

)(4.2.25)

In Dirac representation, each eigenfunction takes the simpleexpression ψn(ξ) ⇒ |n〉 corresponding to a quantum state composedof n harmonic oscillators.

It can be shown that the actions of the creation and annihilationoperators a+ and a respectively on the eigenfunction are expressedas following

a+|n〉 =√n+ 1|n+ 1〉

a|n〉 =√n|n− 1〉 (4.2.26)

Consequently, the simplified Schrodinger equation in Dirac repre-sentation is

HHO|n〉 = E|n〉 → ω

(a+ a+

12

)|n〉

= ω

(n+

12

)|n〉 (4.2.27)

The successive application of a first and a+ next, i.e., a+ a on the|n〉 state translates the number n of the quantum oscillators havingenergy ω. Thus, a+ a is also called the number operator.

• The Dirac equation

In the last sections we have resumed basic principles of quantummechanics and the Schrodinger equation (4.1.13) has been established



by using a non-relativistic Hamiltonian. For a relativistic free particlewith non-zero rest mass the energy is

E2 = (c p)2 + (mc2)2 (4.2.28)

Introducing in the last expression the operator definitions (4.1.12)for the energy and momentum we obtain the relativistic Schrodingerequation

−2 ∂

2

∂t2Ψ(r, t) = −

2c2∇2Ψ(r, t) + (mc2)2Ψ(r, t) (4.2.29)

The last equation is usually called Klein-Gordon equation and isemployed mainly for the description of a relativistic free particle withzero-spin. The presence of the second derivative with respect to timein (4.2.29) was rather bothering for Dirac who was looking for a firstorder differential equation. He managed to overcome the difficultyrelated to the expression of the Hamiltonian through the square rootof the energy by linearizing the equation. For that, he assumed thata vector α and a scalar matrix β, both independent of E, p,r and t

should exist for the following equation to hold

E =√

(c p)2 + (mc2)2 = c α · p+ βmc2 (4.2.30)

Using again (4.1.12) we get directly the Dirac equation(i∂

∂t+ ic α · ∇− βmc2

)Ψl = 0 (4.2.31)

where Ψl is now a column matrix with l components.This is a first order time differential equation for which the Dirac

Hamiltonian HD writes

HD = −ic α · ∇ + βmc2 (4.2.32)

Obviously, for HD to be Hermitian the matrices α and β haveto be Hermitian too. On the other hand, each component of the Ψl

column matrix has to satisfy the equation (4.2.29). Hence, it has beendemonstrated that these conditions are fulfilled by α and β matricesof minimum dimensions 4×4. This corresponds to the description of aspin 1/2 particle whose wave function involve two spin states, but also



of its anti-particle. Consequently, the Dirac equation has the meritto be capable of predicting the existence of spin 1/2 antiparticles.For the description of the electron and positron i.e., α and β takethe form

αi =(

0 σi

σi 0

)(4.2.33)

with

σi=1 =(

0 11 0

), σi=2 =

(0 −ii 0

), σi=3 =

(1 00 −1

)(4.2.34)

where σi are known as the Pauli matrices, and

β =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

(4.2.35)

We can easily verify that α and β are Hermitian, α = α+ and β = β+.Thus, the Dirac equation is well appropriate for the description

of spin 1/2 particles and antiparticles and is considered as one of thegreatest achievements of modern physics.

4.3. Quantum Electrodynamics (QED)and the Photon Description

Brief description of selected experiments that have historicallyplayed an important role for the introduction of the photon concept.

We have seen in 4.1 that for the interpretation of black bodyradiation, Planck was the first to introduce the revolutionary idea ofthe quantized absorptions and emissions of the electromagnetic fieldby oscillators.

In this chapter, we will discuss the main experiments that haveled to the particle concept of the electromagnetic field, and then wewill describe the theoretical methods that have been developed, inorder to transit from the continuous electromagnetic waves to thequantum electromagnetic entity called the photon.



• The photoelectric effect and the quantum interpretation

By the end of the 19th century, it has been observed by manyphysicists like Hertz, Lenard, Stoletov and others, that metal surfaceseject charged particles when irradiated by ultraviolet light. In a seriesof experiments measuring the charge to mass ratio of the ejectedcharged particles, Lenard demonstrated that these were electronsand the effect was called photoelectric.

Based on Planck’s ideas on the quantized emission of radiationby the equally quantized energy levels of atomic oscillators for theinterpretation of the black body radiation, Einstein proposed in 1905,an explanation for the photoelectric effect. He advanced that lightconsists of point particles, called photons, carrying an amount ofenergy proportional to the frequency of the light E = hν, where h isPlanck’s constant.

Hence, considering that a minimum work W is required for anelectron to escape from the surface of the metal, he deduced that themaximum kinetic energy of the ejected electron is simply obtained by

Ekin =mu2

2= hν −W (4.3.1)

From 1914 to 1917, Millikan carried out many experiments onthe photoelectric effect using the visible light on lithium, potassiumand sodium, all of these characterized by low values of the extractionwork W and confirmed the last relation.

• Compton scattering

Modifications of the wavelength of X-rays when scattered by elec-trons have been reported by many experiments between 1912 and1920. In 1922, for the interpretation of the experimental observations,Compton, who had carried out many similar experiments himself,proposed the revolutionary concept that X-rays whose wavelength ismodified are scattered in crystals by electrons almost at rest.

The theoretical aspect of this effect is described by considering notonly electrons but also photons as integral particles and by applyingthe energy and momentum conservation laws.



Consider an X-ray photon with frequency ν colliding with an elec-tron with rest mass m0 which after the collision acquires momentum.

The energy conservation law writes

hν +m0c2 = hν ′ +mc2 ⇒ h(ν − ν ′) = (m−m0)c2 (4.3.2)

and the momentum conservation

hν

c=hν ′

ccos θ + pe cosφ⇒ hν

c− hν ′

ccos θ = pe cosφ (4.3.3)

hν ′

csin θ = pe sinφ⇒

(hν ′

c

)2

sin2 θ = p2e sin2 φ (4.3.4)

where θ is the angle of the scattered photon trajectory from thecollision axis and φ that of the scattered electron.

Taking the square of (4.3.3) and using (4.3.4) and (4.3.2), we get

p2e =

(hν

c

)2

− 2hν

c

hν ′

c+(hν ′

c

)2

+ 2m0ch(ν − ν ′)

c(4.3.5)

The energy conservation for the electron writes

m2c4 = m20c

4 + p2ec

2 ⇒ p2e = (m2 −m2

0)c2 (4.3.6)

Combining (4.3.3) and (4.3.4), we obtain

p2e =

(hν

c

)2

− 2hν

c

hν ′

ccos θ +

(hν ′

c

)2

(4.3.7)

Equating (4.3.5) and (4.3.7), we get(hν ′

c

)2

+(hν

c

)2

− 2hν

c

hν ′

ccos θ

=(hν

c

)2

− 2hν

c

hν ′

c+(hν ′

c

)2

+ 2m0ch(ν − ν ′)

c

getting the frequency of the scattered photon by the electron

ν ′ =ν

1 + hν(1−cos θ)m0c2

(4.3.8)

The last relation obtained by particle dynamics interprets quitesatisfactorily the experimental results demonstrating the particle



nature of the photon, which is characterized by energy, momentumand directionality.

Finally, what is also extremely important here is that the Comp-ton scattering demonstrates that the energy of a single photon, bythe same token, its wavelength and frequency, can be modified duringa collisional process.

• Low intensity Young’s double-slit interferences

As we have seen in the historical survey, Young performed thefamous double-slit experiment in 1802, obtaining an interference pat-tern that could be only explained by the wave theory of the light.At that time, it was the strongest argument supporting the waveconcept of the light against Newton’s particle hypothesis.

However, in 1909, Taylor repeated the same experiments by usingextremely low intensity light sources (Fig. 4.3.1) and observed that“spots” were appearing on a photographic plate whose graduateaccumulation tended to form interference patterns.

This was readily interpreted by many authors as a demonstra-tion of the photons existence. More recently, using lasers and singlephoton pulse techniques, many experiments have also revealed thecreation of spots on the detection screen and the gradual formationof interference pattern.

Very low intensity light source

Double slit screenSpots forming gradually an interferencepa ern

Fig. 4.3.1. Schematic representation of Taylor’s double slit diffraction experiment inwhich the light source has an extremely low intensity.



• Second quantization

The photoelectric effect, Compton’s scattering and Taylor’sexperiments, among others, have given a decisive impulse toward thegeneral acceptance of the particle nature of light and consequently,a theoretical effort had started to model the quantized electromag-netic field based upon the harmonic oscillator quantum mechani-cal model described in (4.2) satisfying Schrodinger’s equation. Alongthese developments the question of the photon wave function ariseas well that of the photon position operator.

• Quantization process of the electromagnetic field

We will give now the main lines of the electromagnetic fieldquantization procedure, which is generally described in the Coulombgauge.

Considering that the electromagnetic field vector potential A(r, t)consists of a superposition of plane waves, the classical electromag-netic field energy in a given volume V can be written according to(3.4.8) as

EV = 2ε0V∑k,λ

ω2k|Akλ|2 (4.3.9)

where ε0 is the vacuum permittivity and |Akλ| the vector potentialamplitude corresponding to the mode with a wave vector k, angularfrequency ωk and polarization λ. Caution has to be taken here not toconfuse two different physical entities, number of polarizations andwavelength, which are both denoted in QED with λ.

Notice that (4.3.9) derives from (3.4.8), which is a mean value overa period, in other words over the wavelength. It is generally acceptedthat it is difficult to conceive a photon along the propagation axiswith dimensions less than the wavelength. On the other hand, all thecalculations on the guided propagation of the electromagnetic wavesdeveloped in Chapter 3 (3.5) show that a minimum volume has tobe considered for a given mode to subsist. Consequently, cautionhas to be taken when using the formalism (4.3.9) in a volume V ofgiven dimensions for the total energy calculation based on the modessummation.



In fact, for a fixed volume V we can write more rigorously

EEM = 2ε0V∑

k≥kc,λ

ω2k|Akλ|2 (4.3.10)

where the summation on the wave vectors has a lower cut-off limitdefined by the boundary conditions of the box dimensions V . For thesake of simplicity we will drop this lower limit in the next calcula-tions, which comes out to consider that the volume V is much biggerthan the wavelengths of the modes considered.

The equivalence with the harmonic oscillator Hamiltonian isobtained by introducing equivalent expressions with the definitions(4.2.21) and (4.2.22), relating Akλ to the canonical variables of posi-tion Qkλ and momentum Pkλ

Akλ =1

2ωk

√ε0V

(ωkQkλ + iPkλ)

A∗kλ =

12ωk

√ε0V

(ωkQkλ − iPkλ)

Qkλ =√ε0V (Akλ +A∗

kλ)

Pkλ = −iωk

√ε0V (Akλ −A∗

kλ) (4.3.11)

Using the above relations in (4.3.9), the energy of the electromag-netic field writes as a function of Qkλ and Pkλ

EEM =12

∑k,λ

(P 2kλ + ω2

kQ2kλ) (4.3.12)

Comparing the last relation with (4.2.25), it can be deduceddirectly that each mode of the radiation field can be considered equiv-alent to a harmonic oscillator.

In order to express the amplitude of the vector potential withrespect to the quantized energy of the photons ω, we start fromthe monochromatic electromagnetic wave energy density equation(3.4.7). The energy density of N photons in a box V equals the meanenergy density over a period 2π/ω of the classical electromagnetic



field

〈4ε0ω2A20(ω) sin2(k ·r − ωt)〉period = 2ε0ω2A2

0(ω) =Nω

V(4.3.13)

Consequently, for N = 1 in the relation (4.3.13), we obtainthe link between the classical electromagnetic wave issued fromMaxwell’s equations and the quantized field operators used in QED,which is expressed by the intermediate of the vector potential ampli-tude in a volume V as follows

Akλ =

√

2ε0ωkVakλ A∗

kλ =

√

2ε0ωkVa+

kλ (4.3.14)

Where akλ and a+kλ are respectively the annihilation and creation

operators of a k mode and λ polarization photon composing thephoton number operator Nkλ = a+

kλakλ.Using the above expressions for the amplitude of the vector poten-

tial the classical formalism of equation (3.4.2) becomes in QED, whentaking into account all k modes and polarizations λ,

A (r, t) =∑k,λ

√

2ε0ωkV[akλεkλe

i(k·r−ωkt+θ) + a+kλε

∗kλe

−i(k·r−ωkt+θ)]

(4.3.15)According to (4.2.26), the creation and annihilation operators of

a k mode and λ polarization photon have the properties

a+kλ|nkλ〉 =

√nkλ + 1|nkλ + 1〉

akλ|nkλ〉 =√nkλ|nkλ − 1〉

a+kλakλ|nkλ〉 = Nkλ|nkλ〉 = nkλ|nkλ〉 (4.3.16)

where the eigenvalue of the photon number operator expresses thenumber of photons.

Using the link expressions (4.3.14) in the equations of Qkλ and Pkλ

given in (4.3.11), one obtains the position and momentum operatorsexpressed with respect to akλ and a+

kλ

Qkλ =

√

2ωk(a+

kλ + akλ) Pkλ = i

√ωk

2(a+

kλ − akλ) (4.3.17)



Replacing the canonical variables Qkλ and Pkλ in equation(4.3.12) by the expressions of Qkλ and Pkλ (4.3.17) and consideringthe commutation relation [akλ, a

+kλ] = 1 we get the Hamiltonian of

the quantized electromagnetic field

HEM =12

∑k,λ

(P 2kλ + ω2

kQ2kλ) =

∑k,λ

ωk

(Nkλ +

12

)(4.3.18)

Hence, in QED, photons are considered as point particles createdand annihilated by the operators a+

kλ and akλ respectively which obeythe properties of (4.3.16), and having the energy of an ensemble ofharmonic oscillators given by (4.3.18).

Generally, the polarization λ takes two values corresponding toRight- and Left-hand circularly polarized photons.

The term 1/2 of the established radiation Hamiltonian is an occu-pation operator, which rigorously means that in complete absenceof photons, i.e., when Nkλ = 0, empty space is filled permanentlyby “half” photons of all frequencies and polarizations. Hence, thequantization procedure of the classical expression of the energy ofthe electromagnetic waves to get the radiation Hamiltonian leadsto the zero-point radiation field, also called the quantum vacuum,expressed by the term

∑k,λ

12ωk. This term is mainly responsible

for the QED singularities related to the infinite energy of the vacuum,while at the same time it is used in many semi-classical descriptionsfor the explanation of various experimentally observed effects, suchas spontaneous emission, atomic levels energy shifts (Lamb shift),the Casimir effect etc.

• The Photon Wave Function

In the first quantization process that we have briefly describedin 4.1 and 4.2 particles with non-zero mass in a quantum stateare characterized by a wave function Ψ(r, t), depending on spaceand time, which satisfies Schrodinger’s equation (4.1.13) with eigen-values the energy corresponding to the given quantum state. Thewave functions i.e. for the stationary quantum states of a particlein a square or harmonic potential well are given by (4.1.26/27) and



(4.2.13) respectively. In the second quantization process the wavefunctions for massive particles are expressed through operators.

The photon, as a relativistic massless particle, constitutes a par-ticular case and the proposed wave functions published in the litera-ture until now are not generally accepted. An important theoreticaldifficulty, among others, lays in the fact that the photon is notconsidered as a “localizable particle” and consequently a positionoperator cannot be defined. Furthermore, even the concept of a time-dependent wave function for the photon is rejected by some authors.

Nevertheless, various fruitful attempts have been carried out inorder to establish a full photon wave function and can be found in thebibliography of this Chapter. The most pertinent theoretical devel-opments are based on the complex vector function F (r, t) initiallyintroduced by Riemann

F (r, t) =1

2√ε0D(r, t) + i

12√µ0

B(r, t) (4.3.19)

where the real functions of the electric displacement flux densityD(r, t) and the magnetic field flux density B(r, t) have been definedin (3.1.4).

Using F (r, t) the Maxwell’s equations for homogeneous mediumcan be written in the following compact form

i∂

∂tF (r, t) = c∇× F (r, t) (4.3.20)

∇ · F (r, t) = 0 (4.3.21)

In this way, a proposed photon wave function can be defined as

Φ(r, t) =

12√ε0D(r, t) + i

12√µ0

B(r, t)

12√ε0D(r, t) − i

12√µ0

B(r, t)

(4.3.22)

which is a six component vector satisfying Schrodinger’s equation inthe form

i∂

∂tΦ(r, t) = c

−i∇ · S 0

0 i∇ · S

Φ(r, t) (4.3.23)



where S are the spin matrices for a spin 1 particle with the Cartesiancomponents

Sx =

0 0 0

0 0 −i0 i 0

, Sy =

0 0 i

0 0 0−i 0 0

, Sz =

0 −i 0i 0 00 0 0

(4.3.24)The photon wave function Φ(r, t) defined in (4.3.22) does not

involve a position operator but due to the presence of the complexRiemann field F (r, t) its square modulus provides the electromag-netic field energy density (3.2.29) at an instant t and at a givencoordinate r on the propagation axis

|Φ(r, t)|2 = Wω(r, t) =12

(ε0| E(r, t)|2 +

1µ0

| B(r, t)|2)

(4.3.25)

Obviously, the energy can be expressed directly by integrating inspace the bilinear form of the complex vector function F (r, t) getting

E =∫

F ∗(r, t) · F (r, t) d3r (4.3.26)

Some different versions from the one described above have beenadvanced for the photon wave function but without introducing a realprogress in the problem of position operator which still remains anopen question in the field of the photon localization representationformalism which is tightly related to the experimental conditions.Mandel was the first to introduce the definition of a “detection oper-ator” for the interpretation of the photon detection experiments thathe was carrying out in the 60s

M(r) =∫d3k

8π3

∑λ

εkλaλ(k) eik·r (4.3.27)

where λ takes two values corresponding to the Left and Right circularpolarizations.

By the intermediate of this “detection operator” a local “photonnumber operator” n(V ) which is volume dependent can be defined

n(V ) =∫

VM∗(r) · M(r)d3r (4.3.28)



It has been thus demonstrated that the localization region for ak-mode photon cannot be shorter than its wavelength λk and conse-quently the localization volume has to satisfy the condition

V photonlocalization

∝ λ3k (4.3.29)

This is an important theoretical and experimental result that willbe quite useful for the development of a non-local description for thephoton which will be discussed in the following chapters.

4.4. Interaction between Electromagnetic Wavesand Charged Particles, Reality of the Vector Potential

• Interaction Hamiltonian between an electromagnetic wave and acharged particle

The interaction Hamiltonian of a charged particle of charge q andmass m, having a momentum p, in an electromagnetic field charac-terized by a vector potential A(r, t) and a scalar potential Φ(r, t) iswritten as

H =1

2m(p− q A (r, t))2 + qΦ(r, t) (4.4.1)

Using the Coulomb gauge∇ · A(r, t) = 0 and Φ(r, t) = 0 (4.4.2)

and considering the momentum operator (4.2.7), the time dependentSchrodinger equation for an electron, where q = −e, in the electro-static Coulomb potential of an hydrogenic atom − Ze2

4πε0r , interactingwith the electromagnetic field writes

i∂

∂tΨ(r, t)

=(−

2

2m∇2 − Ze2

4πε0r− ie

mA (r, t) · ∇ +

e2

2mA (r, t)2

)Ψ(r, t)

(4.4.3)

In fact, it can be easily shown that the Coulomb gauges makes A

and ∇· to commute∇ · ( AΨ) = A · (∇Ψ) + (∇ · A)Ψ = A · (∇Ψ) (4.4.4)



In the case of weak fields, the term of the square of the vectorpotential in Schrodinger equation is quite small compared to thelinear one and consequently, the main part of the interaction Hamil-tonian between the bounded electron and the electromagnetic fieldwrites

Hint(t) ∼= − iem

A(r, t) · ∇ (4.4.5)

which is time dependent because of the vector potential.From the established interaction Hamiltonian, arises the question

whether the vector potential represents a mathematical artifact or ifit corresponds to a physical entity.

• Reality of the vector potential, Ehrenberg–Siday or Aharonov–Bohm effect.

Obviously, according to the Hamiltonian (4.4.1), the vector poten-tial directly affects the modulus and the direction of the momentumof a charged particle.

In 1949, W. Ehrenberg and R.E. Siday were the first to deducethe effect of the vector potential of the electromagnetic fields oncharged particles. The Ehrenberg–Siday effect is a fine demonstra-tion of the real existence of the scalar and vector fields issued fromthe electromagnetic theory. In fact, before Ehrenberg’s and Siday’sworks, the vector and scalar fields, A(r, t); Φ(r, t) respectively, werebelieved to be simply mathematical constructs for the developmentof the gauge theories in electromagnetism. In reality, both of thesepotentials interact with a charged particle even in absence of electricand magnetic fields.

The same effects were later rediscovered by Aharonov and Bohmin 1959 and were confirmed experimentally by Chambers in 1960 andOsakabe in 1986 using a quite advanced experimental apparatus.

From a theoretical point of view the behavior of a charged particlewith charge q and mass m in the vicinity of a solenoid is describedby the Hamiltonian (4.4.1). In the case of a very long solenoid alongthe z axis the magnetic field is uniform in the region inside andzero outside. Considering that the solenoid is not charged the scalarpotential Φ can be put to zero. However, the vector potential in the



outside region is not zero and depends on the magnetic field flux inthe solenoid

A =1

2π r

∫s

B · dS eθ (4.4.6)

where r is the radial distance from the z axis, S is the surface enclosedin a circle with radius r perpendicular to z and eθ the angular unitvector in cylindrical coordinates.

Using the expression of the momentum operator from (4.1.12)Schrodinger’s equation for the charged particle in the vector potentialregion and in complete absence of any external potential (V (r, t) = 0)writes

i∂

∂tΨ(r, t) =

12m

(i∇ + q A

)2Ψ(r, t) (4.4.7)

where A is given by (4.4.6).In the absence of magnetic field in the outer region, B = 0, the

solutions of the last equation are wave functions that can be written

Ψ(r, t) = Ψp(r, t) eiq

R r0

A(r′)·dr′ (4.4.8)

where Ψp(r, t) is solution of the Schrodinger’s equation without thepresence of the vector potential

i∂

∂tΨp(r, t) = −

2

2m∇2 Ψp(r, t) (4.4.9)

The exponential part of the wave function (4.4.8) entails that twoidentical particles with equal charge, propagating both at the samedistance from the solenoid axis but the first in the same directionwith the vector potential A and the second in the opposite direction,will suffer a phase difference

δΘ =q

∫s

B · dS (4.4.10)

Indeed, interference patterns for electrons in analogues condi-tions have been observed experimentally with the phase differencedescribed by the relation (4.4.10) demonstrating that the vectorpotential is not a mathematical artefact and interacts directly withcharged particles in full absence of magnetic and electric fields.



4.5. Transition Rates and Vacuum InducedSpontaneous Emission

For describing the transitions between the quantized energy levelsof the atoms, we have started in the previous chapter, defining theinteraction Hamiltonian (4.4.1) of a charged particle in an electro-magnetic field characterized by a vector potential A(r, t) and a scalarpotential Φ(r, t).

Hence, the interaction Hamiltonian is a time dependent pertur-bation and can be studied within the time dependent perturbationtheory according to which the wave function Ψ(r, t) is expended interms of the unperturbed eigenfunctions ϕl(r), solutions of the timeindependent Schrodinger equation

H0ϕl(r, t) =(−

2

2m∇2 − Ze2

4πε0r

)ϕl(r, t) = Elϕl(r, t) (4.5.1)

The expansion of the wave function writes

Ψ(r, t) =∑

l

cl(t)ϕl(r)e−iElt/ (4.5.2)

where the summation runs over all the discrete and continuous setsof the eigenfunctions ϕl(r). The time dependence is included in theexponential argument and in the coefficients cl(t), which in the timedependent perturbation theory satisfy the coupled equations

idc

(n+1)m (t)dt

=∑

l

∫ϕ∗

m(r )Hint(t)ϕi(r )d3r c(n)l (t)eiωmlt

(4.5.3)

where n = 0, 1, 2, . . . and Em −El = ωml

Assuming that the system is initially at t = 0 in a defined sta-tionary state ϕi(r) corresponding to the energy Ei, for n = 1, thecoefficient c(0)l (t) = δli for the discrete states or c(0)l (t) = δ(l − i) forthe continuous ones.

Where, δli is Kronecker delta symbol, δli = 1 if l = i and δli = 0if l = i and δ(l − i) is Dirac delta function, δ(0) = 1 (if l = i) andδ(l − i) = 0 if l = i.



In this case the last equation gives

idc

(1)m (t)dt

=∫ϕ∗

m(r)Hint(t)ϕi(r) d3reiωmit (4.5.4)

Using the interaction Hamiltonian (4.4.5) and the expression forthe vector potential (3.4.2), we obtain the transition probabilityamplitude at a given instant t by integrating within an angular fre-quency interval δω and over the time interval from 0 to t

c(1)m (t) = − e

m

∫δωA0(ω)dω

×[eiθ〈ϕm|eik·rε · ∇|ϕi〉

∫ t

0ei(ωmi−ω)t′dt′

+ e−iθ〈ϕm|e−ik·rε∗ · ∇|ϕi〉∫ t

0ei(ωmi+ω)t′dt′

](4.5.5)

The square modulus of c(1)m (t) represents the probability of thesystem to be in the stationary state m at time t.

|c(1)m (t)|2 = 2( em

)2∫

δω

|A0(ω)|2Θ(t, ω − ωmi)|Mmi(ω)|2dω(4.5.6)

where the matrix element |Mmi(ω)| has the form

|Mmi(ω)| =∣∣∣∣∫ϕ∗

m(r)eik·rε · ∇ϕi(r)d3r

∣∣∣∣ (4.5.7)

The trigonometric function Θ(t, ω − ωmi) involved in the integrationof (4.5.6) results from the time integration of the exponentials andhas the form

Θ(t, ω − ωmi) =1 − cos [(ω − ωmi)t]

(ω − ωmi)2(4.5.8)

Since Θ(t, ω − ωmi) takes its maximum value for ω = ωmi andits integration over all the frequencies gives π t we can consider ina first approximation that for a large time interval ω = ωmi in theexpressions of the vector potential amplitude A0(ωmi) and the matrix



element between the states ϕm(r) and ϕi(r), Mmi(ωmi) and the inte-gration of (4.5.6) now gives

|c(1)m (t)|2 = 2π( em

)2 |A0(ωmi)|2|Mmi(ωmi)|2t (4.5.9)

which increases linearly with time.We obtain immediately the transition rates T for absorption

Tabs =d

dt|c(1)m (t)|2 = 2π

( em

)2|A0(ωmi)|2|Mmi(ωmi)|2 (4.5.10)

and stimulated emission

T stimulatedemi =

d

dt|c(1)m (t)|2 = 2π

( em

)2 |A0(ωmi)|2|Mim(ωmi)|2(4.5.11)

for which the main difference relies on the order inversion of thematrix element between the initial and final state.

It is worth noting that for the above calculations, we have usedthe classical expression of the vector potential given by (3.4.2).

• Photoelectric effect and the semi-classical interpretation

At that level, within a chapter that deals mainly with the photon,we will make a parenthesis to present briefly the way Wentzel andBeck in 1926, Lamb and Scully in 1960s demonstrated that the photo-electric effect may quite well be interpreted using the electromagneticwave nature of the light.

In fact, we can calculate the transition rate in the case of thephotoelectric effect from (4.5.6) and (4.5.8), exactly in the same wayas in the previous paragraph

|c(1)m (t)|2 = 2( em

)2∫

|A0(ω)|2 1 − cos[(ω − ωmi)t](ω − ωmi)2

|Mmi(ω)|2dω(4.5.12)

On the other hand we have∫ +ω

−ω

1 − cos[(ω − ωmi)t](ω − ωmi)2

dω = πt (4.5.13)



And for sufficiently long time-periods

limt→∞

1 − cos[(ω − ωmi)t](ω − ωmi)2

= πtδ(ω − ωmi) (4.5.14)

Consequently (4.5.12) becomes

|c(1)m (t)|2 = 2πt( em

)2|A0(ω)|2|Mmi(ω)|2δ(ω − ωmi) (4.5.15)

And the transition rate writes

Tphotoelectric =d

dt|c(1)m (t)|2 = 2π

( em

)2 |A0(ω)|2|Mmi(ω)|2δ(ω − ωmi)

(4.5.16)

Between an initial bounded state and the continuum level, Diracdelta function guarantees that the energy of the photoelectron sat-isfies the photoelectric equation. The rate of the emission is directlyproportional to the square modulus of the radiation vector potential,and consequently to the field intensity. Obviously, because of (4.5.14),the calculated transition rate for the photoelectric effect is valid fortime periods t ω−1 and consequently for visible and UV light, itpredicts the emission of electrons without any delay.

As remarked by many authors, a semi-classical calculation, inwhich the electromagnetic field is strictly considered with theclassical wave expressions, gives an excellent interpretation of thephotoelectric effect without the introduction of the photon con-cept. Recalling that the photoelectric effect is used experimentallyto detect photons, this is a quite upsetting result.

• Spontaneous emission rate

As we have discussed in (4.3.18) in the QED description, thevacuum state is composed by an infinite quantity of photons of allfrequencies and polarizations

∑k,λ

12ωk. Consequently, in order to

describe the spontaneous emission we have to use the expression(4.3.15) for the vector potential, since in the classical electromagnetictheory the vacuum has no electromagnetic fields and consequently(3.4.2) is zero.



The part of the quantum expression of the vector potential respon-sible for the creation of a photon is

〈 AC(r, t)〉 = 〈. . . nkλ, nkλ + 1 . . . |

×√

2ε0ωkV

[a+

kλε∗kλe

−i(k·r−ωkt+θ)]| . . . nkλ . . .〉

=

√(nkλ + 1)2ε0ωkV

[ε∗kλe

−i(k·r−ωkt+θ)]

(4.5.17)

Of course, we can use the same expression for the transition rate asin (4.5.16), which writes here

Temi = π( em

)2 (nkλ + 1)ε0ωkV

|Mim|2δ(ωk − ωmi) (4.5.18)

In absence of k mode and λ polarization photon, i.e., when nkλ =0, the transition rate does not vanish and develop within a direction

T spontaneousemi =

( em

)2 π

ε0ωkV|Mim|2δ(ωk − ωmi) (4.5.19)

which depends on the volume V of the system, but if we considerthe density of states (3.5.42) and integrate over ω and all spatialdirections of emission, while summing on both possible photon polar-ization (λ = 1, 2), then the parameter V is eliminated and we get afinal expression of the total transition rate which is independent of V .

T spontaneousemi (total) =

( em

)2

8π2ε0c3

∫dΩ∑

λ

|Mim|2λωmi (4.5.20)

The special interest on the spontaneous emission arises from thefact that it results from the coupling of the atomic levels to thequantum vacuum state.

• Dipole approximation and spontaneous emission

Within the dipole approximation, according to which when thewavelength of the electromagnetic field is too big with respect tothe atomic dimension, i.e., when kr 1, the exponential in (4.5.7)



can be considered to be close to unity and using (4.2.7) the matrixelement writes

Mmi =∫ϕ∗

m(r)ε · ∇ϕi(r)d3r = ε ·(i

)∫ϕ∗

m(r)pϕi(r)d3r

(4.5.21)

According to Heisenberg equation of motion for a vector observ-able F

d

dtF = − i

[F ,H0] (4.5.22a)

p = md

dtr = −m i

[r,H0] (4.5.22b)

where H0 is the unperturbed Hamiltonian of the system.So (4.5.21) becomes

Mmi = ε ·(m

2

)∫ϕ∗

m(r)[r,H0]ϕi(r)d3r

= ε ·(m

2

)(Ei − Em)

∫ϕ∗

m(r)rϕi(r)d3r

= −(m

)ωmiε ·

∫ϕ∗

m(r)rϕi(r)d3r

= −(m

)ωmiε ·rmi = −

(m

)ωmi|rmi| cos θ (4.5.23)

where θ is the angle between r and the polarization vector ε.Thus, the spontaneous emission rate writes

T spontaneousemi (total) =

e2

8π2ε0c3ω3

mi

∫dΩ∑

λ

|rmi|2λ cos2 θ

=e2

3πε0c3ω3

mi|rmi|2 (4.5.24)

where we have considered two polarizations and cos2 θ has beenreplaced by the mean value

〈cos2 θ〉 =14π

∫cos2 θ dΩ =

14π

∫ 2π

0dφ

∫ +1

−1cos2 θd(cos θ) =

13

(4.5.25)



Finally, it can be demonstrated that the interaction Hamilto-nian between an electron and the electromagnetic field in the dipoleapproximation can be simply written

Hint = − D · E(r, t) = −er · E(r, t) (4.5.26)with D = er the electric dipole moment of the electron with chargee at the position r from the nucleus and E (r, t), the electric field ofthe electromagnetic wave obtained by the vector potential accordingto (3.3.15).

• The photon spin

Conservation of angular momentum during spectroscopy exper-iments has shown that each photon has a component of angularmomentum along the propagation axis of magnitude ±. This cor-responds to the intrinsic spin of photons whose component alongthe propagation axis is also called helicity. A left-hand circularlypolarized photon has an helicity of + while a right-hand circularlypolarized photon has an helicity of −.

Consequently, the λ subscript in the creation and annihilationoperators takes two values corresponding to L and R circular polar-izations and the total spin angular momentum of an ensemble ofphotons propagating along the z axis writes in a the general operatorform

S =

∑k

(a+k,Lak,L − a+

k,Rak,R)ez (4.5.27)

According to (3.4.12) the polarizations Πκλ for L and R k-modephotons when considering the propagation function along z areexpressed as follows

|Πk,L〉 =1√2

1i

0

e−i(ωkt−k·z+φ) |Πk,R〉 =

1√2

1−i0

e−i(ωkt−k·z+φ)

(4.5.28)The combination of an ensemble of photons with circular polar-

izations can give rise to an elliptical polarization which is the onemainly observed from the classical sources of light in nature. Theuse of appropriate filters permits the selection of R,L or even linearpolarization for experimental purposes.



4.6. Lamb Shift

• Nonrelativistic calculations: Bethe’s approach

The energy difference between the energy levels 2S1/2 and 2P3/2

of the atomic hydrogen was measured precisely by W.E. Lamb andR.C. Retherford in 1947 using microwave technics. They obtained avalue of 4.5 × 10−5 eV, which was not explained by Dirac’s theory,which predict these two levels to have the same energy.

With the development of QED and the introduction of the notionof the quantum vacuum, many scientists advanced the hypothesisthat the experimentally observed shifts of the atomic levels, namedLamb shifts, are due to the interaction of the bounded electrons withthe vacuum. The first to propose a theoretical interpretation wasH.A. Bethe in 1947 by introducing an astonishing way for extractingfinite quantities from singularities. We reproduce briefly here thosecalculations.

Since the basis of the hypothesis is the interaction of an elec-tron with the vacuum photons, we go back to Schrodinger equation(4.4.3) and start by examining the contributions of the vector poten-tial terms.

The quadratic term of the vector potential contributes in thesame manner to all atomic states since it does not contain atomicoperators

∆E( A2) = 〈nkλ, vacuum| e2

2mA2|nkλ, vacuum〉

=e2

2m

∑k,λ

2ε0ωkV=

e2

4π2ε0mc3

∫ ∞

0ωdω (4.6.1)

where we have replaced the discrete summation by a continuous oneaccording to the well- known transformation introduced in (3.5.44)and taking into account the polarization λ

∑k,λ

→∑

λ

V

8π3

∫d3k =

V

8π3

∑λ

∫4πk2dk =

V

2π2c3

∑λ

∫ω2dω

(4.6.2)



The (4.6.1) expression is infinite. However, it is fully neglectedbased on the argument that it does not induce “measurable energyshifts”.

Now, starting again from (4.4.3) and considering (4.5.15), the sec-ond order perturbation theory gives the energy shifts to the atomiclevels, due to the coupling with the quantum vacuum resulting fromthe linear term of the vector potential in the interaction Hamiltonian

∆El( A) =∑

f

∑k,λ

× e2

m2

(

2ε0ωkV

) |〈f, nk,λ = 1|a+k,λ(εk,λ · p)|l, nkλ = 0〉|2

El − Ef − ωk

=∑

f

∑k,λ

e2

m2

(

2ε0ωkV

) |εk,λ · pfl|2El − Ef − ωk

=∑

f

e2

4c3ε0m2π2

∑λ

|pfl|2〈cos2 θ〉∫

ω dω(El−Ef

)− ω

=∑

f

(e2

6m2c3ε0π2

)|pfl|2

∫E dE

El − Ef − E

=∑

f

(2αFS

3πm2c2

)|pfl|2

∫ ∞

0

E dE

El − Ef − E(4.6.3)

where we have used the relations (4.5.25) and (4.6.2) and consideredtwo polarizations (right- and left-hand circular). Also αFS is the finestructure constant

αFS =e2

4πε0c=

1137.036

; e2 = 2αFS (ε0hc) (4.6.4)

As in the case of the quadratic term of the vector potential, theobtained expression (4.6.3) for the energy shift ∆El( A) induced bythe linear part of the vector potential is also infinite.

To circumvent this difficulty, Bethe introduced a renormalisationprocedure. In fact, for a free electron in the vacuum field, all thevalues of the atomic levels energy differences El−Ef are zero so that



∆El( A) writes

∆El( A)free =∑

f

(2αFS

3πm2c2

)|pfl|2

∫ ∞

0dE (4.6.5)

in which the momentum operator between the atomic levels is pre-served.

∆El( A)free is equally infinite. However, the experimentallyobserved shift for an atomic level |l〉 should be the difference

∆El( A) − ∆El( A)free

=∑

f

(2αFS

3πm2c2

)|pfl|2(El − Ef )

∫ ∞

0

dE

El − Ef − E(4.6.6)

This operation is named Bethe’s renormalization and results tothe relation (4.6.6) which is still infinite but through a slow logarith-mic divergence.

Next, an upper level for the integration has been introduced andfixed to mc2, m being the electron mass at rest, getting


=∑

f

(2αFS

3πm2c2

)|pfl|2(Ef −El) log

∣∣∣∣ mc2

El − Ef

∣∣∣∣ (4.6.7)

In order to get the logarithm out from the discrete summation,the mean value can be considered


=(

2αFS

3πm2c2

)⟨log∣∣∣∣ mc2

El − Ef

∣∣∣∣⟩∑

f

|pf l|2(Ef − El) (4.6.8)

where the average is calculated over all the discrete and continuousatomic levels⟨

log∣∣∣∣ mc2

El − Ef

∣∣∣∣⟩

≈ log(mc2) −∑

f |pfl|2(Ef −El) log |Ef −El|∑f |pfl|2(Ef − El)

(4.6.9)



It can be easily shown that for a hydrogen atom with atomicnumber Z, we get∑

f

|pfl|2(Ef − El) = 2πZ|eϕl(o)|2 (4.6.10)

entailing that the shift should be more important for the s states forwhich |ϕl(o)|2 = 0.

Thus (4.6.8) becomes

∆El( A) − ∆El( A)free ≈(

4αFSZe2

2

3m2c2

)⟨log∣∣∣∣ mc2

El − Ef

∣∣∣∣⟩|ϕl(o)|2

(4.6.11)

which is the calculated energy shift for the level |l〉.Using a numerical estimate of the (4.6.9) along all the energy

levels |f〉 up to the continuum spectrum, Bethe found for the 2Shydrogen level, an energy shift corresponding to a frequency of about1000 MHz, in good agreement with the experimental value.


Some quite important concepts and calculation methodologies havebeen developed in this chapter.

The second quantization processes introduced the link equations(4.3.14) for the vector potential amplitude between the electromag-netic wave expressions and those of the QED creation and anni-hilation operators. This permitted to establish the correspondencebetween the electromagnetic wave and the harmonic oscillators inquantum mechanics establishing the Hamiltonian of the quantizedelectromagnetic field (4.3.18). The validity of this process will bediscussed in the next chapters. Note that the link equations (4.3.14)etc., obtained by considering the mean value of the energy densityof the electromagnetic wave over a period, in other words over awavelength, are put equal to the value of a point photon with energyω in a given volume V .

Furthermore, we have seen that the interaction Hamiltonianbetween the electromagnetic field and the charged particles is



expressed through the vector potential, a real physical entity, firstdemonstrated by Ehrenberg and Siday. The same interaction Hamil-tonian has been employed by Wentzel, Beck, Lamb and Scully todemonstrate that the photoelectric effect, generally advanced as ademonstration of the particle nature of the light, can quite well beinterpreted through the electromagnetic wave representation.

Finally, we have briefly presented two important effects of thevacuum-electron interactions, the spontaneous emission and theLamb shift, which will extensively discussed in the next chapters.Many authors have discussed and commented Bethe’s nonrelativis-tic calculations for the Lamb shift. P.A.M. Dirac was the first toargue that intelligent mathematics consists of neglecting negligi-ble quantities and not infinite quantities because, simply, you don’twant them. In fact, the original method that Bethe introduced wasto calculate finite quantities by subtracting and neglecting infinitequantities. However, as many authors remarked, what is extremelypuzzling is that after manipulating and ignoring infinite quanti-ties, after imposing arbitrary integration limits, after consideringmean logarithmic values over an infinity of atomic energy levels nearthe continuum, “the final result compares “remarkably well” to theexperiment”.

We have not presented in this chapter, the relativistic calculationof the Lamb shift, developed and published in 1949 by Kroll et al.because the calculations are extremely tedious. Nevertheless, we haveto mention that even in this approach various approximations areintroduced and a “different” mean logarithmic value is employed.Furthermore, it seems that the upper integration limit appears natu-rally in the relativistic approach, however, the author, as the major-ity of the authors in the literature, has not attempted to reproducecompletely these calculations.

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Chapter 5

Theory, Experiments and Questions

In what follows, we analyze the questions raised from the quantumtheory of radiation and we point out the mathematical difficultiesencountered mainly from the second quantization procedure whichhas been developed to provide photons description from the quantumpoint of view. We first make a theoretical analysis of some particularaspects of QED, including the well known singularities, and then wepresent some experiments which strongly support the simultaneouswave-particle nature of photons.

5.1. Planck’s Constant and the Vacuum IntrinsicElectromagnetic Properties

The correspondence for the energy between the electromagnetic wavetheory and the Quantum electrodynamics (QED) describing the sameentity, the light, is given by the well-known expression

E (ω) = ω ↔ E (ω,A0(ω)) =∫

V2ε0ω2A2

0(ω,r) d3r (5.1.1)

In QED, the energy of the photon ω depends only on the angu-lar frequency of the electromagnetic wave ω and it is proportional toPlanck’s reduced constant , while the vacuum electric permittivityconstant ε0 and the vector potential amplitude A0(ω), which bothplay a major role in the classical electromagnetic wave description(3.4.8), are totally ignored. On the other hand, it is obvious that theknowledge of the angular frequency alone is not sufficient in the clas-sical electromagnetic theory in order to deduce the electromagneticfield energy. The vector potential amplitude is indispensable.

77



Consequently, for (5.1.1) to be physically coherent, should berelated to both the vacuum electric permittivity ε0 and the vectorpotential amplitude A0(ω).

Indeed, is related to ε0 through the fine structure constant αFS

(4.6.4) according to the relation:

=(

e2

4πε0

) (1

cαFS

)(5.1.2)

where e is the electron charge.From the last equation one can also express in QED, the energy

of a photon depending on ε0 as follows

E(ω) = ω =(

e2

4πε0

) (1

cαFS

)ω (5.1.3)

According to (5.1.3), one may assume that the absolute value ofthe energy of a photon with an angular frequency ω corresponds tothe Coulomb interaction between an electron and a positron sepa-rated by the characteristic distance lω

lω =cαFS

ω= αFS

λ

2π(5.1.4)

where λ is the wavelength.However, it is puzzling to note that the expression (5.1.3), intro-

duces the charge of the electron and the fine structure constant whichessentially characterize the atomic energy levels. Consequently, onecould naturally wonder why these physical quantities should be nec-essary to describe the energy of light in vacuum, in complete absenceof charges and matter? In fact, the electron (positron) and the elec-tromagnetic field are strongly related physical entities. Under par-ticular conditions, in the presence of a strong electric field like inthe vicinity of a heavy nucleus, a high energy gamma photon can beannihilated, giving birth to an electron and a positron. Of course,the probability of such a phenomenon is extremely low and becomessignificant for a nucleus of Z ∼ 140, which is physically instable.Nevertheless, Fulcher et al. have argued that such a nuclear statecould be created during the collision of two nuclei of large Z. Thus,we can say that (5.1.3) is a simple relation which implies directly thatthe electron (positron) and the electromagnetic waves are physical


Theory, Experiments and Questions 79

entities that might be issued from the same quantum field related tovacuum. This aspect will be further analyzed in the next chapter andan exact relation will be established expressing precisely the electroncharge with respect to the constants characterizing the vacuum.

Now, we have seen that we can easily establish a relation between and the vacuum permittivity ε0 through the fine structure con-stant αFS but what would be the dependence of the vector potentialamplitude A0(ω) on Planck’s constant for a single photon guessedby (5.1.1)? In a coherent description of the photon as a quantumof the electromagnetic field, this relation should be independent ofany external parameter such as the volume V involved in the linkequations (4.3.14).

We will also explore the possibility of such a relation in the nextchapter.

5.2. Hamiltonian Issued from the Quantizationof the Electromagnetic Field

Let us go back to the quantization procedure of the electromagneticfield. When we introduce the link expressions (4.3.14), directly in therelation of the energy of the classical electromagnetic field (4.3.9), weobtain the following radiation Hamiltonians

HEM =∑k,λ

ωkNkλ (5.2.1)

if the “normal ordering” expression is considered in (4.3.9), i.e.,

2ε0V∑k,λ

ω2kA

∗kλAkλ (5.2.2)

and

HEM =∑k,λ

ωk(Nkλ + 1) (5.2.3)

when the anti-normal ordering is used

2ε0V∑k,λ

ω2kAkλA

∗kλ (5.2.4)

It is very important to note that it is impossible to get HEM =∑k,λ ωk (Nkλ + 1

2) when starting from the electromagnetic fieldenergy expression (4.3.9), the “normal ordering” is used or not.



This mathematical ambiguity, which leads to a quite different phys-ical interpretation of the radiation zero-level state, demonstrates thelack of a coherent mathematical transition from the classical to thequantum mechanical formalism of the electromagnetic field energy.

Consequently, a strong doubt may arise in whether the term∑k,λ

12ωk, corresponding to the vacuum energy in (4.3.18) represents

a real physical state since it is not in reality a natural consequence ofthe electromagnetic field quantization. Hence, we will analyze in nextchapters, the reasons for which one obtains different expressions forthe zero-point energy. Also we will show how the relation (4.3.12)has to be transformed to ensure a coherent transition between theclassical electromagnetic wave energy and the quantum mechanicalHamiltonian, in order to eliminate this mathematical ambiguity.

5.3. QED Singularities

Let us consider the Hamiltonian of the electromagnetic field (4.3.18)in complete absence of photons, that is when Nk,λ = 0, whichbecomes

Hvacuum =∑k,λ

12

ωk (5.3.1)

The summation runs over all modes k and polarizations λ. Hence,as it has been mentioned in Chapter 4 (4.3.1), the vacuum in the QEDdescription consists of an infinite sea of “photons” corresponding tothe state |Nk,λ = 0,∀ k, λ〉 corresponding to infinite energy.

In this context, the vacuum electromagnetic energy density is

Wvacuum =1V

∑k,λ

12

ωk (5.3.2)

using (4.6.2) and considering two polarizations, we get

Wvacuum =

2π2c3

∫ω3dω (5.3.3)

which is also infinite. However, when considering “reasonable” cut-offs for the upper limit of the integration, that is an angular frequencyof ∼1043 Hz rad corresponding to Planck’s energy 1019 GeV, the



values obtained in QED for the vacuum energy density are generallyof the order 10110 J/m3.

When limited in the visible region, situated in the wavelengthsrange between 4000 and 7000 Angstroms, the energy density calcu-lated using (5.3.3) corresponds roughly to 22 J/m3, which representsa tremendous amount of energy in space which is fully unobservableexperimentally. Recent astronomical observations have demonstratedthat the measured vacuum energy density in space is 10−9 J/m3, thatis roughly a hundred of orders of magnitude less than that predictedby QED theory. The huge discrepancy between the theoretical andthe measured values of the vacuum energy has been called the “quan-tum vacuum catastrophe.”

On the other hand, the quantized radiation electric field E(r, t) isobtained using the vector potential expression (4.3.15) and (3.3.15)in the Coulomb gauge

E(r, t) = −∂A(r, t)∂t

= i∑k ,λ

√ωk/(2ε0V )

×[akλεkλei(k·r−ωkt+ϕ) − a+

kλε∗kλe

−i(k·r−ωkt+ϕ)]

(5.3.4)

where εkλ is a polarization vector. Consequently, the expectationvalue of the photons electric field in vacuum within a volume V isgiven by the expression

〈E(r)2〉vacuum =∑k,λ

ωk,

2ε0V(5.3.5)

which is also infinite.Although the summation of (5.3.5) is divergent we can deduce the

electric field |εk| of a single k−mode photon state

|εk| =√

ωk

2ε0V(5.3.6)



Using classical plane wave considerations the magnetic field |βk| =√ε0µ0|εk| of a single photon can be written

|βk| =

√ωkµ0

2V(5.3.7)

As quoted by many authors in the literature, the last relationssignify that the electric and magnetic fields of a single photon dependdirectly on an external arbitrary volume parameter V . However,this is physically an intriguing result because the volume V is notinvolved explicitly in the classical expression of the electromagneticfield energy density (3.4.7) even when it is space localized within awavelength (3.4.8). From the quantum point of view it is difficultto conciliate the concept of a point particle having precise energyand momentum with that of an electromagnetic plane wave whoseelectric and magnetic fields amplitudes are dependent on an externalvolume. This particular concept will be further discussed in the nextchapter.

5.4. Electron-Vacuum Interactions and the Associated Effects

The presence of the vacuum energy singularity in the quantised radi-ation field Hamiltonian Hvacuum =

∑k,λ

12ωk constitutes the fun-

damental argument generally found in the literature, in order toexplain some physical effects such as the spontaneous emission, Lambdisplacements in atomic levels as well as the Casimir effect. The ter-minology usually employed for that purpose uses the term “vacuumfluctuations”, though in most cases no mathematical representationof any kind of fluctuation is explicitly presented. Some authors pre-fer using the term “virtual photons” for the vacuum Hamiltonian.However, we could naturally wonder why the first term of the Hamil-tonian (4.3.18) is considered as real while the second one as “virtual”.

Nevertheless, it is absolutely certain that the last effects are dueto the electron-vacuum interactions and cannot be calculated using asemi-classical model, in which the atomic energy levels are quantizedwhile the electric field is represented by a classical wave equationin the interaction Hamiltonian. The reason is simply that in theclassical description, the vacuum has no electromagnetic waves and



consequently, E(r, t) in the interaction Hamiltonian Hint in (4.5.26)is zero. Conversely, in QED, according to the radiation Hamiltonian(4.3.18), the vacuum is considered to be composed by photons of allmodes and polarizations, permitting one to consider an interactionHamiltonian.

Now, the above effects are due to the interaction between anelectron in a given atomic state and the vacuum. Consequently, fortheir description the term

∑k,λ

12ωk of the Hamiltonian (4.3.18),

which is the eigenvalue of the fundamental radiation eigenstate| . . . , 0k,λ, . . . , 0k′,λ′ , . . .〉, corresponding to the vacuum state, shouldbe of high importance.

However, as we will see in the following, this singularity has noimplication in the mathematical interpretation of those effects.

• Spontaneous emission

We recall the total Hamiltonian of an atom in the presence of anelectromagnetic field in the dipole approximation:

Htot = HEM +Hint +Hatom

=∑k,λ

ωk

(Nkλ +

12

)− D · E(r, t) +

∑i

ωi|Ψi〉〈Ψi|

(5.4.1)

where D = er is the dipole moment of an atomic electron of charge eaccording to (4.5.26) and |Ψi〉 the atomic levels with the correspond-ing energies ωi.

The spontaneous emission rate is always calculated in QEDneglecting

∑k,λ

12ωk in Htot and using the expression (5.3.4) for

the electric field E(r, t).Therein, we recall that according to Heisenberg’s equation of

motion (4.5.22.a), the electric field writes

E(r, t) = −∂A(r, t)∂t

=i

A(r, t),

∑k,λ

ωk

(Nkλ +

12

) (5.4.2)

Consequently, for the calculation of the electric field attributedto the vacuum state, the term

∑k,λ

12ωk should be the principal



contribution to the Hamiltonian. But the commutation operator ofthe vector potential and the vacuum Hamiltonian cancels

A (r, t),∑k,λ

12

ωk

= 0 (5.4.3)

Hence, the term∑

k,λ12ωk in this representation has absolutely

no contribution in the general expression of the quantized electricfield used to calculate the spontaneous emission.

Obviously, the vacuum effect in the quantized vector potential isinvolved in the commutation relation [akλ, a

+kλ] = 1 and not in the∑

k,λ12ωk term.

Indeed, in a rigorous calculation, the interaction Hamiltonianbetween an atomic electron and the vacuum writes

Hint = −d · i

A (r, t),

∑k,λ

12

ωk

= 0 (5.4.4)

which also vanishes in QED, as in the semi-classical description.The physical reason is that in QED, the vacuum state Hamil-

tonian is not described by a function of akλ and a+kλ operators and

consequently it is impossible to define an interaction Hamiltonian forthe description of the electron-vacuum interaction processes.

• Lamb shift

The mathematical treatment of the quantum vacuum effects uponthe atomic energy levels described in Section 4.6 is without any doubtamong the most revolutionary ones introduced in modern physics. Itconsists of neglecting infinite quantities for calculating finite energydifferences which are measurable experimentally.

We can now have a close look to a detailed analysis of this calcu-lation, resuming the comments and arguments of many authors.

First, the quadratic term of the vector potential of equation(4.6.1) is dropped out, though it’s infinite. Despite the fact thatthe linear term of the vector potential of equation (4.6.3) is alsoinfinite, it is not dropped out as the quadratic one, but instead itsuffers a renormalization procedure which equally results in an infi-nite quantity (4.6.6). At that level one should expect that since the



quadratic term, being infinite, does not induce a measurable shift,by the same token the linear term, being also infinite, should alsonot induce a measurable shift. Nonetheless, the linear term is keptin the calculation and in order to “overcome” the difficulty, an arbi-trary upper limit for the integration of (4.6.6) is introduced in thenonrelativistic approach whose choice, at that level of calculation, isnot based on any experimental or physical argument. Conversely, inthe relativistic approach developed much later, the upper integrationlimit seems to appear “naturally”. Consequently, one may wonderwhy the same upper limit is not equally applied in the integration ofthe quadratic term of the vector potential. Furthermore, the intro-duction of a mean value for the logarithmic term (4.6.9), calculatednumerically over an indefinite number of discrete states, in a differentmanner in the nonrelativistic than in the relativistic approach, givesrise to the upsetting feeling that under these calculation conditions,the final obtained displacement energy is accurate compared to theexperiment.

Finally, once again, in these calculations the vacuum effects onthe atomic energy levels is not due to the zero-point level of theHamiltonian (4.3.18). Except Bethe’s quantum mechanical approach,various semi-classical calculations of the Lamb shift have been pub-lished, each one by considering the influence of the vacuum upon theatomic orbitals through different aspects; Welton’s vacuum electricfield perturbation, Feynman’s gas interpretation, Stark shift due tothe vacuum electric field and so forth. It is quite interesting noticingthat almost all these physical mechanisms used for the interpretationof the atomic energy shifts do no require any mass renormalizationprocess.

• Casimir effect

The behavior of two perfectly conducting and uncharged parallelplates with similar surfaces S = L × L separated by a very shortdistance d(d L) has been investigated first by Casimir in 1948.This physical situation has been then extensively studied by manyauthors over half a century and of course it is out of the scope ofthis book to present all these works in details. In an analogue way



with the boundary conditions expressed in two dimensions for theelectromagnetic waves guided propagation in (3.5.2) the frequenciespermitted in a rectangular cavity with dimensions L×L×d in vacuumare given by

ωn1n2n3 =

√1

ε0µ0

((n1π

L

)2+(n2π

L

)2+(n3π

d

)2)

(5.4.5)

where n1, n2, n3 = 0, 1, 2, . . . are integers with of course two simulta-neous zero values excluded. Based on (5.3.1) the zero-point energybetween the plates writes

Evacuum =∑

n1,n2,n3,λ

12

ωn1n2n3 (5.4.6)

The summation is infinite entailing that the vacuum energy inthe QED description is infinite in every finite volume. However,when introducing appropriate cut-off, justified by the fact thatany material becomes transparent at extremely high frequencies,then the vacuum energy difference between that contained in acubic box Evacuum(L3) and that in the rectangular configurationEvacuum(L2 × d) becomes finite inducing an attractive force betweenthe plates which after a slightly tedious calculation writes

F (d) = −π2c

240L2

d4(5.4.7)

Consequently, an explanation advanced for the origin of the Casimirforce is that the density of the vacuum photons outside the plates ishigher than that in the region inside inducing a radiation pressurepushing the plates together to come closer.

From the experimental point of view, the measurement of Casimirforce presents many technical difficulties laying mainly in the align-ment of perfectly parallel plates at microscopic separation distances.The first experiments carried out by Sparnaay in 1958 and then byvan Blokland and Overbeek in 1978 in order to measure Casimir’sforce were not quite successful and the experimental errors were verysignificant, nearly 100% and 30% respectively. However, they permit-ted to put in evidence an attractive force between the plates.

More recent experiments mentioned by Lambrecht and Reynaudin 2002 attained a precision of the order of 1% but they were still



unable of confirming neither the theoretical predictions at larger dis-tances nor the temperature effects.

On the theoretical front, it is important underlying that Schwingeret al. in 1978, Milonni in 1982 and Ezawa in 1997 obtained the sameexpression for the Casimir attraction force without invoking at all thezero-point energy of the electromagnetic field. In fact, as quoted byMilonni in 1992, a simple summation of the Van der Waals pairwiseintermolecular forces on the surfaces of the two plates gives a quiteclose result to that of the Casimir force. If the interactions withthird neighboring atoms are also taken into account in the calcula-tions then the exact Casimir force is obtained. Hiroshi Ezawa, KoichiNakamura and Keiji Watanabe advanced an excellent demonstrationof the Casimir effect from Lorentz’s forces issued on the surfaces ofthe plates by the charge currents. There in, on the question if we stillneed the vacuum zero-point energy the authors give a straightforwardanswer “We wish to make it clear that we don’t”.

Consequently, the previous mentioned studies demonstrate thatthe Casimir force can be established from a classical electrodynamicsapproach without considering at all the hypothetical energy densitydifference of the vacuum photons in the regions between and outsidethe plates. Despite this and despite the experimental uncertainties,the Casimir effect is considered by many authors as the principledemonstration of the existence of the vacuum zero-point level pho-tons involved in the Hamiltonian (4.3.18).

Finally, a further fundamental difficulty for the conceptual com-prehension of the Casimir effect comes from the fact that the expres-sion of the force given in (5.4.7) is completely independent on theelectron charge which is always involved in the interactions withthe electromagnetic field. Many authors have commented this factconcluding that the interpretations of the Casimir effect are puz-zling. Indeed, we have to underline that the Casimir effect is fairlywell interpreted through the prism of two different physical aspectsdescribed above, introducing a certain ambiguity in the explanationsfound in the literature. Hence, it appears that higher accuracy exper-iments are further required in order to fully confirm the agreement ofthe experimental measures with the theoretical interpretations andconfirm the real physical origin of the Casimir effect.



5.5. Simultaneous Wave-particle Nature of the PhotonRevealed by the Experiments — Discussions

• Photoelectric effect, Compton’s scattering and Young’s double-slitexperiment

The photoelectric effect was initially considered as a direct demon-stration of the particle nature of light and historically was thestrongest argument of the photon concept, although Einstein had notadvanced any explanation on what could correspond the frequencyν for a point particle.

Nevertheless, as we have seen in Chapter 4 (4.5), Wentzel in1926 and Beck in 1927, as well as much later, Mendel, Lamb andScully in the 1960s, demonstrated that the photoelectric effect canbe interpreted remarkably well by considering the electromagneticwave nature of the light issued from Maxwell’s equations withoutreferring at all to the photon concept.

We also have to note that Klein and Nishina had advanced anexcellent semi classical interpretation of the Compton scattering byconsidering a classical plane wave approach for the incident lightand Dirac’s equation (4.2.31) for the electron. Consequently, sinceboth wave and particle models interpret satisfactorily the photoelec-tric effect and Compton’s scattering, they cannot be considered as adecisive experiments for the introduction of the photon concept.

On the other side, Young’s double-slit experiment, which was ini-tially interpreted with the wave nature of light and was used as thestrongest argument against the particle theory, was used by Taylorat very low intensities to demonstrate the particle concept.

The interpretation of Taylor’s experiment by some authors wasbased on the wave nature of a single photon permitting to passthrough both slits. Many scientists were skeptical about this inter-pretation, as in recent similar experiments, interference patterns havebeen observed with heavy atoms and molecules. Indeed, Nairz et al.obtained recently interferences with spherical fullerenes, which arealmost six orders of magnitude bigger than a nucleon particle. It isimpossible to conceive, at least in the present stage of our knowledgethat fullerenes can pass through both slits simultaneously.



In order to get a comprehensive picture of all these interferenceexperiments carried out with light and material particles, Jin et al.in 2010 advanced a particle-based description of Young’s double slitinterferences without even referring to the wave theory.

Consequently, what can be essentially deduced from theexperimental evidence during more than two centuries of strugglingbetween the wave and particle theories, is that experiments simplyconfirm both natures for light (and for material particles).

Bohr, although initially in favor of the wave concept, understoodquite rapidly that the only way out of this frustrating dilemma is thewave-particle complementarity, even if this concept is hardly com-prehensive for the human common sense.

We will now present selected recent experiments and discuss thefinal results with respect to the second quantization procedure.

• Mizobuchi and Ohtake double-prism experiment

Since Bohr announced his famous complementarity principleexpressing that light has simultaneously both the wave and particlenature, appearing mutually exclusive in every physical situation, theopposition between the defenders of the wave or particle theories hadceased. However, the majority of scientists were skeptical about thewave-particle simultaneity which is not at all comprehensive for thehuman mind.

With the development of lasers and the revolutionary parametricdown converters techniques, it becomes possible to realize conditionsin which, with a very high degree of statistical accuracy, only onephoton is present in the experimental apparatus. Hence, in order totest Bohr’s complementarity principle, Ghose, Home and Agarwal,proposed in 1991, the double prism experiment which was carriedout by Mizobuchi and Ohtake in 1992.

A single photon pulse goes through two prisms separated byan air gap whose thickness is less than the incident photon wave-length (Fig. 5.5.1). In these conditions, classical electromagnetic wavetheory predicts a tunneling effect for the wave-like photon, accordingto which the wave will traverse the air gap straightforward and will bedetected by Detector 1. On the other hand, a particle-like photon will



Single photon hv

Detector 1 Transmission

Detector 2Reflec on

Double Prisme

Gap

Coincidence

Counter forTransmission

Counter forReflec on

Fig. 5.5.1. Schema of the double prism experiment initially proposed by Ghose, Homeand Agarwal and carried out by Mizobuchi and Ohtake. With kind permission fromSpringer Science+Business Media: <Foundations of Physics, The Two-Prism Experimentand Wave-Particle Duality of Light, 26(7), 943–953, Fig. 1 1996, Partha Ghose andDipankar Home, and any original (first) copyright notice displayed with material>.

be reflected by the inner wall of the first prism and will be detectedby Detector 2. Both detectors are connected to a coincidence counterin order to detect any eventual coincidence.

The experiment results have shown very strong anti-coincidenceentailing that a single photon state goes either through the gapby wave tunneling or is reflected by the first prism like a particle.Consequently, both the wave and the particle natures of the photonare exhibited in the same experimental conditions, confirming thewave-particle concept and at the same time contradicting the mutualexclusiveness of Bohr’s complementarity principle.

Finally, it is worth noting that simultaneous wave-particle dualityfor neutrons in similar interferometer have also been reported byGreenberger and Yasin.

• Grangier’s et al. experiments, photon indivisibility

The purpose of this experiment is to test the indivisibility of thephoton. It is based on the two emission lines of the lower energylevels of Ca atoms 1S0 →1 P1 and 1P1 →1 S0 (Fig. 5.5.2). The firsttransition gives a photon with angular frequency ω1, whose detection



PM1 Ca atoms

ω1 ω2

Beam Spli er PM Transmission

PM Reflec on

Rectangular Gate

Ν(ω1)

Ν(Τ)Ν(R) Coincidence

γ

ω1

ω2

1P1

1S0

1S0

Fig. 5.5.2. Experimental disposition of Grangier’s et al. experiment for the demonstra-tion of the indivisibility of a single photon state. Reproduced with permission fromRef. 23.

opens a rectangular gate to control the detection of the second photonemission at ω2. The last one passes through a beam splitter behindwhich photomultipliers can detect the transmitted and the reflectedω2 photons. The intermediate level has a life time τ and in order todecrease the probability to get coincidence, counting due to two ω2

photons issued from two different Ca atoms, the gate width γ hasbeen chosen to be much smaller than τ .

The final results of the experiment revealed extremely low levelsof coincidences, demonstrating without any doubt, that the photonsare integral particles transmitted or reflected by the beam splitter asa whole.

Although the issue of the experiment shows the indivisibility ofthe photons, it does not exclude the wave nature of them. It isimportant adding here that A. Aspect’s experiments on the enta-gled photon states, followed by a large number of other experimentsworld around, have shown the permanent violation of Bell’s inequal-ities implying the absence of hidden variables for the point particlerepresentation of photons. Furthermore, they have demonstrated thenecessity of the development of a non-local representation with a realwave function in which new variables are not explicitly excluded.Consequently, we will examine in the next Chapter the possibilityof establishing a quantum equation for the photon with a quantized



vector potential amplitude within a non-local representation with thevector potential function playing the role of a real wave function.

• Hunter–Wadlinger experiments, the photon spatial expansion

Akhiezer and Berestetskii, among many others, have pointed outthat it is impossible to consider a photon within a length shorterthan the wavelength. In fact, a point-photon should have a zerowavelength and infinite energy. Nowadays, this concept is generallyaccepted. However, the lateral expansion of the photon considered asan integral particle was always an intriguing part of physics.

For many decades, it has been well known that Faraday’s metalliccage is a quite efficient shield toward electromagnetic waves, providedthat the dimensions of the holes of the grid are roughly five timessmaller than the wavelength of the incident radiation, which is ∼ λ/5.That was rather an empirical rule and most of the constructed grillsto screen the radiation of a wavelength λ imposed the dimensions ofthe holes to be within the interval λ/4 to λ/8.

Robinson in 1953 and Hadlock in 1958 carried out experimentsusing microwaves crossing small apertures, and deduced that noenergy is transmitted through apertures whose diameters are lessthan roughly ∼λ/4.

In 1986, Hunter and Wadlinger, realized a very simple experimentinspired by those carried out by Robinson and Hadlock (Fig. 5.5.3).They used x-band microwaves with λ = 28.5 mm and measured thetransmitted power through rectangular or circular apertures of dif-ferent dimensions.

The purpose was to define the smaller dimensions of the slitbeyond which the transmitted power is zero. All the results, correctedwith the subtraction of the first harmonic of the emitted beamby the microwave antenna, have shown that the cut-off dimension(diameter for the circular aperture, width for the rectangular one) isroughly ∼λ/π.

Consequently, taking also into account that the longitudinalextension of the photon along the propagation axis is the wavelengthλ itself, it implies that the intrinsic “photon volume” depends on itswavelength and it should be proportional to λ3.



Transmitted power

Microwave antennaPower detector

Screen with variable slitSlit width (mm)

++

+

++

hc/λ

∼λ/π

Fig. 5.5.3. Schema of Hunter’s and Wadlinger’s experiment for the investigation of thelateral spatial extension of the electromagnetic waves. Reproduced with permission fromRef. 27.


We have seen that the equivalence of the photons energy with that ofthe classical electromagnetic wave description implies that a relation-ship between Planck’s constant and the vector potential quantizedamplitude should exist. We have also analyzed the QED singularitiesand shown that the zero-point energy term

∑k,λ

12ωk is not involved

in the vacuum-electron interactions.In fact, the vacuum effects in QED like the spontaneous emis-

sion and the Lamb shift are interpreted based on the commutationproperties of the creation and annihilation operators a+

kλ and akλ

respectively, without invoking at all the term∑

k,λ12ωk in the cal-

culations. Indeed,∑

k,λ12ωk is a constant and in the QED repre-

sentation commutes with all Hermitian operators corresponding tophysical observables.

Conversely, the zero-point energy term∑

k,λ12ωk is used in var-

ious semi-classical descriptions for the interpretation of the Casimireffect and for the Lamb shift (Welton’s and Feynman’s models) inwhich the appropriate cut-offs, introduced in order to get physicalresults, are taken from the QED calculations in which. . . the term∑

k,λ12ωk is ignored.



In other words, the zero-point energy in QED results fromthe mean value of the normal ordering and anti-normal orderingHamiltonians, leading to a harmonic oscillator Hamiltonian and hasno influence in the quantum calculations for the vacuum effects sinceit is a constant but it is used in the semi-classical descriptions forthe interpretation of the vacuum effects.

On the experimental front, we may essentially deduce that thephotoelectric effect, Compton’s scattering and Young’s interferenceshave been interpreted by both the wave and particle nature of light.This has been further established by Mizobuchi and Ohtake’s exper-iment, which also demonstrates that the wave and particle naturesappear simultaneously and are not necessarily exclusive.

On the other hand, Robinson, Hadlock, Hunter and Wadlingerhave demonstrated that the photon should have a lateral extensionof the order λ/4 to λ/π and cannot be a “point particle,” while theexperiments of Grangier et al. demonstrated the photon indivisibility.

During the Compton scattering experiments, the photonexhibits a very particular property; it conserves its integrity asparticle, but may concede a fraction of its energy, entailing a modi-fication of its wavelength and consequently its energy, momentumand spatial extension. Bothe’s experiments demonstrated this partic-ular property of light showing further the conservation of energy andmoment during each event of a x-ray photon scattered by an electron.

To sum up the experimental evidence, it turns out that thephoton appears to be an indivisible quantum (segment) of the elec-tromagnetic field over a period, thus extended along a wavelength,with intrinsic wave properties, emitted and detected as a whole andcapable of interacting with charged particles, increasing (or evendecreasing) its wavelength and consequently decreasing (or evenincreasing) its energy.

In the next chapter, we will consider further theoretical elabo-rations conform to the experiments and advance a non-local photonrepresentation based on a precise quantization of the electromagneticfield vector potential, linking coherently the electromagnetic theoryand quantum mechanics and entailing a well-defined description ofthe quantum vacuum.



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Chapter 6

Analysis of the ElectromagneticField Quantization Process

and the Photon Vector Potential.Non-Local Photon

Wave-Particle Representationand the Quantum Vacuum

6.1. Quantized Vector Potential Amplitude of a SinglePhoton State

Frequency and vector potential imply electromagnetic wave propertieswhile energy and momentum imply particle properties. In a full wave-particle description of light the challenge consists of establishing acoherent relation among all these physical quantities.

• Dimension analysis from Maxwell’s equations. Vector potentialamplitude proportional to the frequency.

The relations (4.1.9) attribute a precise value of energy andmomentum to a photon related to the electromagnetic wave onlyby the angular frequency, while no information is supplied about thephoton vector potential which is basically indispensable in the clas-sical electromagnetic description. As we have discussed in Chapter5 (5.1), in the electromagnetic theory it is impossible to define theenergy of an electromagnetic wave simply by the frequency whileomitting the vector potential. Hence, if the photon is an integral

99



particle with definite quantum energy then the exact quantization ofits vector potential amplitude should be a natural consequence.

Obviously, the quantization procedure of the electromagneticfield, described briefly in Section (4.3) does not result effectively tothe description of the vector potential of individual photons. Indeed,examining the link relations (4.3.14), obtained from (4.3.13) whenN = 1, the amplitude of the vector potential for a single k-modephoton with angular frequency ωk used in quantum electrodynamics(QED) description is given by

α0k(ωk) =

√

2ε0ωkV(6.1.1)

Of course, for a large number of photons in a volume V , with veryshort wavelengths compared to the characteristic dimensions of V ,the summation transformation (3.5.44) established in the density ofstates theory serves to eliminate directly the external volume parame-ter V involved in the expression of the square amplitude of the vectorpotential. Consequently, one may argue that it is needless to enhancefurther the analysis of (6.1.1) to a single photon since it is sufficientfor the description of physical systems studied in QED. However,the extraordinary evolution of the experimental technics, such as theparametric down conversion, permit the realization of single photonstate experiments opening possibilities for direct measurements ofthe photon physical properties like the electric and magnetic fields.Hence, without going beyond the level of equation (6.1.1) we missfundamental information on the photon vector potential amplitudethat might ensure the link with the electromagnetic theory leadingto a comprehensive representation of the wave-particle single photonstate.

In fact, it is intriguing to deduce from (6.1.1) that the vectorpotential of a single photon depends on an independent free spacevariable V and that, for a given finite volume, it is inversely propor-tional to the square root of the angular frequency. This is also puz-zling from physical point of view because, knowing that the currentdensity j(r, t) has units of charge (C) times m−2 s−1, a dimensionanalysis of the general expression of the vector potential amplitude


Analysis of the Electromagnetic Field Quantization Process 101

in equation (3.3.10) issued from the Maxwell’s equations shows thatit should be proportional to the frequency and consequently for ak-mode photon with angular frequency ωk we should have

α0k(ωk) ∝ ωk → α0k(ωk) = ξωk (6.1.2)

where ξ is a constant.This is contradictory with the frequency dependence of the vector

potential amplitude in equation (6.1.1) implying mathematically thatin the case of a single photon the volume V cannot be a free externalparameter but it has to be proportional to ω−3

k

Vk ∝(

1ω3

k

)(6.1.3)

Thus, according to the density of modes analysis in chapter 3 andthe relations (3.5.45) and (4.3.29) the quantization volume of a singlephoton state seems to be naturally an intrinsic property.

The possible spatial extension of the photon will be discussedlater. First, we are going to have a close look to the vector potentialfunction and the associated quantization constant ξ of the amplitude.

• Quantum equation for the photon vector potential.

Based on (6.1.2), the fundamental physical properties characteriz-ing both the particle (energy and momentum) and the wave (vectorpotential, wave vector and dispersion relation) nature of a singlek-mode photon are related linearly to the angular frequency

Ek

=

|pk|/c

=α0k

ξ=

|k|√ε0µ0

= |k|c = ωk (6.1.4)

An interesting feature results when considering that the photonmomentum |Pk| has the units of a mass mk times the velocity c

in vacuum so we may write |Pk| = k = mkc. Of course, mk is aparameter that has simply a dimensional value here as the photonis generally considered to be a massless particle. Using Ek = ωk = k c one gets directly Ek = mkc

2 showing that the energy-massequivalence is a direct result of the wave-particle double nature oflight since we have used the wave dispersion relation ωk = |k|c andthe momentum |Pk| = |k| which is a particle property.



Now, according to (6.1.2) the expression of the k-mode photonvector potential may be written in a general plane wave expressionwith the quantized amplitude

αkλ(r, t) = ωk

(ξελe

i(ωkt−k·r+ϕ) + cc)

= ωkΞkλ(ωk, r, t) (6.1.5)

with λ the polarization (right or left circular), ελ the polarization unitvector, ϕ a phase parameter and cc denotes the complex conjugate.

It is natural for the photon vector potential function αkλ(r, t) tosatisfy the wave propagation equation in vacuum

∇2αkλ(r, t) − 1c2∂2

∂t2αkλ(r, t) = 0 (6.1.6)

The second derivative with respect to time is proportional to thesquare of the amplitude and using (6.1.4) we have

∂2

∂t2αkλ(r, t) = −ω2

kαkλ(r, t) = −(α0k

ξ

)2

αkλ(r, t) (6.1.7)

The propagation equation now writes[α2

0k + ξ2c2∇2]αkλ(r, t) = 0 (6.1.8)

entailing that the photon vector potential amplitude can be expressedas an operator α0k proportional to the constant ξ

α0k = −iξc∇ (6.1.9)

which is valid for every mode k. Obviously, the established vectorpotential amplitude operator is quite symmetrical with the relativis-tic Hamiltonian operator for a massless particle

H = −ic∇ (6.1.10)

Application of α0k upon αkλ(r, t) along the propagation axis inducesa phase shift getting

α0kαkλ(r, t, ϕ) = iξck αkλ

(r, t, ϕ+

π

2

)(6.1.11)

But for the time variation we also get

∂

∂tαkλ(r, t, ϕ) = ωkαkλ

(r, t, ϕ+

π

2

)(6.1.12)



Combination of the last two equations yields a linear time differ-ential equation for the photon vector potential

iξ∂

∂tαkλ(r, t) = α0kαkλ(r, t) (6.1.13)

Notice that the last equation for the vector potential with quan-tization constant ξ is analogue to the Schrodinger equation for theenergy with quantization constant .

Of course, the photon vector potential function αkλ(r, t)also sat-isfies Schrodinger’s equation with the relativistic massless particleHamiltonian (6.1.10). Hence, we can write the vector potential-energy(wave-particle) equation for the photon

i

(ξ

)∂

∂tαkλ(r, t) =

(α0k

H

)αkλ(r, t) (6.1.14)

with corresponding eigenvalues ξωk for the vector potentialamplitude and ωk for the energy.

Consequently, in a non-local representation the vector potentialfunction αkλ(r, t), which also satisfies the wave propagation equation(6.1.6), may play the role of a real wave function for the photon thatcan be suitably normalized. The square modulus of ωk

√2ε0αkλ(r, t)

gives the energy density of the electromagnetic field at the coordinater at an instant t.

Furthermore, we also remark that when considering Heisenberg’senergy-time uncertainty principle

δEkδt ≥ (6.1.15)

we deduce directly from the relations (6.1.4) a vector potential-timeuncertainty according to the relation

δα0kδt ≥ ξ (6.1.16)

It is worth noting the symmetrical correspondence between thepairs Ek(ωk), and α0k(ωk), ξ for a single photon characterizingits particle and wave nature respectively.

When considering the propagation of a k-mode photon with awavelength λk along the z axis, unlike the momentum operator whichis defined by Pz = kz → Pz = −i ∂

∂z , it is not obvious how to define



a position operator and this difficulty has been widely commented inthe literature. In fact, Heisenberg’s uncertainty for the position andmomentum for a photon propagating along the z-axis can be written

δz δPz → δz δ

(1λk

)≥ 1 (6.1.17)

The physical meaning of this relation is that the photon as anintegral entity cannot be localized precisely along z except with ahigh degree of uncertainty which is of the order of the wavelengthλκ. This is a natural consequence that lays in the intrinsic wave-particle nature of the photon as expressed by the condition (4.3.29).As quoted by many authors, despite the fact that the point-particlemodel used in the QED formalism is an efficient mathematical tool,in reality, the photon cannot be defined within a length shorter thanits wavelength. Consequently, according to the density of states the-ory (3.5.45), the experimental evidence expressed in (4.3.29) and theanalysis leading to (6.1.3), a single photon state occupies a minimum“quantization volume”.

Thus, when a k-mode photon is emitted at a coordinate ze at aninstant te and propagates in vacuum along the z-axis the probabilityΠk to be localized at time t and at the coordinate z = ze+c(t−te) hasto be considered within a volume of roughly λ3

k (that is the quan-tization volume) around the point z. Considering that the vectorpotential can be a wave function for the photon then Πk is propor-tional to the square of the angular frequency

Πk = |αkλ(z, t)|2 ∝ ξ2ω2k (6.1.18)

showing naturally that the higher the angular frequency (the shorterthe wavelength) of the photon the higher the localization possibilityon the propagation axis.

Following the expression (6.1.5), the photon vector potential canbe written in general wave-particle formalism

αkλ(r, t) = ωk

[ξελe

i(ωkt−k·r+ϕ) + ξ∗ε∗λe−i(ωkt−k·r+ϕ)

]= ωk

Ξkλ(ωk, r, t) (6.1.19)



αkλ = ωk

[ξakλελe

i(ωkt−k·r+ϕ) + ξ∗a+kλε

∗λe

−i(ωkt−k·r+ϕ)]

= ωkΞkλ(akλ, a+kλ) (6.1.20)

where for (6.1.20) we have used the creation and annihilation opera-tors a+

kλ and akλ respectively for a k-mode and λ polarization photon.The general equation for the vector potential of an electromag-

netic wave, considered as a superposition of plane waves, now writes

A(r, t) =∑k,λ

ωk

[ξελe

i(ωkt−k·r+ϕ) + ξ∗ε∗λe−i(ωkt−k·r+ϕ)

]

=∑k,λ

ωkΞkλ(ωk, r, t) (6.1.21)

and that of a large number of photons in the quantum electrodynam-ics description is

A =∑k,λ

ωk

[ξakλελe

i(ωkt−k·r+ϕ) + ξ∗a+kλε

∗λe

−i(ωkt−k·r+ϕ)]

=∑k,λ

ωkΞkλ(akλ, a+kλ) (6.1.22)

The vector potential with the quantized amplitude ξ in the waveexpression (6.1.19) translates the individual wave property of thephoton while that of (6.1.20) its particle property, reflected throughthe vector potential amplitude operator. The representations of thewave and particle expressions for the vector potential amplitude oper-ator write

wave: αw0k = −iξc∇; αw∗

0k = iξ∗c∇particle: αp

0k = ξωkakλ; αp∗0k = ξ∗ωka

+kλ (6.1.23)

We also deduce the amplitude of the electric field |εk| for a singlek-mode photon which writes

|εkλ| =∣∣∣∣− ∂

∂tαkλ(r, t)

∣∣∣∣ ∝ ξω2k (6.1.24)

The last relation lifts the external volume dependence involvedin (5.3.6) and expresses the dependence of the single k-mode pho-ton electric field on the square of the angular frequency. It is worth



investigating experimentally this result by using modern technics.For example, the dependence of the photon electric field on the angu-lar frequency may be explored by high resolution laser spectroscopyon the linear Stark shifts δEStark induced by the electric field of asingle photon εkλ(ωk) on the atomic or molecular energy levels. Infact, linear δEStark is precisely proportional to the electric field whichitself depends on the angular frequency, δEStark ∝ εkλ(ωk) ∝ ωp

k. Theratio of the measures of the Stark displacements δE1 and δE2 inducedindependently by two photons with quite different energies, ωk1 andωk2 could permit to define the power dependence p of the photonelectric field on the angular frequency, that is δE1

δE2∝ (ωk1

ωk2)p. Such

an experiment could also confirm the validity of the relation (5.3.6)and whether an external arbitrary macroscopic box with dimensionsmuch bigger than the photon wavelength has any influence on thesingle photon electric field.

To resume, the quantization of the vector potential amplitudeα0k = ξωk has permitted the development of a wave-particle for-malism for the photon expressing its double nature put in evidenceby the experiments. The vector potential amplitude and the elec-tric/magnetic field for a single photon are intrinsic properties. It isimportant to keep in mind that the quantum mechanical part ofthe above wave-particle formalism is developed on the basis of theQED frame in which photons are assumed to be point particles. Thishypothesis permits to use a remarkable mathematical formalism inQED in order to study the interaction of light with matter. However,many experiments like those described in the previous chapter haveshown that photons are not really point particles and that they arespatially extended over a wavelength. Apparently, the real nature ofindividual photons seems to be more complicated than the simplepoint particle representation and we deal with this aspect in thefollowing.

• Photon quantization volume.

It is useful to recall that, in order to get the link equations (4.3.14)one has to consider in (4.3.13) the mean value of a monochromaticplane wave energy density over a period T (or in space representation



over a wavelength), while puttingN = 1 for the photon number in thequantum relation in the right hand side. In other words the operationconsists in QED to take the mean value of the classical electromag-netic wave energy over a wavelength and attribute it to a single pointparticle. Notice also that in classical theory the energy density of theelectromagnetic field 〈Wω〉T and the energy flux obtained by thePoynting’s vector 〈Sω〉T in (3.4.8) are both calculated over a periodT but they are completely independent on any external volume.

It comes out that both these physical quantities, 〈Wω〉T and〈Sω〉T , describing the energy density and flow of a “fraction” of theelectromagnetic wave limited to its wavelength, depend on the angu-lar frequency ω, on the amplitude of the vector potential α0(ω), andon the vacuum electric permittivity ε0.

Consequently, following the above analysis and considering thephoton as an integral particle with a given energy we may writeusing (3.2.29), (3.4.4) and (3.4.5) for a single k-mode photon∫

ε0ω2kα

20k(ωk)[ελei(ωkt−k·r+ϕ) + c.c]

2d3r = ωk (6.1.25)

For the last relation to hold at any instant t we deduce thatthe photon polarization unit vector ελ should have at least twoorthogonal components, e1 and e2, such as ελ = σ1e1 + σ2e2 with|σ1|2 + |σ2|2 = 1 and e1 · e2 = 0.

According to (3.4.12) right (R) and left (L) hand circular polar-ization unit vectors, εL,R = 1√

2(e1 ± ie2) are naturally appropriate

to satisfy this condition and equation (6.1.25) becomes∫2ε0ω2

kα20k(ωk)d3r = ωk (6.1.26)

Indeed, it is generally accepted that the photons are characterizedby Left or Right circular polarization and consequently λ takes onlytwo values.

The equation (6.1.26) is equivalent to the normalization ofthe energy over a wavelength of a classical electromagnetic planewave with circular polarization, issued from Maxwell’s equations, toPlanck’s expression of the quantized radiation energy. At that point,we recall the energy density equivalence between the classical and



the quantum mechanical formulations given by (4.3.13), which forN = 1 is simply reduced to

2ε0ω2kα

20k(ωk)Vk = ωk (6.1.27)

where α0k(ωk) is the vector potential amplitude that may beattributed to a k−mode photon and Vk the corresponding quanti-zation volume.

Comparison of the last two equations implies that Vk should cor-respond to an intrinsic property of a k-mode photon. From (6.1.2)and (6.1.27) we get

Vk =(

2ε0ξ2

)ω−3

k (6.1.28)

where the dependence on the angular frequency is in agreement withthe previous analysis that led to the relations (3.5.45), (4.3.29) and(6.1.3).

Since it has been demonstrated that a k-mode photon cannotbe considered as an integral entity within a region smaller than itswavelength λκ (4.3.29) and since the directional character of photonshas been demonstrated experimentally by Compton, the integrationof (6.1.26) may be carried out in cylindrical coordinates (ρ, θ, z),where z is the propagation axis along which the vector potentialrotates perpendicularly. In a full period interval, 0 ≤ z ≤ λk, theangular coordinate θ is related to z by θ = 2πz/λk and the integrationmay be reduced to the variables, z and ρ in the limits 0 ≤ z ≤ λk and0 ≤ ρ ≤ ηλk , with η a positive dimensionless constant characterizingthe transverse spatial extension of the photon and permitting thefollowing relation to hold

2ε0ω2kα

20k(ωk)

(η2

2λ3

k

)= ε0ξ

2η2 (2πc)3

ω3k

ω4k = ωk (6.1.29)

The product of the two constants η and ξ has the value

ηξ =1

(2π)3/2

√

ε0c3(6.1.30)

In a first approximation, the last relation is almost independentof the integration coordinates of equation (6.1.26). For instance,



integrating in spherical coordinates with 0 ≤ r ≤ λk2 one gets

2ε0ω2k(ξωk)2

∫ λk/2

04πχ2r2dr = ωk (6.1.31)

where χ2 is equally a positive constant for the last equation to hold.We obtain

χξ =1

(2π)3/2

√3

πε0c3(6.1.32)

Recalling that c =√

1ε0µ0

, the physical meaning of the relations

(6.1.30) and (6.1.32) is that Planck’s constant is related to the vac-uum electric permittivity ε0 and magnetic permeability µ0 by theintermediate of the photon’s intrinsic physical parameters namely,the transverse extension η and the vector potential amplitude quan-tization constant ξ.

We can now deduce the particle properties of the photon by usingthe wave characteristics and the quantization volume.

In fact, the particle properties, energy, momentum and spin ofa k-mode photon are not carried by a point particle and can beexpressed in terms of the quantization volume Vk and the wave prop-erties namely the dispersion relation, the vector potential and theelectric and magnetic fields.

The energy of the photon can be obtained by (6.1.26) using thequantization volume Vk and the quantized vector potential amplitude

Ek =∫

Vk

2ε0α20kω

2kd

3r =∫

Vk

2ε0ξ2ω4kd

3r = 2ε0ξ2ω4kVk = ωk

(6.1.33)With the same token considering a circular polarization (3.4.13

and 3.4.14) the momentum writes

Pk =∫

Vk

ε0εkλ × βkλd3r = ε0(

√2ωkα0k)

(1c

√2ωkα0k

)Vk

k

|k|= k

(6.1.34)where εkλ and βkλ denote the electric and magnetic fields amplitudesof a single mode k.



According to the classical electromagnetic theory the spin can bewritten through the electric and magnetic fields components

|S| =∫

Vk

ε0|r × (εkλ × βkλ)|d3r

= ε0

(c

ωk

)(√

2ωkα0k)(

1c

√2ωkα0k

)Vk = (6.1.35)

where we have considered the property calculated by Rybakov andSaha that the mean value of the length in a k-mode photon state isequivalent to c/ωk.

The fact that the energy, momentum and spin of the photoncan be expressed through Vk supports that the photon is not inreality a point but has a minimum spatial extension correspondingto the quantization volume in which the quantized vector potentialoscillates over a period. Consequently instead of “wave-particle” weshould more properly employ the term “wave-corpuscle”. It is worthnoting that according to the relation (6.1.28) for micro-waves andradio-wave frequencies the spatial extension of a single photon cor-responds to macroscopic dimensions.

The quantization volume Vk appears to be an intrinsic propertyfor a single photon and defines the minimum space in which thequantized electric and magnetic fields oscillate at the frequency ωk

over a wavelength along the propagation axis. It may also ensure thelink with the particle representation. In fact, when introducing theparticle operators αp

0k and αp∗0k expressed in (6.1.23) into the quantum

electrodynamics position and momentum operators relations (4.3.11)one gets directly the corresponding position Qkλ and momentumPkλ operators depending on Vk

Qkλ =√ε0Vk(α

p0k + αp∗

0k) Pkλ = −iωk

√ε0Vk(α

p0k − αp∗

0k)(6.1.36)

Using (6.1.28) the demonstration of Heisenberg’s commutationrelation, a fundamental concept in quantum theory, is immediate

[Qkλ, Pk′λ′ ] = −iε0ω2k′ωk

√VkVk′ [(ξakλ + ξ∗a+

kλ), (ξak′λ′ − ξ∗a+k′λ′)]

= iδkk′δλλ′ (6.1.37)



showing that the origin of the uncertainty in the quantum mechanicsformalism lays in the dimensional nature of quanta as we have alreadyseen in (6.1.17) .

• Relation between the photon vector potential and the electroncharge.Definition of the vector potential amplitude quantization con-stant ξ.

Following the above concepts on the wave-particle nature of thephoton we may write the energy density equivalence (6.1.27) with aslight rearrangement as follows

4πc(

14πc

2ε0ω2kα0k(ωk)Vk

)α0k(ωk) = 4πcQα0(ωk) = ωk

(6.1.38)making to appear a Q parameter which has charge units. Using(6.1.2) and (6.1.30) we obtain

Q2 =(

14πc

2ε0ω2kα0(ωk)Vk

)2

=η2

4(ε0hc) (6.1.39)

where h is Planck’s constant.Using for η the experimental approximate value of 1/4 discussed

in Chapter 5.5 we obtain immediately Q ∼ 1.6 10−19 Coulomb,which is the electron charge, a physical constant appearing naturallyhere when considering the equivalence of the classical and quantummechanical energy density of the electromagnetic field beyond thepoint particle concept.

This signifies that the physical origin of the electron charge isstrongly related to the quantum of the electromagnetic field over aperiod, the photon. A similar result has been obtained by Williamsonand van der Mark following a semiclassical approach leading to theexpression Q2 = 3

(2π)3(ε0hc) ≈ 1.4 10−19C. The better numerical

value obtained with (6.1.39) is due to the experimental estimationof η.

Based on this result we can go further in the definitions of both ηand ξ constants by the intermediate of the electron charge expressed



through the fine structure constant αFS as defined in the equation(4.6.4), by equating to (6.1.39)

η2

4(ε0hc) = 2αFS(ε0hc) (6.1.40)

getting

η =√

8αFS (6.1.41)

Consequently, we can have an approximate idea of the numericalvalue of the vector potential amplitude quantization constant from(6.1.38) and the coupling of (6.1.30) to (6.1.41)

4πcQα0(ωk) ≈ 4πceξωk ≈ ωk (6.1.42)

|ξ| ≈∣∣∣∣∣ 1

(2π)3/2

√

8αFSε0c3

∣∣∣∣∣ =∣∣∣∣

4πec

∣∣∣∣ = 1.747 10−25 Volt m−1s2

(6.1.43)

where e is the electron change. Appropriate experiments couldattribute more precise values to η and ξ.

Note that according to the numerical value obtained for ξ andwithout yet considering any normalization factor the probability Πk

in (6.1.18), expressing the possibility of localizing a photon at thecoordinate z within approximately the quantization volume, tendsnaturally to 1 for high values of frequencies, of the order 1024 Hz,which correspond to photon wavelengths shorter than the electronradius.

Finally, it is important remarking that when introducing in therelation (4.6.4) the expression ≈ 4πceξ issued from (6.1.42) andtaking into account that c2 = 1/ε0µ0 we obtain the interesting resultthat the electron charge can be expressed naturally through the finestructure constant and the vacuum constants ξ and µ0.

e = (4π)2αFSξ

µ0(6.1.44)

The physical meaning of the last relation is that the elementarycharges might also be issued from the quantum vacuum like photonsas we will see in the following section. The question of topologicalstructures of the quantum vacuum that might lead to the elementarycharges arises naturally from the last relation.



6.2. Ground Level of the Electromagnetic Field. QuantumVacuum Representation

• Analysis of the origin of the zero-point energy in the QEDformalism

Let us go back to the two expressions of the Hamiltonians, (5.2.1)and (5.2.3) which result simply from the commutation of A∗

kλ andAkλ in equation (4.3.9) and let us deal with the fundamental problemof the mathematical origin of the zero-point energy term ω/2.

As it is well known, a simple commutation between the vectorpotential amplitude and its complex value, which has absolutelyno influence in the classical expression of the electromagnetic fieldenergy (4.3.9), proves to be of crucial importance for the quan-tum mechanical Hamiltonian. We can now analyse the reason forwhich the harmonic oscillator radiation Hamiltonian (4.3.18) cannotbe obtained when the link equations (4.3.14) are introduced in theexpression of the electromagnetic field energy (4.3.9).

We recall that for a harmonic oscillator of a particle with massm = 0 and momentum p = mdq/dt, the transition from the classicalexpression of energy (4.2.20) to the Hamiltonian (4.2.25), i.e.,

EHO =12(P 2 + ω2Q2) → HHO =

12(P 2 + ω2Q2) = ω

(a+a+

12

)(6.2.1)

where P = i√

ω2 (a+ − a) and Q =

√

2ω (a+ + a), is immediate andneeds no commutation operations between the canonical variables ofposition and momentum.

Consequently, the harmonic oscillator Hamiltonian HHO for anon-zero mass particle is a direct result expressing a perfect cor-respondence between the classical canonical variables of momentumand position, P and Q respectively, and the quantum mechanicalHermitian operators P and Q.

Conversely, this is not the case for the electromagnetic fieldbecause commutations between Qkλ and Pkλ occur during thetransition from the expression (4.3.9) to (4.3.12). Therein, we recall



that Heisenberg’s commutation relation [Qkλ, Pk′λ′ ] = iδkk′δλλ′ isa fundamental concept of quantum mechanics which should not beignored when transiting from classical variables to quantum mechan-ical operators. This is exactly the reason for which the equation(4.3.12) cannot be considered as the electromagnetic field energyexpression for establishing the quantized radiation Hamiltonian sinceit is not equivalent, from the quantum mechanical point of view, toequation (4.3.9).

In fact, introducing the vector potential amplitude expressions of(4.3.11) in the equation (4.3.9)

Akλ =(ωkQkλ + iPkλ)

2ωk

√ε0V

; A∗kλ =

(ωkQkλ − iPkλ)2ωk

√ε0V

→

E = 2ε0V∑k,λ

ω2k|Akλ|2 (6.2.2)

and keeping the commutation terms appearing in the calculation onegets the complete expression for the energy of the electromagneticfield

EEM =12

∑k,λ

(P 2kλ + ω2

kQ2kλ) ± iωk[Qkλ, Pkλ] (6.2.3)

where the (+) sign is obtained when the starting equation is consid-ered to be in the “normal ordering”, 2ε0V

∑k,λ ω

2kA

∗kλAkλ, and the

(-) one when in the “antinormal ordering”2ε0V∑

k,λ ω2kAkλA

∗kλ.

Equation (6.2.3) is a rigorous mathematical result which using(4.3.17) leads to the radiation Hamiltonians

HEM =12

∑k,λ

(P 2kλ + ω2

kQ2kλ) ± iωk[Qkλ, Pk′λ′ ]

⇒

H+

EM =∑k,λ

Nkλωk

H−EM =

∑k,λ

(Nkλ + 1

)ωk

(6.2.4)

where H+EM is the Hamiltonian corresponding to the (+) sign, hence

to the normal ordering of the vector potential amplitude and its



conjugate, while H−EMthe Hamiltonian corresponding to the (-) sign

thus to the antinormal ordering.This is the correct mathematical way to get identical results

with the Hamiltonian (6.2.4) when the link equations (4.3.14) areintroduced directly in the classical energy expression (4.3.9). Indeed,the introduction of the fundamental relations of the vector poten-

tial amplitude operators Akλ =√

2ε0ωkV akλ; A∗kλ =

√

2ε0ωkV a+kλ

in the classical electromagnetic field energy expression EEM =2ε0V

∑k,λ ω

2k|Akλ|2 leads directly to the “normal ordering” Hamil-

tonian HNO =∑

k,λNk,λωk and the “antinormal ordering” one

HANO =∑

k,λ (Nk,λ + 1)ωk.Consequently, the harmonic oscillator Hamiltonian for the electro-

magnetic field can only be obtained by considering the mean valuesof the normal ordering and antinormal ordering Hamiltonians

HEM(harmonic oscillator)

=12(HNO + HANO) =

∑k,λ

(Nk,λ +

12

)ωk (6.2.5)

Of course, the same result is obtained from (6.2.3) and (6.2.4)using the position and momentum representation

HEM(harmonic oscillator)

=12

1

2

∑k,λ

(P 2kλ + ω2

kQ2kλ) + iωk[Qkλ, Pk′λ′ ]

+12

∑k,λ

(P 2kλ + ω2

kQ2kλ) − iωk[Qkλ, Pk′λ′ ]

=12

∑k,λ

(P 2kλ + ω2

kQ2kλ) =

12(H+

EM + H−EM)

=∑k,λ

(Nkλ +

12

)ωk (6.2.6)



The last operations clearly show that considering the mean valueof the normal ordering and antinormal ordering Hamiltonians impliesthat the Heisenberg commutation relation is eliminated suppressingby this way an important quantum property involved in both Hamil-tonians. However, the physical reasons for considering this meanvalue of the Hamiltonians in order to get a harmonic oscillator oneare quite elusive since no experiment has ever demonstrated that asingle photon state is a harmonic oscillator.

As it has been analyzed in section (5.4), the zero-point energy ofthe electromagnetic field Hamiltonian as a constant value is not reallyinvolved directly in the electron-vacuum interactions because it com-mutes with all Hermitian operators corresponding to physical observ-ables. Therein, it has been pointed out by many authors that themathematical and physical basis for the description of the electron-vacuum interactions in QED issues from the properties of the pho-ton creation and annihilation operators,a+

kλ and akλrespectively. Onthat point, K.Symanzik was the first to argue that the zero-pointenergy singularity in the quantized fields theory is intriguing leadingto unphysical situations.

Indeed, one may wonder whether the contribution of the term∑k,λ

12ωkto the quantized electromagnetic field Hamiltonian of

equation (4.3.18), obtained by suppressing indirectly Heisenberg’scommutation relation±iωk[Qkλ, Pkλ] in the calculations, can reallyrepresent a physical state.

Considering the normal ordering (+) in the radiation Hamiltonianwithout dropping Heisenberg’s commutation relation, the Hamilto-nian of the Electromagnetic field writes

H(+)EM =

12

∑k,λ

(P 2kλ + ω2

kQ2kλ) + iωk[Qkλ, Pk′λ′ ] =

∑k,λ

Nkλωk

(6.2.7)

This is not an harmonic oscillator Hamiltonian and it is generallyemployed in QED casting aside the quantum vacuum energy singu-larity. Now, we may naturally wonder that if the Hamiltonian (6.2.7)represents the physical state of a system of photons then what could



be the quantum vacuum in complete absence of photons? We willinvestigate this particular situation in the following.

• Ground level of the electromagnetic field and the vacuum

According to the relations (6.1.4), for ωk → 0 all the physicalproperties of the photon vanish. However, even when ωk = 0, thatis in complete absence of energy and vector potential, the field Ξkλ

expressed in (6.1.19 to 6.1.22) does not cancel and reduces to thevacuum field Ξ0λ, involving the polarization, which can be describedas a vector field as well as a quantum mechanical operator

Ξ0λ = ξελeiϕ + ξ∗ε∗λe

−iϕ (6.2.8)

Ξ0λ = ξakλελeiϕ + ξ∗a+

kλε∗λe

−iϕ (6.2.9)

whereas the presence of the creation operator means that the vac-uum field Ξ0λis a dynamic entity capable of inducing electronictransitions.

Equations (6.2.8) and (6.2.9) represent the ground state of theradiation field (ωk = 0,∀ k), in complete absence of energy and vectorpotential and can thus be assimilated to the vacuum state. Obvi-ously, the field Ξ0λ is a real entity having the units Volt m−1s2

implying an electric potential nature of the quantum vacuum asin the case of the classical electromagnetic description. In fact, aswe have seen in (3.2.13) and (3.2.15), since the very beginning ofthe electromagnetic theory the vacuum has been characterized by animpedance, Rvacuum =

√µ0

ε0≈ 120π Ω, a magnetic permeability µ0 =

4π× 10−7Hm−1 and an electric permittivity ε0 = 8.85× 10−12Fm−1.These physical quantities play an important role in Maxwell’s equa-tion independently on the system of units used because they definethe celerity of the electromagnetic field in vacuum, c =

√1/ε0µ0.

The vacuum field Ξ0λ is the generating function for photons. Con-sequently, the energy and by the same token the mass appear to bethe direct result of the real vacuum field vibrations.

Furthermore, following the relation (6.1.43) the electric potentialcharacter of the vacuum expressed through the constant ξ entails that



every charge moving in space with an acceleration γ will experiencean electric potential Uvacuum due precisely to the field Ξ0λ

Uvacuum = ξ |γ| (6.2.10)

This could play an important role in the cosmic vacuum energyand consequently contributes to the cosmological constant as we willdiscuss later.

Finally, introducing (6.1.28) in (5.3.5) we obtain the expectationvalue of the electric field in vacuum which tends to zero in the absenceof photons

〈E(r)2〉vacuum =∑k,λ

(ξω2k)

2 → 0 for ωk(∀k) → 0 (6.2.11)

Hence, in this description the QED singularities, related to theinfinite vacuum energy and infinite expectation value of the photonselectric field in the vacuum state are lifted.

6.3. The Vacuum Field Effects

We can examine now in which manner the effects due to the quan-tum vacuum, described briefly in Section 4 and whose difficulties areencountered in Section 5, can be described through the formalismdeveloped in the previous chapter.

• Spontaneous emission

For the description of the electron-vacuum interaction and inorder to overcome the difficulty arising from the (5.4.4), a “vacuumaction” operator corresponding to an interaction Hamiltonian perangular frequency between the vacuum field Ξ0λ and an electron ofmass me and charge e can now be defined like the third term of theHamiltonian (4.4.3)

Hωk= −i e

meΞ0λ · ∇ (6.3.1)

Thus, the amplitude of the transition probability per angular fre-quency between an initial state |Ψi, 0〉 with total energy Ei = ωi,corresponding to an atom in the state |Ψi〉 with energy Ei in vacuum| . . . , 0kλ, . . . , 0k′λ′ , . . .〉, and a final state |Ψf , nkλ〉 with total energy



Ef,nkλ= ωf + nkλωk representing the atom in the state |Ψf 〉 with

energy Ef = ωf in the presence of nkλ photons, can be expressedin first order time dependent perturbation theory using (4.5.5)

c(1)ωk(t) = − e

me

∫ t

0[〈Ψf , nkλ|ξεkλ · ∇akλ|Ψi, 0〉ei(ωi−ωf−nkλωk)t′

+〈Ψf , nkλ|ξ∗ε ∗kλ · ∇a+kλ|Ψi, 0〉ei(ωi−ωf−nkλωk)t′ ]dt′

(6.3.2)

Since akλ|Ψi, 0〉 = 0, using the fundamental definition of the momen-tum operator (4.1.12) as well as Heisenberg’s equation of motion(4.5.22b) the scalar product of the equation (6.3.2) corresponding tothe creation operator a+

kλ writes

〈Ψf , nkλ|ξ∗ε ∗kλ · ∇a+kλ|Ψi, 0〉 = −ξ∗δ1,nkλ

meωifε∗kλ · rif/ (6.3.3)

with rif = 〈Ψf |r|Ψi〉, ωif = ωi − ωf and considering the expressionof ξ =

4πec from (6.1.43) one gets with an equivalent calculation asin (4.5.24) the spontaneous emission rate in the elementary solidangle dΩ

Wif =1

3c3e2

4πε0ω3

if |rif |2dΩ (6.3.4)

The calculation shows that the spontaneous emission is mainlydue to the creation operator a+

kλ, which here is involved in the quan-tum vacuum expression Ξ0λ permitting to establish an interactionHamiltonian.

• Lamb shift

In section 4.6 we have calculated the energy level shift due to thevacuum interaction by employing Bethe’s famous calculation consist-ing of neglecting infinite quantities and imposing arbitrary integra-tion limits. Whatever the strongly negative comments of Dirac andthose of many authors in the literature Bethe’s calculations havethe merit to be the first demonstrating the influence of the creationoperator onto the atomic energy levels obtaining precise results. It isworth mentioning that when introducing the vector potential opera-tor (6.1.23) in the interaction Hamitonian inducing the energy shift



(4.6.3) in Bethe’s calculation and considering the same assumptionswe get identical result showing that the energy shift is indeed due tothe interaction with the vacuum.

However, we can now have a different look to the Lamb shiftby applying a method for the description of the interaction of thevacuum state with the atomic energy levels based on the vacuumradiation pressure.

Within a first approximation, the energy shifts of the electronbounded states can be seen as a topological effect of the vacuumradiation pressure upon the electronic orbitals. In fact, the motion ofa bounded state electron with charge e, whose energy is Enlj, in thevacuum field Ξ0λ can be characterised by Bohr’s angular frequency

ωnlj =Enlj

=R∞n2

(6.3.5)

where n, l, j are the quantum numbers of the corresponding orbitalentailing the rise in the electron frame of a vector potential amplitude

α0(nlj) = ξω(nlj) (6.3.6)

corresponding to “vacuum photons” with energy

4πecξωnlj = ωnlj (6.3.7)

Hence, despite the fact that the vacuum state has no photons inthe normal ordering Hamiltonian representation, an electron experi-ences in its frame photons due to its periodic motion in the vacuumfield. The radiation pressure per unit surface on the atomic orbitalswrites

dPvacuum =∑

λ

ε0

∣∣∣E(R,L)0

∣∣∣2 dΩ (6.3.8)

where |E(R,L)0

| is the electric field amplitude of the left and righthand polarized photons seen by the electron in its frame due to thevacuum.

Using (3.4.13) and (3.4.14) to express the circular polarizationcomponents we obtain for the square amplitude of the electric field

|E(R)0

|2 = ω2nlj(|A(R)

x0|2 + |A(R)

y0|2) = ω2

nlj(|A(L)x0

|2 + |A(L)y0

|2)= ω2

nlj(2ξ2ω2

nlj) (6.3.9)



Consequently, using (6.3.9) and (6.1.43) the summation of (6.3.8)over the two circular polarizations, R and L, and over the whole solidangle gives the total vacuum radiation pressure

Pvacuum = 4πε0(4ξ2ω4nlj) ≈

ω4nlj

4αFSπ2c3(6.3.10)

Hence, if the electron subsists in the effective volume Vnlj(eff ) ofthe nlj orbital the corresponding energy shift due to vacuum radiationpressure writes

δE = PvacuumVnlj(eff ) ≈ ω4nlj

4αFSπ2c3Vnlj(eff ) (6.3.11)

A delicate operation consists of defining V(nlj)(eff). The nS elec-tronic orbitals for example are characterised by a probability den-sity distribution decreasing smoothly with an exponential expression.Furthermore, for the excited electronic nS states the radial wavefunctions contain negative values and the electron probability den-sity distribution involves a significant space area with zero values.Hence, in the case of atomic hydrogen for the spherically symmetri-cal electronic orbitals nS, in order not to take into account the spaceregions where the probability density is zero, the effective volumemay be written in a first approximation

V(nS)(eff ) ≈ 1n

43π|〈ψns|r|ψns〉|3 (6.3.12)

and following the last two equations the corresponding frequency ofthe energy shift

νnS =δE(nS)

h≈ ω4

nS

6αFSπ2c31n|〈ψns|r|ψns〉|3 (6.3.13)

Putting ωnS = R∞n2

with R∞ = 13.606 eV, and considering thecube of the mean value of the distance in the hydrogen electronicorbitals nS

|〈r〉|3n,l=0 =∣∣∣∣∫

Ψ(n,l=0)rΨ∗(n,l=0)d

3r

∣∣∣∣3

= a30

(3n2

2

)3

(6.3.14)



with a0 = 0.53 10−10 m being the first Bohr radius, the frequency ofthe energy shift writes

νnS ≈ 1n3

9R4∞a30

16αFSπ2c34(6.3.15)

Significant Lamb shifts have been observed for the nS orbitals(n being the principal quantum number) having a spherical prob-ability density distribution and zero orbital momentum l = 0. Weobtain the Lamb shifts frequencies for the hydrogen nS levels

1S: ∼ 7.96 GHz(8.2), 2S: ∼ 1000 MHz (1040),

3S: ∼ 296 MHz (303), 4S: ∼ 124 MHz

where the experimental values are given in parenthesis.Consequently, the energy shifts of the nS levels of the hydrogen

atom interacting with the vacuum can be estimated with a rathergood approximation for such a simple calculation by considering theradiation pressure of the vacuum “seen” in the frame of the electronsdue to their periodic motion in the vacuum field Ξ0λ.

By the same token the Casimir effect may be interpreted as theradiation pressure difference between the inner and outer space ofthe conducting plates due to photons “seen” by the electrons, atomsand molecules of the inner and outer surfaces of the plates due tothe field Ξ0λ.

• Vacuum energy density and the cosmological constant

In Chapter 5.3 we have pointed out that recent astronomicalobservations revealed the vacuum energy density in the universe tobe roughly 10−9 J/m3, about 120 orders of magnitude lower than thevalue predicted by the QED theory which is 10110 J/m3 when consid-ering cut-offs for the integration of equation (5.3.3) at about 1043 Hzrad corresponding to Planck’s energy ∼1019 GeV. Not surprisingly,some authors have characterized this result as the “worst theoreticalprediction in the history of physics”. This huge discrepancy betweenthe theoretical and the observed values of the vacuum energy den-sity has been called “vacuum catastrophe”, in analogy with the “UVcatastrophe” which a century ago revealed the necessity for the devel-opment of a new theory, Quantum Mechanics. We recall again that



the vacuum catastrophe” is a direct result of the zero-point energy∑k,λ

12ωk of the fields which results when considering the mean

value of the normal ordering and antinormal ordering Hamiltoniansin the quantization process.

Recent studies on the dark energy and the arising constraintshave stimulated the creation of new models in order to remedy thetheoretical difficulties in cosmology. Carroll et al. have shown thatthe modelling of the dark energy as a fluid has led to the classi-cal phantom description which revealed to be instable from bothquantum mechanical and gravitational point of view. Some propos-als based on phantom cosmology have introduced more consistentdark energy models such as Fanaoni’s quantum effects in generalrelativity, Nojiri’s non-linear gravity-matter interaction, Abdalla’srevised gravity and Elizalde’s effective phantom phase and hologra-phy among many others.

On the other hand, various studies on the contribution of thequantum vacuum state to the dark energy have been carried outbased on both quantum chromodynamics (QCD) and QED descrip-tions like those of Frieman, Silvestri, and more recently Labun’sworks. However, it has been pointed out by many authors that ourunderstanding of the extremely complex QCD vacuum state is veryblurred. Conversely, QED theory offers a quite comprehensive frame-work and it is worthy to analyse here the eventual contribution of theelectromagnetic field ground state (zero energy, zero vector potential)to the cosmic vacuum energy density.

According to the relation (6.1.28), for photons with very low angu-lar frequencies the corresponding spatial extension gets extremely big.For ωk → 0 the volume of a k-mode photon tends to cosmic dimen-sions while the energy, the vector potential and the momentum tendto extremely low values. For a finite, universe behaving as a cosmiccavity, ωk in (6.1.28) gets infinitely small but not zero. Hence, we mayassume the universe to be filled by electromagnetic waves with cosmicdimensions and extremely low energy. Under these conditions a quitelarge number of cosmic photons issued from the fluctuations of thebackground vacuum field Ξ0λ could contribute to the vacuum energydensity. Furthermore, in particular situations the various distributions



of mass and charges in the universe may also induce fluctuations ofΞ0λ giving rise to higher frequency photons that might be in the rangeof long wavelength radio waves up to microwaves contributing to themicrowaves irregularities observed in space.

Another possible contribution to the cosmic vacuum energy den-sity could come from the interaction of particles with the vacuumfield Ξ0λ. In fact, as we have seen in Chapter 6 for ωk → 0 (∀k)thefield Ξkλ(ωk, r, t) does not vanish and reduces to the electromagneticground field Ξ0λ filling the whole space and having electric poten-tial units Volt m−1 s2. Consequently, any charge q submitted to anacceleration γ in the field Ξ0λ experiences an electric potential pro-portional to the acceleration, given in (6.2.10). Thus, the amplitudeof the interaction energy in this case is

Evacuum = |qξγ| (6.3.16)

As a result, a part of this interaction energy with the vacuum isradiated in space by Bremsstrahlung effect. The energy density dueto the interaction of N accelerated charges in a volume V with thevacuum field Ξ0λ writes

Wvacuum =1V

N∑i

|qiξγi| (6.3.17)

In the case of solar conditions e.g., in which the charge densityis roughly ∼1030q/m3 (q ∼ 10−19C), considering that the meanacceleration of the charges vary from 104 m/s2 up to relativisticvalues 107 m/s2 we obtain from (6.3.17) energy densities in therange Wvacuum ∝ 10−10 − 10−7 J/m3 which compare well with theastrophysical observations. Obviously, the total energy density issuedfrom the vacuum interactions in all the cosmic bodies withinin thegalaxies, in the intergalactic gaz and dusts as well as in the inter-stellar medium should participate to the cosmic background energydensity. Consequently, the interaction energy of charged particlesin the universe with the electromagnetic field ground state Ξ0λ,as well as the low frequency electromagnetic waves that may beissued from the fluctuations of the last field may have a significant



contribution to the cosmological constant and is worthy to be inves-tigated experimentally.


In this chapter we have advanced theoretical elaborations in order toensure a coherent link between the electromagnetic wave theory andquantum electrodynamics with the purpose to establish a non-localwave-particle description of the photon with a real wave function asrequired by the experiments. We have started from the fact that thevector potential amplitude of a single photon state is not physicallydefined in (6.1.1) since it depends on an arbitrary external volumeparameter V . Hence, the essential concept of the elaborations is basedon the fact that the k-mode electromagnetic field vector potentialamplitude α0k is proportional to the angular frequency ωk. This isnot a hyphothesis but a direct result of the dimension analysis ofthe vector potential issued from the resolution of Maxwell’s equa-tions. Consequently, since the energy of a k-mode photon consideredas an integral and indivisible entity corresponds to a definite valueωk, then the vector potential amplitude has to be quantized at asingle photon level α0k = ξωk, where ξ is the quantization constant.The value of ξ has been evaluated in a first approximation to be|ξ| ≈ 1.747 10−25 Volt m−1s2.

This operation complements the wave-particle relations of thephoton by introducing the vector potential amplitude in the equa-tions (6.1.4) and leads to a linear time-dependent equation (6.1.13)for the vector potential amplitude operator α0 = −iξc∇ with theeigenvalue ξωk, which is analogue to the Schrodinger’s equation forthe energy with the relativistic Hamiltonian for a massless particleH = −ic∇ and the eigenvalue ωk. The coupling of (6.1.13) withthe Schrodinger’s equation yields the vector potential–energy (wave-particle) equation for the photon expressed in (6.1.14). In additionto the wave propagation equation (6.1.6), the vector potential func-tion αkλ(r, t) (6.1.5) with the quantized amplitude ξωκ satisfies thecoupled equation (6.1.14) and consequently behaves as a real wavefunction for the photon that can be suitably normalized.



On the other hand attempting to understand the real physicalnature of single photons, the equivalence between the classical andthe quantum mechanical energy density (4.3.13) results inevitablyto a spatial extension for a k-mode photon according to the relationVk ∝ ω−3

k ∝ λ3k. The particle properties of a single photon, energy,

momentum and spin can be expressed through the quantization vol-ume Vk which is revealed to be proportional to the cubic power of thewavelength Vk = /2ε0ξ2ω3

k. This entails that the photon cannot be a“point” but is rather a “wave-corpuscle” with a minimum quantizedvolume extended over a period of the electromagnetic field. Hence,the probability of detecting a photon has to be considered within avolume λ3

k around a given coordinate r along the propagation axisand is proportional to the square modulus of the vector potentialamplitude which is proportional to ξ2ω2

k. It comes out that the elec-tric and magnetic fields of individual photons are intrinsic physicalproperties and they are directly proportional to ξω2

k.Next we have analyzed the origin of the vacuum energy singularity

in QED which is responsible for the quantum vacuum catastrophe.As quoted previously by many authors the electromagnetic field zero-point energy results from the mathematical ambiguity regarding thecommutation relation of canonical variables of position and momen-tum and that of the corresponding quantum mechanics operators.Disregarding this ambiguity the harmonic oscillator Hamiltoniancan only be obtained by considering the mean value of the nor-mal ordering and antinormal ordering Hamiltonians which automati-cally casts aside the fundamental Heisenberg’s commutation relation[Qkλ, Pk′λ′ ] = iδkk′δλλ′ .

In the experimental frame no experiments have demonstrated thatindividual photons are harmonic oscillators. In the theoretical frame,the normal ordering Hamiltonian, which is not a harmonic oscilla-tor one, is mainly used in QED. Photons appear to have a morecomplicated structure and are neither point particles nor harmonicoscillators.

Within the normal ordering Hamiltonian description, the vacuumstate appears from the electromagnetic field ground state Ξ0λ whichcan be expressed as a vector function Ξ0λ = ξελe

iϕ+ξ∗ε∗λe−iϕ as well



as a quantum operator Ξ0λ = ξakλελeiϕ + ξ∗a+

kλε∗λe

−iϕ involving thephoton creation and annihilation operators. It is worth noting thatthe circular polarization is present in both expressions of the vacuumfield entailing a direct influence on the entangled photon states.

Thus, photons appear to be “wave-corpuscles” issued from theoscillation of the electromagnetic field ground state Ξ0λ. The waveproperties result from the oscillation over a period of the quantizedvector potential with circular polarization within a minimum quan-tization volume Vk extended along a wavelength and repeated suc-cessively along the propagation axis. In this description the vectorpotential with the quantized amplitude αkλ(r, t) behaves as the pho-ton wave function that can be complemented to take into accountthe lateral extension of the vector potential and then suitably nor-malized.

We have shown that Heisenberg’s commutation relations[Qkλ, Pk′λ′ ] = iδkk′δλλ′ as well as the exact electron charge e =(4π)2αFS ξ/µ0, a physical constant, result naturally from the non-local expression of the photon energy.

Finally, the quantum vacuum field effects such as the spontaneousemission, Lamb’s shift and the cosmological vacuum energy densityhave been discussed showing the consistency of the elaborations.

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Chapter 7

Epilogue

The mathematical expressions of a valid theory should correctlydescribe the physical mechanisms involved in a complete ensembleof experimentally observed phenomena. When many theories, eachbased on different concepts, interpret only partially some measurableobservables then it becomes obvious that they cannot be epistemo-logically valid entailing that they have to be improved, if possible, orabandoned. Hence, the historical evolution over twenty five centuriesof the concepts on light’s nature followed the sequence Corpuscles,Ray Optics (without having any idea on the nature of the rays), WaveOptics (with no idea about the nature of the waves), ElectromagneticWaves Theory and finally Quantum Point Particle Theory. Despiteof all these evolutions, the general picture of the nature of light isnot satisfactory and Bohr’s Complementarity Principle was, and stillremains, the artificial mask beyond which is hidden the difficulty ofour comprehension of the simultaneous wave-particle concept. How-ever, the mutual exclusiveness of Bohr’s Complementarity Principle,according to which light has both wave and particle natures butonly one of those is expressed explicitly in a given physical situa-tion, is strongly put in doubt by recent experiments. Following theelaborations presented in the last Chapter, the next sequence on thecomprehension of the nature of light conform with the experimentalevidence might be: “wave-corpuscle”.

In this book we have tried to resume selected chapters of bothclassical electromagnetism and quantum electrodynamics in orderto help the reader forming an opinion on how the nature of lightis represented through each theory. It is of outstanding importance

131



to note that in both representations, light and vacuum are stronglyrelated entities.

Classical electrodynamics issued from Maxwell’s equationsrevealed the necessity of introducing the notion of volume for anelectromagnetic wave to stand entailing precise values of cut-off wave-lengths for each mode in order to account with the shape and dimen-sions of the surrounding space. Conversely, QED theory considerslight to be composed of point particles, disregarding the conceptualquestion on how the frequency of oscillating electric and magneticfields may be attributed to a point particle. The concept of the pho-ton being a massles point particle with energy and momentum cannotbe epistemologically valid and conflicts with experimental evidence.The energy and momentum of an electromagnetic wave are carriedby the electric and magnetic fields oscillating over a period whichfor a given mode need a minimum volume to ocupy and cannotbe conceived as a point. The theoretical basis for establishing theQED mathematical expressions for a point particle is the classicaland the quantum mechanical energy density equivalence (4.3.13) inwhich the mean value of the energy of an electromagnetic plane waveover a wavelength is attributed to a point particle. This operation haspermitted the development of a powerfull and efficient mathematicalformalism in QED for the study of the interactions of light with mat-ter but revealed unable of providing a physical and comprehensivepicture of the photon compatible with the experiments. Obviously,the incapacity of QED to advance a physical description of a singlephoton state lays in the point particle concept. Furthermore, the vec-tor potential for a single photon as well as the electric and magneticfields are not precisely defined in QED. Based on Maxwell’s equationswe have thus developed the quantized vector potential descriptionfor a single photon that led to the vector potential — energy (wave-particle) equation (6.1.14). The photon electric and magnetic fieldsare revealed to be proportional to the square of the angular frequency(6.1.24).

On the experimental front we have seen that the photoelectriceffect, traditionally presented as strong demonstration of the par-ticle nature of light, is pretty well interpreted in a semi-classical


Epilogue 133

formalism by applying classical electromagnetism using the waveconcept. A similar approach using Dirac’s equation for the electronand plane wave classical formalism for the incident light has beenapplied successfully for the interpretation of Compton’s scaterringwhich was equally considerd as a demonstration of the particle natureof light. On the other hand, Young’s diffraction experiments, initiallysupporting the wave theory, were quite well interpreted using particleformalism.

We also presented some particular single-photon experimentsrequiring for their interpretation the notion of the simultaneouswave-particle nature of light and which can be hardly interpretedeither by classical electromagnetism alone or the QED theory.Using a remarkable experimental setup Grangier et al. demonstratedthe indivisibility of photons. Various experiments carried out byRobinson, Hadlock, Hunter and Wadlinger confirmed the spatialextension of the photons over a wavelength while Mandel’s exper-iments and theoretical developments showed that a photon cannotbe detected in a volume smaller than the cube of its wavelength.

All these strongly suggest that the natural way out of the wave-particle dilemma and the QED shortcomings is to go beyond the stan-dard model and consider that a k-mode photon is not a point particlebut rather a quantum, “segment”, of the electromagnetic field withina minimum volume Vk, corresponding to a three dimensional expan-sion over a wavelength, composed of a quantized vector potentialwith definite amplitude proportional to the angular frequency, rotat-ing (with R or L circular polarization) over a period and giving riseto local oscillating electric and magnetic fields. Hence, for a physicaland comprehensive picture it should be more appropriate to considerthe photon as an indivisible “wave-corspuscle” composing the elec-tromagnetic field and propagating along the vector potential wavefunction. By this way, the QED formalism should evolve towards a3-dimensional representation i.e., creation and annihilation operatorsof individual 3-dimensional photons a+

Vkλ and aVkλ respectively.In Chapter 5 we have analyzed the QED main theoretical diffi-

culties and singularities related to the zero-point energy, that is theelectromagnetic quantum vacuum.



The zero-point energy of the electromagnetic field as deducedin the harmonic oscillator Hamiltonian (4.3.18) issues from themathematical ambiguity related to the drop of the permutationof the canonical variables of position and momentum during thequantization procedure casting aside Heisenberg’s commutationrelations which are of crucial importance because the final result istransformed into the quantum mechanics operator formalism. In fact,the harmonic oscillator Hamiltonian for the electromagnetic field canonly be obtained by considering the mean value of the normal order-ing and antinormal ordering Hamiltonians, although the necessityof this operation is not supported by any experimental evidence.Furthermore, it yields an infinite energy quantum vacuum, an unob-served experimentally singularity which leads to serious cosmologicalproblems, among which the very challenging quantum vacuum catas-trophe related to the vacuum energy density. In fact, the contributionof the electromagnetic field vacuum energy to the cosmological con-stant in QED is in contradiction with the astronomical observationsby rougly 120 orders of magnitude. On the other hand, the vacuumeffects like the spontaneous emission and the Lamb shift are inter-preted in QED based on the commutation properties of the creationand annihilation operators, a+

kλ and akλ respectively, without invok-ing the zero-point energy term

∑k,λ

12ωk in the calculations. In fact,∑

k,λ12ωk is a constant and in the QED formalism commutes with all

Hermitian operators corresponding to physical observables. Finally,we have seen that the Casimir effect has been quite well interpretedby many authors using aspects of electrodynamics and without invok-ing at all the vacuum photons. All these definetly indicate that thezero-point energy of the harmonic oscillator Hamiltonian of the elec-tromagnetic field does not correspond to a real physical state.

We have then analysed what could be the vacuum state whenconsidering naturally the normal ordering Hamiltonian, which is inreality the only one used in QED. In the extension of the calcula-tions developed in Chapter 6, the vacuum appears naturally to havean electric potential nature, composed of a universal electromagneticfield ground state Ξ0λ with amplitude ξ which is also the essenceof the photon vector potential. Under these conditions, the photon


Epilogue 135

can be seen as a quantum vacuum vibration. Consequently, the vac-uum is not a sea of photons with all frequencies and polarizations,which leads to the well-known QED singularity of infinite vacuumenergy, but the electromagnetic waves (photons) are oscillations ofthe quantum vacuum sea which is composed of a real electric poten-tial field Ξ0λ.

The calculations show that the field Ξ0λ is a real entity, capableof interacting with the matter since it is expressed as a function ofphoton creation and annihilation operators a+

kλ and akλ respectively.An interaction Hamiltonian between the vacuum and the electronscan be defined in this way in order to describe the spontaneous emis-sion effect. Ξ0λ can also be used in Bethe’s calculation for the Lambshift getting identical results. In a different way, the atomic levelenergy shifts as well as the Casimir effect can also be interpreted asa result of the vacuum radiation pressure “seen” by the electrons intheir frame due to their periodic motion in the vacuum field.

Analysis of the equation (6.1.13) with the wave function (6.1.5)indicates physically that a vibration of the vacuum field at an angu-lar frequency ωκ gives rise to a k-mode photon with vector potentialamplitude ξωk oscillating over a wavelength and yielding the quan-tum energy ωk according to (6.1.26/27). Point photons (and parti-cles) do not exist and the mass-energy equivalence appears to be adirect result of the “wave-corpuscle” nature of the physical entities(photons and particles).

As shown in (6.1.39) the electron charge, a physical constant,appears naturally when considering the equivalence between theenergy of an electromagnetic field restricted to the wavelength andthe quantum expression ω. It yields that the electron charge may beexpressed through the vacuum parameters only e = (4π)2αFSξ/µ0.The electron, and perhaps all the elementary particles, might alsobe issued from the quantum vacuum field vibrations. Thus, energy,mass and charge may originate from the vacuum field.

This representation coherently links the Electromagnetic theoryto QED opening possibilities for further developments beyond thestandard theories. Also, it attributes well defined physical prop-erties to single photons and to the vacuum state and shows the



complementarity of the wave-corpuscle nature of light as requiredby the experimental evidence. The natural space interval for a pho-ton is the wavelength. Despite the fact that the point particle rep-resentation of the photon in QED is a practical model permittingthe association of a quite efficient mathematical formalism, the realnature of a single photon occupying a characteristic minimum quan-tization volume of wavelength dimensions is quite different and needsto be further explored experimentally. These concepts open perspec-tives for new experiments which are indispensable in order to fixethe value of ξ, investigate the spatial expansion of the single photonstate and finally the dependence of the photon electric field on itsangular frequency ξω2

k. The theoretical formalism in QED has also toevolve consequently in order to reflect a more realistic representationof the 3-dimensional photon nature.

Finally, with an objective point of view we can frankly confessthat at the present state of knowledge our comprehension of thephoton is still unsatisfactory and the purpose of this manuscript isto raise questions and give hints of answers in order to stimulatefurther theoretical and experimental investigations.

February 20, 2017 11:23 Light and Vacuum (Second Edition) 9in x 6in b2812-index page 137

Index

Abdalla, 123Agarwal, 89Aharonov, 61Akhiezer, 92Ampere’s, 11, 15Ampere’s equation, 14Ampere’s laws, 12, 15amplitude of the vector potential, 26,

56, 66, 100–102, 108, 114, 115, 126,133, 135

ancient Greeks, 3, 4angular momentum, 69Angstrom, 81annihilation and creation operators,

47, 48, 56, 69, 84, 105, 119, 127,133, 134

anti-coincidence, 90anti-normal ordering, 79, 114–116,

126, 134Arab mathematicians, 4Archimedes, 3astronomical observations, 81, 122,

134astrophysical, 7atomic energy levels, 7, 70, 72, 74, 120atomic hydrogen, 70, 122atomic level energy shifts, 71, 73, 122atomic levels energy, 74, 122

background vacuum energy, 122beam splitter, 91Beck, 7, 65, 74Berestetskii, 92Bessel function, 32

Bethe, 70, 73Bethe’s approach, 70, 85, 119, 120Bethe’s renormalization, 72black body, 5blackbody radiation, 39, 50Blokland, 86Bohm, 61Bohr, 6, 7Bohr radius, 122Bohr’s complementarity principle, 7,

89, 90, 131Bohr’s angular frequency, 120Boltzmann’s constant, 39boundary conditions, 28–33, 44, 55Boyle, 4Byzantine emperors, 4Bremsstrahlung, 124

Ca atoms, 90cage, 92canonical variables of position and

momentum, 47, 55, 57, 113Carroll, 123Cartesian coordinates, 30, 42Casimir, 85, 122Casimir effect, 85, 87, 122Casimir’s force, 86, 87Casimir’s theory, 85Chambers, 61charge conservation, 12charge density, 12, 13, 25charged particles, 63Cheops pyramid, 3circular aperture, 92

137



circular polarizations, 59, 69, 107,109, 127

circular waveguide, 31, 32circularly polarized electromagnetic

waves, 27circularly polarized photons, 57circularly polarized waves, 27, 28coincidences, 91colors, 4commutation operator, 42, 84commutation relation, 48, 57, 114,

115, 126, 127complementarity principle, 7, 89, 90,

131Compton, 7, 51, 108Compton scattering, 51, 53, 54, 88,

132continuity equation, 12, 13continuum spectrum, 73corpuscles, 3, 110, 126, 127, 133, 135cosmological constant, 134cosmology, 123Coulomb, 11, 12, 14Coulomb Gauges, 24, 25, 54, 60, 81Coulomb interaction, 78Coulomb potential, 60Coulomb’s law, 13creation and annihilation operators,

47, 48, 56, 69, 73, 93, 105, 116, 127,133, 134, 135

creation operator, 105current and charge densities, 24current densities, 11, 13, 14, 23, 34, 90cut-off, 53cut-off angular frequency, 30, 32cut-off wave vector, 21, 22, 30cut-off wavelength, 30–32, 36, 122

dark energy, 123Davisson, 41De Broglie, 7density of modes, 101density of states, 34, 35, 67, 100, 104density probability distribution,

121Descartes, 4

detection operator, 59

diffraction, 5

diffraction patterns, 4

dipole approximation, 67–69

dipole moment, 69, 83

Dirac, 7, 48, 49, 70, 74, 88, 116, 119

Dirac delta function, 63, 66

Dirac equation, 48–50, 88, 132

Dirac Hamiltonian, 49

Dirac’s representation, 47, 48

Dirac’s theory, 70

directional character of photons, 108

discrete and continuous atomic levels,63, 72, 85

dispersion relation, 14, 16, 17, 35,101, 109

effective phantom phase, 123

Ehrenberg–Siday effect, 61, 74

eigenfunction, 45, 46, 48, 63

Einstein, 6, 51, 88

electric and magnetic fields, 5, 11, 12,19–21, 23–25, 28, 29, 36, 61

electric and magnetic fields of a singlephoton, 82, 100, 109, 110, 126, 132,133

electric charges, 13, 112

electric dipole moment, 69, 83

electric displacement, 12, 13

electric field, 11, 14, 15, 18, 19, 25,26, 29–31, 33, 78, 81–85, 105, 106,118, 120

electric field intensity, 11, 13, 14

electric permittivity, 11–13, 15, 16,20, 36, 77–79, 107, 109, 117

electric polarization, 14

electromagnetic field, 7, 13, 16, 26,35, 40, 41, 49, 50, 54, 55, 57, 59, 60,63, 66, 67, 69, 73, 77, 79, 80, 83, 87,94, 99, 100, 101, 103, 107, 111, 113,116, 117, 121, 123, 125, 126

electromagnetic field energy, 54, 59,77, 79, 80, 82, 113–115

electromagnetic field Hamiltonian,116


Index 139

electromagnetic field polarization, 25,27, 28

electromagnetic field quantization,54, 80, 99

electromagnetic field quantizationprocedure, 54

electromagnetic properties, 77electromagnetic wave energy density,

26, 55, 59, 73, 80, 103, 106, 107, 111electromagnetic wave propagation,

14, 18, 19, 23–25, 102, 125electromagnetic waves theory, 3, 11,

71, 111electron, 41, 50–52, 60–62, 66, 69–72,

78, 79, 82, 83, 87, 88, 94, 111, 112,116, 120, 121, 127, 132, 135

electron charge, 87, 111, 112, 127electron–vacuum interaction, 82, 84,

116, 118electronic orbitals, 85, 120, 121Elizalde, 123energy and momentum conservation

laws, 51energy and momentum operators, 41energy and momentum, 42energy conservation, 52energy density of the electromagnetic

wave, 26, 55, 59, 73, 80, 103, 106,107, 111

energy flux of electromagnetic waves,19, 36, 107

energy quanta, 6energy shifts, 57, 70, 71, 85, 106, 120,

122, 135energy spectrum, 46, 47energy–mass equivalence, 101Erwin Schrodinger, 41Euclid, 3expectation value of the electric field

in vacuum, 81, 118expectation value of the photons

electric field, 81Ezawa, 87

Fanaoni, 123Faraday, 11, 12, 123

Faraday’s cage, 92Faraday’s laws, 11, 15Fermat’s principle, 4Feynman, 85fine structure constant, 71, 78, 79, 112Fresnel, 4, 5Frieman, 123Fulcher, 78fullerenes, 88fundamental fields, 12

Gassenti, 4Gauge Invariance, 23gauge theories, 61Gauss’s laws, 13Germer, 41Ghose, 89Grangier, 90, 94, 133Greenberger, 90Grimaldi, 4ground level electromagnetic field,

117guided electromagnetic waves, 44guided propagation, 28, 34, 36, 37,

54, 86

Hadlock, 92, 94, 133Hamiltonian, 48, 49, 55, 60, 61, 63,

69, 71, 73, 118, 123, 134, 135Hamiltonian of the quantized

electromagnetic field, 55, 57, 79,80, 82, 84

Hamiltonian operator for a masslessparticle, 102

harmonic oscillators, 7, 48, 57, 73, 126harmonic oscillator Hamiltonian, 47,

55, 113–116, 126harmonic oscillator in quantum

mechanics, 44Heisenberg, 7, 103, 104Heisenberg’s commutation relation,

110, 114, 116, 127Heisenberg’s equation of motion, 68,

83Heisenberg’s uncertainty, 103, 104helicity, 69



Helmholtz equation, 18, 22Hermite polynomials, 46Hermitian operators, 49, 50, 93, 113,

116Heron of Alexandria, 3Hertz, 5, 49historical survey, 3Hirocho Ezawa, 87Home, 89Hook, 4Hooke’s law, 44Hunter and Wadlinger, 94, 133Hunter–Wadlinger experiments, 92Hypatia of Alexandria, 4Huygens, 4hydrogen nS levels, 121, 122hydrogen atom, 60, 70, 73, 121hydrogen electronic orbitals, 121hydrogenic atom, 60

impendence, 17, 36indivisibility of the photons, 90, 91,

94, 133Intensity, 20interaction Hamiltonian, 60, 61, 63,

69, 70, 71, 73, 74, 83, 84, 118, 119interference experiments, 53, 89, 94interference patterns, 53, 62, 88interference phenomena, 6interferences, 5, 62, 88interferometer, 90irradiance, 20

James Clerk Maxwell, 5, 13Jin, 89Jordan, 7

k mode photon, 81, 100–103, 105,107–110, 125, 126, 133, 135

Klein-Gordon equation, 49Kroll, 74Kronecker delta, 63Kronecker symbol, 42

Labun, 123Lamb, 65, 70, 74

Lamb shifts, 57, 70, 72, 74, 84, 119Lambrecht, 86Laplace equation, 21, 22lasers, 8, 53Lenard, 6, 51Lewis, 7light corpuscles, 4light intensity, 6light quanta, 6, 7light ray, 6light’s properties, 4linear harmonic oscillator, 45linear polarization, 28linear term of the vector potential,

71, 84, 85link equations, 56, 73, 79, 100, 106,

113, 115link expressions, 56link relations, 100logarithmic divergence, 72, 85Lorentz Gauge, 23Lorentz’s, 6Lorentz’s force, 87Louis de Broglie, 41

magnetic field, 5, 11, 12, 19–21,23–25, 28, 29, 36, 61

magnetic field flux intensities, 12, 13,15, 18

magnetic field induction, 18magnetic field intensity, 11magnetic induction, 12, 14, 26magnetic monopoles, 13magnetic permeability, 11–13, 15, 16,

20, 36, 109, 117magnetization, 14Mandel, 59, 60mass–energy equivalence, 101, 135matrix element, 64, 65, 68matrix-mechanics, 7Max Planck, 6, 40, 41Maxwell’s equations, 11, 13–15, 21,

24, 28, 29, 35, 36, 42, 56, 58, 88, 99,101, 107, 117, 125, 132

Maxwell’s theory, 35Maxwell’s wave theory, 5, 7


Index 141

Mendel, 88

Michelson, 6

microwave antenna, 84

microwaves irregularities, 124

Millikan, 49

Milonni, 87

Mizobuchi, 89, 90, 94Mizobuchi and Ohtake double-prism

experiment, 89, 90

momentum, 6, 52, 53

momentum and position operators,47, 56, 110, 113

momentum conservation, 50

momentum operators, 42, 45, 52, 56,103, 119

monochromatic, 19, 20, 24, 28, 55,106

moonlight, 4

Morley, 6

mutual exclusiveness, 89, 90, 131

mutually exclusive, 7

Nairz, 88

Nakamura, 87

neutrons, 90

Newton, 4, 51

Niels Bohr, 41

Nojiri, 123

nonrelativistic approach, 85normal ordering, 79, 94, 114, 115,

120, 123, 126, 134

number of modes, 34, 39

number of photons in a volume V , 35number of states, 35, 36

number operator, 48, 56, 59

Ohtake, 89, 90, 94

Optica, 3, 4Opticks, 4

optics, 3, 4

Osakabe, 61

oscillating electric and magneticfields, 15, 20, 132, 133

oscillators, 6, 40, 49

Overbeek, 86

parametric down converters, 89, 100particle nature of the light, 7, 54, 74,

88–90, 94,particle properties, 99, 101, 109, 126particle theories, 6, 89Pauli, 7Pauli matrices, 50periodic boundary conditions, 34periodic relief, 4phantom cosmology, 123phantom description, 123phase parameter, 25Phos, 2, 7photoelectric effect, 6, 7, 51, 54, 65,

88photoelectric effect and the

semi-classical interpretation, 65, 66photoelectric equation, 51photon, 7photon concept, 6, 88photon electric field, 105photon indivisibility, 90, 91, 94, 133photon momentum, 101photon number operator, 56, 59photon spatial expansion, 83, 101photon spin, 69, 110photon vector potential, 99–104, 111photon volume, 92, 101, 104, 106,

108, 110, 111photon wave function, 57photon wavelength dimensions, 92, 93Pierre Gassendi, 4Planck, 6, 49, 93Planck’s constant, 40, 51, 77, 93, 109,

111Planck’s energy, 122Planck’s hypothesis, 41Planck’s ideas, 40, 49Planck’s radiation law, 41plane wave solution, 19plane waves, 23, 34, 54, 105Poincare’s, 6point particles, 51, 57, 73, 82, 88, 91,

94, 104, 106, 109, 111, 126, 131–133point photon, 92Poisson’s equation, 25, 33



polarization, 5, 26, 28, 35, 54, 56, 57,59, 66–71, 80, 83, 102, 105, 107,109, 117, 120, 127, 133, 134

polarization vector, 66–71, 80polarized plane wave, 27position Qkλ and momentum Pkλ

operators, 110position and momentum operators,

47, 56, 110, 113positron, 50, 78postulate of quantum mechanics, 42potential energy, 42, 45power density, 20, 26Poynting vector, 19, 20, 26, 93, 107principal quantum number, 122Prometheus, 2propagation equations, 18, 19, 24, 25,

32, 102, 103, 125propagation factor, 21, 29, 32propagation of the electromagnetic

field, 15, 18Ptolemy of Alexandria, 3Pyr, 2Pyramide, 3

QED singularities, 57, 70, 77, 80, 93,118

quadratic term of the vectorpotential, 70, 71, 84, 85

quantization of the electromagneticfield, 44

quantization procedure, 44, 54, 57,73, 77, 79, 80, 89, 94, 100, 104, 123

quantization volume, 102, 104, 106,108–110, 112, 126

quantized electromagnetic field, 54quantized radiation electric field, 81quantized vector potential, 84, 99,

109, 110, 127, 133quantum chromodynamics (QCD),

123quantum electrodynamics (QED), 7,

8, 31, 49, 50, 54, 56, 57, 66, 70, 73,77, 78, 80–84, 86, 93, 94, 100, 104,106, 113, 116, 118, 122, 123, 126,132–134

quantum particle theory, 3

quantum theory, 7

quantum theory of radiation, 77

quantum vacuum, 57, 67, 70, 71, 81,84, 94, 99, 112, 113, 116–119, 123,127, 134

quantum vacuum catastrophe, 8, 81,126, 134

quantum vacuum field, 113, 127

quantum vacuum state, 70, 71, 81, 84,94, 99, 112, 113, 116–119, 123, 127

R.C. Retherford, 70

R.E. Siday, 61

radial wave functions, 121

radiation, 5–7, 26, 39

radiation Hamiltonians, 57, 79, 83,113, 114, 116

radiation polarization, 27

radiation pressure, 86, 120–122, 135

radiation vector potential, 66

ray optics, 3, 131

Rayleigh, 5

Rayleigh-Jeans, 40

reality of the vector potential, 60, 61

rectangular aperture, 92

rectangular and circular waveguides,28

rectangular waveguide, 29–31

reduced propagation equation, 30

reflection, 3, 4

refraction, 4

refractive index, 16, 17

relativistic approach, 74, 85

relativistic calculation, 74

relativistic Hamiltonian, 49, 102, 125

renormalisation, 72, 84, 85

resonators, 6

Reynaud, 86

Riemann, 58

right-hand circularly polarizedphoton, 57, 69

Robinson, 92, 94, 133

Roman battle ships, 3


Index 143

scalar and vector potentials, 23scalar potential, 25, 33, 60, 61, 63Schrodinger, 7Schrodinger equation, 42, 43, 45, 47,

48, 61, 62, 103, 125Schwinger, 87Scully, 65, 74second order perturbation theory, 71second quantization, 54, 58, 73, 77, 89Silvestri, 123single k-mode photon, 81, 100–103,

105, 107–110, 125, 126, 133, 135single mode volume, 36single photon pulse techniques, 53single photon state, 90, 99–101, 104,

116, 125, 132, 135singularities, 57, 70, 77, 80, 93, 118Snell, 4space inversion, 13Sparnaay, 86spatial extension of the photon,

92–94, 101, 108, 110, 123, 126, 133special relativity, 6spectral distribution, 39speed of light, 5, 6speed of the electromagnetic waves in

vacuum, 17, 24spin matrices, 59spontaneous emission, 57, 63, 66, 67,

118spontaneous emission rate, 66, 68, 119square potential well, 43Stark shift, 85, 106Stefan, 5stimulated emission, 65Stoletov, 51sunlight, 3, 4Syracuse, 3

TEm,n modes, 29, 31TMm,n components, 30TMm,n modes, 32Taylor, 53Taylor’s experiment, 54, 88TE and TM waves, 33TE field, 33

TE waves, 22TEM fields, 21, 22, 23, 34TEM mode, 28Thales of Miletus, 3thermal equilibrium, 39, 41thermal radiation, 5threshold frequency, 6time delay, 24, 25time dependent perturbation theory,

63, 119time derivative operator, 42time reversal, 13transition probability, 64, 119transition rates, 63, 65, 67transmitted power, 92, 93transverse components, 22Transverse Electric (TE), 19Transverse Electro-Magnetic (TEM),

19, 22, 33, 34Transverse Magnetic (TM), 19, 22Transverse Monochromatic

Electromagnetic (TMEM), 20, 26transverse wave, 21Trigonometric function, 64tunneling effect, 89, 90two prisms, 89, 90

ultraviolet catastrophe, 6, 39, 40uncertainty principle, 103, 104, 111universe, 36, 122–124unperturbed eigenfunctions ϕl, 63unperturbed Hamiltonian, 68UV catastrophe, 6, 8, 122

vacuum, 5, 35, 36, 57, 63, 66, 67, 70,71

vacuum catastrophe, 81, 122vacuum electric nature, 117vacuum electric permittivity, 11–16,

20, 36, 77–79, 107, 109, 117vacuum electromagnetic energy

density, 7, 80, 81, 122–124, 127vacuum field, 116–118vacuum fluctuations, 82vacuum Hamiltonian, 82, 84vacuum impendence, 17



vacuum intrinsic properties, 20, 71vacuum permittivity and

permeability, 11–16, 20, 36, 77–79,107, 109, 117

vacuum photons, 86, 87, 120, 134vacuum properties, 36vacuum radiation pressure, 120–122vacuum’s electromagnetic properties,

17Van der Mark, 111Van der Waals, 87vector potential, 24–26, 54–56, 60–74,

77–85, 94, 99–119, 123, 125–128,132, 133

vector potential amplitude, 26, 56,66, 100–102, 108, 114, 115, 126,133, 135

vector potential amplitude operator,105

vector potential amplitudequantization constant, 101, 103,109, 111, 112, 125

vector potential energy equation, 103vector potential of a single photon,

100, 101vector potential of the

electromagnetic field, 36velocity of light in vacuum, 17, 20,

109,virtual photons, 82visible spectrum, 81

W. Ehrenberg, 61W.E. Lamb, 70Wadlinger, 92, 94Watanabe, 87wave corpuscle, 110, 126, 127, 133,

135wave function, 34, 44, 103, 104, 121

wave nature of the light, 5, 65wave optics, 3, 131wave theory of light, 4, 51, 65wave vector, 16, 17, 19, 26, 34, 52wave-mechanics, 7wave-particle, 77, 88–90, 99–106, 110,

111, 125, 131, 133wave-particle formalism, 104–106wave-particle nature of light, 77,

88–90, 99–106, 110, 111, 125, 131,133

wave propagation equation, 102waveguide, 28wavelengths, 5, 19, 32, 35, 36, 39, 50,

78, 81, 82, 89, 92, 94, 100, 103, 104,106, 108, 110, 112, 124, 126, 127,132, 133

Webers, 12Welton, 85Wentzel, 7, 60, 65, 74, 80Wien, 5Williamson, 111

x-band microwaves, 92X-ray scattering, 51, 52X-rays, 51

Yasin, 90Young, 4, 5, 53Young’s double-slit experiment, 88, 89Young’s double-slit interferences, 53,

94Young’s experiments, 4, 88

zero-point radiation field, 57zero-point energy, 80, 86, 87, 93, 94,

113, 116, 123, 133, 134

Documents

Light and Vacuum: The Wave Particle Nature of the Light and the Quantum Vacuum. Electromagnetic Theory and Quantum Electrodynamics Beyond the Standard Model