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Jean-Pierre GazeauCoherent States in QuantumPhysics

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Jean-Pierre Gazeau

Coherent States in Quantum Physics

WILEY-VCH Verlag GmbH & Co. KGaA

The Author

Prof. Jean-Pierre GazeauAstroparticules et CosmologieUniversité Paris DiderotParis, [email protected]

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V

Contents

Preface XIII

Part One Coherent States 1

1 Introduction 31.1 The Motivations 3

2 The Standard Coherent States: the Basics 132.1 Schrödinger Definition 132.2 Four Representations of Quantum States 132.2.1 Position Representation 142.2.2 Momentum Representation 142.2.3 Number or Fock Representation 152.2.4 A Little (Lie) Algebraic Observation 162.2.5 Analytical or Fock–Bargmann Representation 162.2.6 Operators in Fock–Bargmann Representation 172.3 Schrödinger Coherent States 182.3.1 Bergman Kernel as a Coherent State 182.3.2 A First Fundamental Property 192.3.3 Schrödinger Coherent States in the Two Other Representations 192.4 Glauber–Klauder–Sudarshan or Standard Coherent States 202.5 Why the Adjective Coherent? 20

3 The Standard Coherent States: the (Elementary) Mathematics 253.1 Introduction 253.2 Properties in the Hilbertian Framework 263.2.1 A “Continuity” from the Classical Complex Plane to Quantum States 263.2.2 “Coherent” Resolution of the Unity 263.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane

(as a Euclidean Space) 273.2.4 Analytical Bridge 283.2.5 Overcompleteness and Reproducing Properties 293.3 Coherent States in the Quantum Mechanical Context 303.3.1 Symbols 303.3.2 Lower Symbols 30

Coherent States in Quantum Physics. Jean-Pierre GazeauCopyright © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40709-5

VI Contents

3.3.3 Heisenberg Inequalities 313.3.4 Time Evolution and Phase Space 323.4 Properties in the Group-Theoretical Context 353.4.1 The Vacuum as a Transported Probe . . . 353.4.2 Under the Action of . . . 363.4.3 . . . the D -Function 373.4.4 Symplectic Phase and the Weyl–Heisenberg Group 373.4.5 Coherent States as Tools in Signal Analysis 383.5 Quantum Distributions and Coherent States 403.5.1 The Density Matrix and the Representation “R” 413.5.2 The Density Matrix and the Representation “Q” 413.5.3 The Density Matrix and the Representation “P” 423.5.4 The Density Matrix and the Wigner(–Weyl–Ville) Distribution 433.6 The Feynman Path Integral and Coherent States 44

4 Coherent States in Quantum Information: an Example of ExperimentalManipulation 49

4.1 Quantum States for Information 494.2 Optical Coherent States in Quantum Information 504.3 Binary Coherent State Communication 514.3.1 Binary Logic with Two Coherent States 514.3.2 Uncertainties on POVMs 514.3.3 The Quantum Error Probability or Helstrom Bound 524.3.4 The Helstrom Bound in Binary Communication 534.3.5 Helstrom Bound for Coherent States 534.3.6 Helstrom Bound with Imperfect Detection 544.4 The Kennedy Receiver 544.4.1 The Principle 544.4.2 Kennedy Receiver Error 554.5 The Sasaki–Hirota Receiver 564.5.1 The Principle 564.5.2 Sasaki–Hirota Receiver Error 564.6 The Dolinar Receiver 574.6.1 The Principle 574.6.2 Photon Counting Distributions 584.6.3 Decision Criterion of the Dolinar Receiver 584.6.4 Optimal Control 594.6.5 Dolinar Hypothesis Testing Procedure 604.7 The Cook–Martin–Geremia Closed-Loop Experiment 614.7.1 A Theoretical Preliminary 614.7.2 Closed-Loop Experiment: the Apparatus 634.7.3 Closed-Loop Experiment: the Results 654.8 Conclusion 67

5 Coherent States: a General Construction 695.1 Introduction 69

Contents VII

5.2 A Bayesian Probabilistic Duality in Standard Coherent States 705.2.1 Poisson and Gamma Distributions 705.2.2 Bayesian Duality 715.2.3 The Fock–Bargmann Option 715.2.4 A Scheme of Construction 725.3 General Setting: “Quantum” Processing of a Measure Space 725.4 Coherent States for the Motion of a Particle on the Circle 765.5 More Coherent States for the Motion of a Particle on the Circle 78

6 The Spin Coherent States 796.1 Introduction 796.2 Preliminary Material 796.3 The Construction of Spin Coherent States 806.4 The Binomial Probabilistic Content of Spin Coherent States 826.5 Spin Coherent States: Group-Theoretical Context 826.6 Spin Coherent States: Fock–Bargmann Aspects 866.7 Spin Coherent States: Spherical Harmonics Aspects 866.8 Other Spin Coherent States from Spin Spherical Harmonics 876.8.1 Matrix Elements of the SU (2) Unitary Irreducible Representations 876.8.2 Orthogonality Relations 896.8.3 Spin Spherical Harmonics 896.8.4 Spin Spherical Harmonics as an Orthonormal Basis 916.8.5 The Important Case: σ = j 916.8.6 Transformation Laws 926.8.7 Infinitesimal Transformation Laws 926.8.8 “Sigma-Spin” Coherent States 936.8.9 Covariance Properties of Sigma-Spin Coherent States 95

7 Selected Pieces of Applications of Standard and Spin Coherent States 977.1 Introduction 977.2 Coherent States and the Driven Oscillator 987.3 An Application of Standard or Spin Coherent States in Statistical Physics:

Superradiance 1037.3.1 The Dicke Model 1037.3.2 The Partition Function 1057.3.3 The Critical Temperature 1067.3.4 Average Number of Photons per Atom 1087.3.5 Comments 1097.4 Application of Spin Coherent States to Quantum Magnetism 1097.5 Application of Spin Coherent States to Classical and Thermodynamical

Limits 1117.5.1 Symbols and Traces 1127.5.2 Berezin–Lieb Inequalities for the Partition Function 1147.5.3 Application to the Heisenberg Model 116

VIII Contents

8 SU(1,1) or SL(2, R) Coherent States 1178.1 Introduction 1178.2 The Unit Disk as an Observation Set 1178.3 Coherent States 1198.4 Probabilistic Interpretation 1208.5 Poincaré Half-Plane for Time-Scale Analysis 1218.6 Symmetries of the Disk and the Half-Plane 1228.7 Group-Theoretical Content of the Coherent States 1238.7.1 Cartan Factorization 1238.7.2 Discrete Series of SU (1, 1) 1248.7.3 Lie Algebra Aspects 1268.7.4 Coherent States as a Transported Vacuum 1278.8 A Few Words on Continuous Wavelet Analysis 129

9 Another Family of SU(1,1) Coherent States for Quantum Systems 1359.1 Introduction 1359.2 Classical Motion in the Infinite-Well and Pöschl–Teller Potentials 1359.2.1 Motion in the Infinite Well 1369.2.2 Pöschl–Teller Potentials 1389.3 Quantum Motion in the Infinite-Well and Pöschl–Teller Potentials 1419.3.1 In the Infinite Well 1419.3.2 In Pöschl–Teller Potentials 1429.4 The Dynamical Algebra su(1, 1) 1439.5 Sequences of Numbers and Coherent States on the Complex Plane 1469.6 Coherent States for Infinite-Well and Pöschl–Teller Potentials 1509.6.1 For the Infinite Well 1509.6.2 For the Pöschl–Teller Potentials 1529.7 Physical Aspects of the Coherent States 1539.7.1 Quantum Revivals 1539.7.2 Mandel Statistical Characterization 1559.7.3 Temporal Evolution of Symbols 1589.7.4 Discussion 162

10 Squeezed States and Their SU(1, 1) Content 16510.1 Introduction 16510.2 Squeezed States in Quantum Optics 16610.2.1 The Construction within a Physical Context 16610.2.2 Algebraic (su(1, 1)) Content of Squeezed States 17110.2.3 Using Squeezed States in Molecular Dynamics 175

11 Fermionic Coherent States 17911.1 Introduction 17911.2 Coherent States for One Fermionic Mode 17911.3 Coherent States for Systems of Identical Fermions 18011.3.1 Fermionic Symmetry SU (r) 18011.3.2 Fermionic Symmetry SO (2r) 185

Contents IX

11.3.3 Fermionic Symmetry SO (2r + 1) 18711.3.4 Graphic Summary 18811.4 Application to the Hartree–Fock–Bogoliubov Theory 189

Part Two Coherent State Quantization 191

12 Standard Coherent State Quantization: the Klauder–Berezin Approach 19312.1 Introduction 19312.2 The Berezin–Klauder Quantization of the Motion of a Particle

on the Line 19312.3 Canonical Quantization Rules 19612.3.1 Van Hove Canonical Quantization Rules [161] 19612.4 More Upper and Lower Symbols: the Angle Operator 19712.5 Quantization of Distributions: Dirac and Others 19912.6 Finite-Dimensional Canonical Case 202

13 Coherent State or Frame Quantization 20713.1 Introduction 20713.2 Some Ideas on Quantization 20713.3 One more Coherent State Construction 20913.4 Coherent State Quantization 21113.5 A Quantization of the Circle by 2 ~ 2 Real Matrices 21413.5.1 Quantization and Symbol Calculus 21413.5.2 Probabilistic Aspects 21613.6 Quantization with k-Fermionic Coherent States 21813.7 Final Comments 220

14 Coherent State Quantization of Finite Set, Unit Interval, and Circle 22314.1 Introduction 22314.2 Coherent State Quantization of a Finite Set with Complex 2 ~ 2

Matrices 22314.3 Coherent State Quantization of the Unit Interval 22714.3.1 Quantization with Finite Subfamilies of Haar Wavelets 22714.3.2 A Two-Dimensional Noncommutative Quantization of the Unit

Interval 22814.4 Coherent State Quantization of the Unit Circle and the Quantum Phase

Operator 22914.4.1 A Retrospective of Various Approaches 22914.4.2 Pegg–Barnett Phase Operator and Coherent State Quantization 23414.4.3 A Phase Operator from Two Finite-Dimensional Vector Spaces 23514.4.4 A Phase Operator from the Interplay Between Finite and Infinite

Dimensions 237

15 Coherent State Quantization of Motions on the Circle, in an Interval, andOthers 241

15.1 Introduction 24115.2 Motion on the Circle 241

X Contents

15.2.1 The Cylinder as an Observation Set 24115.2.2 Quantization of Classical Observables 24215.2.3 Did You Say Canonical? 24315.3 From the Motion of the Circle to the Motion on 1 + 1 de Sitter Space-

Time 24415.4 Coherent State Quantization of the Motion in an Infinite-Well

Potential 24515.4.1 Introduction 24515.4.2 The Standard Quantum Context 24615.4.3 Two-Component Coherent States 24715.4.4 Quantization of Classical Observables 24915.4.5 Quantum Behavior through Lower Symbols 25315.4.6 Discussion 25415.5 Motion on a Discrete Set of Points 256

16 Quantizations of the Motion on the Torus 25916.1 Introduction 25916.2 The Torus as a Phase Space 25916.3 Quantum States on the Torus 26116.4 Coherent States for the Torus 26516.5 Coherent States and Weyl Quantizations of the Torus 26716.5.1 Coherent States (or Anti-Wick) Quantization of the Torus 26716.5.2 Weyl Quantization of the Torus 26716.6 Quantization of Motions on the Torus 26916.6.1 Quantization of Irrational and Skew Translations 26916.6.2 Quantization of the Hyperbolic Automorphisms of the Torus 27016.6.3 Main Results 271

17 Fuzzy Geometries: Sphere and Hyperboloid 27317.1 Introduction 27317.2 Quantizations of the 2-Sphere 27317.2.1 The 2-Sphere 27417.2.2 The Hilbert Space and the Coherent States 27417.2.3 Operators 27517.2.4 Quantization of Observables 27517.2.5 Spin Coherent State Quantization of Spin Spherical Harmonics 27617.2.6 The Usual Spherical Harmonics as Classical Observables 27617.2.7 Quantization in the Simplest Case: j = 1 27617.2.8 Quantization of Functions 27717.2.9 The Spin Angular Momentum Operators 27717.3 Link with the Madore Fuzzy Sphere 27817.3.1 The Construction of the Fuzzy Sphere à la Madore 27817.3.2 Operators 28017.4 Summary 28217.5 The Fuzzy Hyperboloid 283

Contents XI

18 Conclusion and Outlook 287

Appendix A The Basic Formalism of Probability Theory 289A.1 Sigma-Algebra 289A.1.1 Examples 289A.2 Measure 290A.3 Measurable Function 290A.4 Probability Space 291A.5 Probability Axioms 291A.6 Lemmas in Probability 292A.7 Bayes’s Theorem 292A.8 Random Variable 293A.9 Probability Distribution 293A.10 Expected Value 294A.11 Conditional Probability Densities 294A.12 Bayesian Statistical Inference 295A.13 Some Important Distributions 296A.13.1 Degenerate Distribution 296A.13.2 Uniform Distribution 296

Appendix B The Basics of Lie Algebra, Lie Groups, and Their Representations 303B.1 Group Transformations and Representations 303B.2 Lie Algebras 304B.3 Lie Groups 306B.3.1 Extensions of Lie algebras and Lie groups 310

Appendix C SU(2) Material 313C.1 SU (2) Parameterization 313C.2 Matrix Elements of SU (2) Unitary Irreducible Representation 313C.3 Orthogonality Relations and 3 j Symbols 314C.4 Spin Spherical Harmonics 315C.5 Transformation Laws 317C.6 Infinitesimal Transformation Laws 318C.7 Integrals and 3 j Symbols 319C.8 Important Particular Case: j = 1 320C.9 Another Important Case: σ = j 321

Appendix D Wigner–Eckart Theorem for Coherent State Quantized SpinHarmonics 323

Appendix E Symmetrization of the Commutator 325

References 329

Index 339

XIII

Preface

This book originated from a series of advanced lectures on coherent states inphysics delivered in Strasbourg, Louvain-la Neuve, Paris, Rio de Janeiro, Rabat,and Bialystok, over the period from 1997 to 2008. In writing this book, I haveattempted to maintain a cohesive self-contained content.

Let me first give some insights into the notion of a coherent state in physics.Within the context of classical mechanics, a physical system is described by stateswhich are points of its phase space (and more generally densities). In quantummechanics, the system is described by states which are vectors (up to a phase) ina Hilbert space (and more generally by density operators).

There exist superpositions of quantum states which have many features (prop-erties or dynamical behaviors) analogous to those of their classical counterparts:they are the so-called coherent states, already studied by Schrödinger in 1926 andrediscovered by Klauder, Glauber, and Sudarshan at the beginning of the 1960s.

The phrase “coherent states” was proposed by Glauber in 1963 in the context ofquantum optics. Indeed, these states are superpositions of Fock states of the quan-tized electromagnetic field that, up to a complex factor, are not modified by theaction of photon annihilation operators. They describe a reservoir with an undeter-mined number of photons, a situation that can be viewed as formally close to theclassical description in which the concept of a photon is absent.

The purpose of these lecture notes is to explain the notion of coherent states andof their various generalizations, since Schrödinger up to some of the most recentconceptual advances and applications in different domains of physics and signalanalysis. The guideline of the book is based on a unifying method of construc-tion of coherent states, of minimal complexity. This method has a substantiallyprobabilistic content and allows one to establish a simple and natural link betweenpractically all families of coherent states proposed until now. This approach em-bodies the originality of the book in regard to well-established procedures derivedessentially from group theory (e.g., coherent state family viewed as the orbit underthe action of a group representation) or algebraic constraints (e.g., coherent statesviewed as eigenvectors of some lowering operator), and comprehensively present-ed in previous treatises, reviews, an extensive collection of important papers, andproceedings.


XIV Preface

A working knowledge of basic quantum mechanics and related mathematicalformalisms, e.g., Hilbert spaces and operators, is required to understand the con-tents of this book. Nevertheless, I have attempted to recall necessary definitionsthroughout the chapters and the appendices.

The book is divided into two parts.

• The first part introduces the reader progressively to the most familiar co-herent states, their origin, their construction (for which we adopt an origi-nal and unifying procedure), and their application/relevance to various (al-though selected) domains of physics.

• The second part, mostly based on recent original results, is devoted to thequestion of quantization of various sets (including traditional phase spacesof classical mechanics) through coherent states.

Acknowledgements My thanks go to Barbara Heller (Illinois Institute of Tech-nology, Chicago), Ligia M.C.S. Rodrigues (Centro Brasileiro de Pesquisas Fisicas,Rio de Janeiro), and Nicolas Treps (Laboratoire Kastler Brossel, Université Pierre etMarie Curie – Paris 6) for reading my manuscript and offering valuable advice, sug-gestions, and corrections. They also go to my main coworkers who contributed tovarious extents to this book – Syad Twareque Ali (Concordia University, Montreal),Jean-Pierre Antoine (Université Catholique de Louvain), Nicolae Cotfas (Universityof Bucharest), Eric Huguet (Université Paris Diderot – Paris 7), John Klauder (Uni-versity of Florida, Gainesville), Pascal Monceau (Université d’Ivry), Jihad Mourad(Université Paris Diderot – Paris 7), Karol Penson (Université Pierre et Marie Curie– Paris 6), Włodzimierz Piechocki (Sołtan Institute for Nuclear Studies, Warsaw),and Jacques Renaud (Université Paris-Est) – and my PhD or former PhD studentsLenin Arcadio García de León Rumazo, Mónica Suárez Esteban, Julien Quéva, PetrSiegl, and Ahmed Youssef.

Paris, February 2009 Jean Pierre Gazeau

Part One Coherent States

3

1Introduction

1.1The Motivations

Coherent states were first studied by Schrödinger in 1926 [1] and were rediscoveredby Klauder [2–4], Glauber [5–7], and Sudarshan [8] at the beginning of the 1960s.The term “coherent” itself originates in the terminology in use in quantum optics(e.g., coherent radiation, sources emitting coherently). Since then, coherent statesand their various generalizations have disseminated throughout quantum physicsand related mathematical methods, for example, nuclear, atomic, and condensedmatter physics, quantum field theory, quantization and dequantization problems,path integrals approaches, and, more recently, quantum information through thequestions of entanglement or quantum measurement.

The purpose of this book is to explain the notion of coherent states and of theirvarious generalizations, since Schrödinger up to the most recent conceptual ad-vances and applications in different domains of physics, with some incursions intosignal analysis. This presentation, illustrated by various selected examples, doesnot have the pretension to be exhaustive, of course. Its main feature is a unifyingmethod of construction of coherent states, of minimal complexity and of proba-bilistic nature. The procedure followed allows one to establish a simple and nat-ural link between practically all families of coherent states proposed until now. Itembodies the originality of the book in regard to well-established constructions de-rived essentially from group theory (e.g., coherent state family viewed as the orbitunder the action of a group representation) or algebraic constraints (e.g., coher-ent states viewed as eigenvectors of some lowering operator), and comprehensivelypresented in previous treatises [10, 11], reviews [9, 12–14], an extensive collectionof important papers [15], and proceedings [16].

As early as 1926, at the very beginning of quantum mechanics, Schrödinger [1]was interested in studying quantum states, which mimic their classical counter-parts through the time evolution of the position operator:

Q (t) = ei� Ht Q e– i

� Ht . (1.1)

In this relation, H = P 2/2m + V (Q ) is the quantum Hamiltonian of the system.Schrödinger understood classical behavior to mean that the average or expected


4 1 Introduction

value of the position operator,

q(t) = 〈coherent state|Q (t)|coherent state〉 ,

in the desired state, would obey the classical equation of motion:

mq(t) +∂V∂q

= 0 . (1.2)

Schrödinger was originally concerned with the harmonic oscillator, V (q) =12 m2ω2q2. The states parameterized by the complex number z = |z|eiϕ, and de-noted by |z〉, are defined in a way such that one recovers the familiar sinusoidalsolution

〈z|Q (t)|z〉 = 2Qo |z| cos(ωt – ϕ) , (1.3)

where Qo = (�/2mω)1/2 is a fundamental quantum length built from the univer-sal constant � and the constants m and ω characterizing the quantum harmonicoscillator under consideration.

In this way, states |z〉 mediate a “smooth” transition from classical to quantummechanics. But one should not be misled: coherent states are rigorously quantumstates (witness the constant � appearing in the definition of Qo), yet they allowfor a classical “reading” in a host of quantum situations. This unique qualificationresults from a set of properties satisfied by these Schrödinger–Klauder–Glaubercoherent states, also called canonical coherent states or standard coherent states.

The most important among them are the following:

(CS1) The states |z〉 saturate the Heisenberg inequality:

〈ΔQ〉z 〈ΔP 〉z = 12 � , (1.4)

where 〈ΔQ〉z := [〈z|Q2|z〉 – 〈z|Q |z〉2]1/2.(CS2) The states |z〉 are eigenvectors of the annihilation operator, with eigen-

value z:

a|z〉 = z|z〉, z ∈ C , (1.5)

where a = (2m�ω)–1/2 (mωQ + iP ).(CS3) The states |z〉 are obtained from the ground state |0〉 of the harmonic oscilla-

tor by a unitary action of the Weyl–Heisenberg group. The latter is a key Liegroup in quantum mechanics, whose Lie algebra is generated by {Q , P , I},with [Q , P ] = i�Id (which implies [a, a†] = I ):

|z〉 = e(za† – za)|0〉. (1.6)

(CS4) The coherent states {|z〉} constitute an overcomplete family of vectors inthe Hilbert space of the states of the harmonic oscillator. This property isencoded in the following resolution of the identity or unity:

Id =1π

∫C

d Re z d Im z |z〉〈z| . (1.7)

1.1 The Motivations 5

These four properties are, to various extents, the basis of the many generaliza-tions of the canonical notion of coherent states, illustrated by the family {|z〉}.Property (CS4) is in fact, both historically and conceptually, the one that survives.As far as physical applications are concerned, this property has gradually emergedas the one most fundamental for the analysis, or decomposition, of states in theHilbert space of the problem, or of operators acting on this space. Thus, property(CS4) will be a sort of motto for the present volume, like it was in the previous,more mathematically oriented, book by Ali, Antoine, and the author [11]. We shallexplain in much detail this point of view in the following pages, but we can say veryschematically, that given a measure space (X , ν) and a Hilbert space H, a family ofcoherent states {|x〉 | x ∈ X }must satisfy the operator identity∫

X|x〉〈x | ν(dx) = Id . (1.8)

Here, the integration is carried out on projectors and has to be interpreted in aweak sense, that is, in terms of expectation values in arbitrary states |ψ〉. Hence, theequation in (1.8) is understood as

〈ψ|∫

Xx〉〈x | ν(dx) |ψ〉 =

∫X|〈x |ψ〉|2 ν(dx) = |ψ|2 . (1.9)

In the ultimate analysis, what is desired is to make the family {|x〉} operationalthrough the identity (1.8). This means being able to use it as a “frame”, throughwhich one reads the information contained in an arbitrary state in H, or in an op-erator onH, or in a setup involving both operators and states, such as an evolutionequation on H. At this point one can say that (1.8) realizes a “quantization” of the“classical” space (X , ν) and the measurable functions on it through the operator-valued maps:

x �→ |x〉〈x | , (1.10)

f �→ A fdef=

∫X

f (x) |x〉〈x | ν(dx) . (1.11)

The second part of this volume contains a series of examples of this quantizationprocedure.

As already stressed in [11], the family {|x〉} allows a “classical reading” of op-erators A acting on H through their expected values in coherent states, 〈x |A|x〉(“lower symbols”). In this sense, a family of coherent states provides the oppor-tunity to study quantum reality through a framework formally similar to classicalreality. It was precisely this symbolic formulation that enabled Glauber and oth-ers to treat a quantized boson or fermion field like a classical field, particularly forcomputing correlation functions or other quantities of statistical physics, such aspartition functions and derived quantities. In particular, one can follow the dynam-ical evolution of a system in a “classical” way, elegantly going back to the study ofclassical “trajectories” in the space X.

6 1 Introduction

The formalisms of quantum mechanics and signal analysis are similar in manyaspects, particularly if one considers the identities (1.8) and (1.9). In signal anal-ysis, H is a Hilbert space of finite energy signals, (X , ν) a space of parameters,suitably chosen for emphasizing certain aspects of the signal that may interest usin particular situations, and (1.8) and (1.9) bear the name of “conservation of en-ergy”. Every signal contains “noise”, but the nature and the amount of noise isdifferent for different signals. In this context, choosing (X , ν, {|x〉}) amounts toselecting a part of the signal that we wish to isolate and interpret, while eliminat-ing or, at least, strongly damping a noise that has (once and for all) been regardedas unessential . Here too we have in effect chosen a frame. Perfect illustrationsof the deep analogy between quantum mechanics and signal processing are Gaboranalysis and wavelet analysis. These analyses yield a time–frequency (“Gaboret”) ora time-scale (wavelet) representation of the signal. The built-in scaling operationmakes it a very efficient tool for analyzing singularities in a signal, a function, animage, and so on – that is, the portion of the signal that contains the most signif-icant information. Now, not surprisingly, Gaborets and wavelets can be viewed ascoherent states from a group-theoretical viewpoint. The first ones are associatedwith the Weyl–Heisenberg group, whereas the latter are associated with the affinegroup of the appropriate dimension, consisting of translations, dilations, and alsorotations if we deal with dimensions higher than one.

Let us now give an overview of the content and organization of the book.

Part One. Coherent States

The first part of the book is devoted to the construction and the description ofdifferent families of coherent states, with the chapters organized as follows.

Chapter 2. The Standard Coherent States: the BasicsIn the second chapter, we present the basics of the Schrödinger–Glauber–Klauder–Sudarshan or “standard” coherent states |z〉 == |q, p〉 introduced as a specific super-position of all energy eigenstates of the one-dimensional harmonic oscillator. Wedo this through four representations of this system, namely, “position”, “momen-tum”, “Fock” or “number”, and “analytical” or “Fock–Bargmann”. We then describethe specific role coherent states play in quantum mechanics and in quantum op-tics, for which those objects are precisely the coherent states of a radiation quantumfield.

Chapter 3. The Standard Coherent States: the (Elementary) MathematicsIn the third chapter, we focus on the main elementary mathematical features ofthe standard coherent states, particularly that essential property of being a continu-ous frame, resolving the unit operator in an “overcomplete” fashion in the space ofquantum states, and also their relation to the Weyl–Heisenberg group. Appendix Bis devoted to Lie algebra, Lie groups, and their representations on a very basic lev-el to help the nonspecialist become familiar with such notions. Next, we state the


probabilistic content of the coherent states and describe their links with three im-portant quantum distributions, namely, the “P”, “Q” distribution and the Wignerdistribution. Appendix A is devoted to probabilities and will also help the readergrasp these essential aspects. Finally, we indicate the way in which coherent statesnaturally occur in the Feynman path integral formulation of quantum mechan-ics. In more mathematical language, we tentatively explain in intelligible terms thecoherent state properties such as (CS1)–(CS4) and others characterizing on a math-ematical level the standard coherent states.

Chapter 4. Coherent States in Quantum InformationChapter 4 gives an account of a recent experimental evidence of a feedback-mediat-ed quantum measurement aimed at discriminating between optical coherent statesunder photodetection. The description of the experiment and of its theoretical mo-tivations is aimed at counterbalancing the abstract character of the mathematicalformalism presented in the previous two chapters.

Chapter 5. Coherent States: a General ConstructionIn Chapter 5 we go back to the formalism by presenting a general method of con-struction of coherent states, starting from some observations on the structure ofcoherent states as superpositions of number states. Given a set X, equipped witha measure ν and the resulting Hilbert space L2(X , ν) of square-integrable func-tions on X, we explain how the choice of an orthonormal system of functions inL2(X , ν), precisely {φ j (x) | j ∈ index set J },

∫X φ j (x)φ j ′ (x) ν(dx) = δ j j ′ , carry-

ing a probabilistic content,∑

j∈J |φ j (x)|2 = 1, determines the family of coherentstates |x〉 =

∑j φ j (x)|φ j 〉. The relation to the underlying existence of a reproduc-

ing kernel space will be clarified.This coherent state construction is the main guideline ruling the content of the

subsequent chapters concerning each family of coherent states examined (in a gen-eralized sense). As an elementary illustration of the method, we present the coher-ent states for the quantum motion of a particle on the circle.

Chapter 6. Spin Coherent StatesChapter 6 is devoted to the second most known family of coherent states, namely,the so-called spin or Bloch or atomic coherent states. The way of obtaining themfollows the previous construction. Once they have been made explicit, we describetheir main properties: that is, we depict and comment on the sequence of prop-erties like we did in the third chapter, the link with SU (2) representations, theirclassical aspects, and so on.

Chapter 7. Selected Pieces of Applications of Standard and Bloch Coherent StatesIn Chapter 7 we proceed to a (small, but instructive) panorama of applications ofthe standard coherent states and spin coherent states in some problems encoun-tered in physics, quantum physics, statistical physics, and so on. The selected pa-

8 1 Introduction

pers that are presented as examples, despite their ancient publication, were chosenby virtue of their high pedagogical and illustrative content.

Application to the Driven Oscillator This is a simple and very pedagogical modelfor which the Weyl–Heisenberg displacement operator defining standard coherentstates is identified with the S matrix connecting ingoing and outgoing states ofa driven oscillator.

Application in Statistical Physics: Superradiance This is another nice example of ap-plication of the coherent state formalism. The object pertains to atomic physics:two-level atoms in resonant interaction with a radiation field (Dicke model andsuperradiance).

Application to Quantum Magnetism We explain how the spin coherent states can beused to solve exactly or approximately the Schrödinger equation for some systems,such as a spin interacting with a variable magnetic field.

Classical and Thermodynamical Limits Coherent states are useful in thermodynam-ics. For instance, we establish a representation of the partition function for sys-tems of quantum spins in terms of coherent states. After introducing the so-calledBerezin–Lieb inequalities, we show how that coherent state representation makescrossed studies of classical and thermodynamical limits easier.

Chapter 8. SU(1, 1), SL(2, R), and Sp(2, R) Coherent StatesChapter 8 is devoted to the third most known family of coherent states, namely,the SU (1, 1) Perelomov and Barut–Girardello coherent states. Again, the way ofobtaining them follows the construction presented in Chapter 5. We then describethe main properties of these coherent states: probabilistic interpretation, link withSU (1, 1) representations, classical aspects, and so on. We also show the relation-ship between wavelet analysis and the coherent states that emerge from the unitaryirreducible representations of the affine group of the real line viewed as a subgroupof SL(2, R) ~ SU (1, 1).

Chapter 9. SU(1, 1) Coherent States and the Infinite Square WellIn Chapter 9 we describe a direct illustration of the SU (1, 1) Barut–Girardello co-herent states, namely, the example of a particle trapped in an infinite square welland also in Pöschl–Teller potentials of the trigonometric type.

Chapter 10. SU(1, 1) Coherent States and Squeezed States in Quantum OpticsChapter 10 is an introduction to the squeezed coherent states by insisting on theirrelations with the unitary irreducible representations of the symplectic groupsS p(2, R) � SU (1, 1) and their importance in quantum optics (reduction of theuncertainty on one of the two noncommuting observables present in the measure-ments of the electromagnetic field).


Chapter 11. Fermionic Coherent StatesIn Chapter 11 we present the so-called fermionic coherent states and their uti-lization in the study of many-fermion systems (e.g., the Hartree–Fock–Bogoliubovapproach).

Part Two. Coherent State Quantization

This second part is devoted to what we call “coherent state quantization”. This pro-cedure of quantization of a measure space is quite straightforward and can be ap-plied to many physical situations, such as motions in different geometries (line,circle, interval, torus, etc.) as well as to various geometries themselves (interval,circle, sphere, hyperboloid, etc.), to give a noncommutative or “fuzzy” version forthem.

Chapter 12. Coherent State Quantization: The Klauder–Berezin ApproachWe explain in Chapter 12 the way in which standard coherent states allow a naturalquantization of a large class of functions and distributions, including tempered dis-tributions, on the complex plane viewed as the phase space of the particle motionon the line. We show how they offer a classical-like representation of the evolutionof quantum observables. They also help to set Heisenberg inequalities concerningthe “phase operator” and the number operator for the oscillator Fock states. By re-stricting the formalism to the finite dimension, we present new quantum inequali-ties concerning the respective spectra of “position” and “momentum” matrices thatresult from such a coherent state quantization scheme for the motion on the line.

Chapter 13. Coherent State or Frame QuantizationIn Chapter 13 we extend the procedure of standard coherent state quantization toany measure space labeling a total family of vectors solving the identity in someHilbert space. We thus advocate the idea that, to a certain extent, quantization per-tains to a larger discipline than just being restricted to specific domains of physicssuch as mechanics or field theory. We also develop the notion of lower and uppersymbols resulting from such a quantization scheme, and we discuss the probabilis-tic content of the construction.

Chapter 14. Elementary Examples of Coherent State QuantizationThe examples which are presented in Chapter 14 are, although elementary, ratherunusual. In particular, we start with measure sets that are not necessarily phasespaces. Such sets are far from having any physical meaning in the common sense.

Finite Set We first consider a two-dimensional quantization of a N-element setthat leads, for N v 4, to a Pauli algebra of observables.

Unit Interval We study two-dimensional (and higher-dimensional) quantizationsof the unit segment.

10 1 Introduction

Unit Circle We apply the same quantization procedure to the unit circle in theplane. As an interesting byproduct of this “fuzzy circle”, we give an expressionfor the phase or angle operator, and we discuss its relevance in comparison withvarious phase operators proposed by many authors.

Chapter 15. Motions on Simple GeometriesTwo examples of coherent state quantization of classical motions taking place insimple geometries are presented in Chapter 15.

Motion on the Circle Quantization of the motion of a particle on the circle (likethe quantization of polar coordinates in a plane) is an old question with so farno really satisfactory answers. Many questions concerning this subject have beenaddressed, more specifically devoted to the problem of angular localization and re-lated Heisenberg inequalities. We apply our scheme of coherent state quantizationto this particular problem.

Motion on the Hyperboloid Viewed as a 1 + 1 de Sitter Space-Time To a certain extent,the motion of a massive particle on a 1 + 1 de Sitter background, which meansa one-sheeted hyperboloid embedded in a 2+1 Minkowski space, has characteristicssimilar to those of the phase space for the motion on the circle. Hence, the sametype of coherent state is used to perform the quantization.

Motion in an Interval We revisit the quantum motion in an infinite square wellwith our coherent state approach by exploiting the fact that the quantization prob-lem is similar, to a certain extent, to the quantization of the motion on the circleS1. However, the boundary conditions are different, and this leads us to introducevector coherent states to carry out the quantization.

Motion on a Discrete Set of Points We end this series of examples by the consid-eration of a problem inspired by modern quantum geometry, where geometricalentities are treated as quantum observables, as they have to be in order for them tobe promoted to the status of objects and not to be simply considered as a substantialarena in which physical objects “live”.

Chapter 16. Motion on the TorusChapter 16 is devoted to the coherent states associated with the discrete Weyl–Heisenberg group and to their utilization for the quantization of the chaotic motionon the torus.

Chapter 17. Fuzzy Geometries: Sphere and HyperboloidIn Chapter 17, we end this series of examples of coherent state quantization withthe application of the procedure to familiar geometries, yielding a noncommutativeor “fuzzy” structure for these objects.


Fuzzy Sphere This is an extension to the sphere S2 of the quantization of the unitcircle. It is a nice illustration of noncommutative geometry (approached in a ratherpedestrian way). We show explicitly how the coherent state quantization of the or-dinary sphere leads to its fuzzy geometry. The continuous limit at infinite spinsrestores commutativity.

Fuzzy Hyperboloid We then describe the construction of the two-dimensional fuzzyde Sitter hyperboloids by using a coherent state quantization.

Chapter 18. Conclusion and OutlookIn this last chapter we give some final remarks and suggestions for future develop-ments of the formalism presented.

13

2The Standard Coherent States: the Basics

2.1Schrödinger Definition

The coherent states, as they were found by Schrödinger [1, 17], are denoted by |z〉in Dirac ket notation, where z = |z| eiϕ is a complex parameter. They are states forwhich the mean values are the classical sinusoidal solutions of a one-dimensionalharmonic oscillator with mass m and frequency ω:

〈z|Q (t)|z〉 = 2l c |z| cos (ωt – ϕ) . (2.1)

The various symbols that are involved in this definition are as follows:

• the characteristic length l c =√

�2mω ,

• the Hilbert space H of quantum states for an object which classically wouldbe viewed as a point particle of mass m, moving on the real line, and sub-jected to a harmonic potential with constant k = mω2,

• H = P 2

2m + 12 mω2Q2 is the Hamiltonian,

• operators “position” Q and “momentum” P are self-adjoint in the Hilbertspace H of quantum states,

• their commutation rule is canonical, that is,

[Q , P ] = i�Id , (2.2)

• the time evolution of the position operator is defined as Q (t) = ei� Ht Qe– i

� Ht .

In the sequel, we present the different ways to construct these specific states andtheir basic properties. We also explain the raison d’être of the adjective coherent.

2.2Four Representations of Quantum States

The formalism of quantum mechanics allows different representations of quantumstates: “position,” “momentum,” “energy” or “number” or Fock representation, and“phase space” or “analytical” or Fock–Bargmann representation.


14 2 The Standard Coherent States: the Basics

2.2.1Position Representation

The original Schrödinger approach was carried out in the position representation.Operator Q is a multiplication operator acting in the space H of wave functionsΨ(x , t) of the quantum entity.

Q Ψ(x , t) = x Ψ(x , t) , P Ψ(x , t) = –i�∂

∂xΨ(x , t) . (2.3)

The quantity P(S) =∫

S |Ψ(x , t)|2 dx is interpreted as the probability that, at theinstant t, the object considered lies within the set S ⊂ R, in the sense that a clas-sical localization experiment would find it in S with probability P(S). Consistently,we have the normalization 1 =

∫R|Ψ(x , t)|2 dx < ∞, and so, at a given time t,

H ~= L2(R).Time evolution of the wave function is ruled by the Schrödinger equation

HΨ(x , t) = i�∂

∂tΨ(x , t) (2.4)

or equivalently Ψ(x , t) = e– i� H (t–t0) Ψ(x , t0).

Stationary solutions read as Ψ(x , t) = e– i� E n t ψn(x), where the energy eigenvalues

are equally distributed on the positive line, E n = �ω(n + 1

2

), n = 0, 1, 2, . . . . To

each eigenvalue corresponds the normalized eigenstate ψn , Hψn = E nψn ,

ψn(x) = 4

√1

2πl2c

1√2nn!

e– x2

4l2c Hn

(x√2l c

),

‖ψn‖2 =∫ +∞

–∞|ψn (x)|2 dx = 1 .

(2.5)

Here, Hn denotes the Hermite polynomial of degree n [18], with n nodes. Thefunctions {ψn , n ∈ N} form an orthonormal basis of the Hilbert space H = L2(R):

δmn = 〈ψm|ψn〉 def=∫ +∞

–∞ψm (x)ψn (x) dx , (2.6)

∀ψ ∈ H, ψ =∑n∈N

cnψn , cn = 〈ψn |ψ〉 . (2.7)

Note that the characteristic length is the standard deviation of the position in the

ground state, n = 0, l c =√

�2mω =

√〈ψ0|x2|ψ0〉.

2.2.2Momentum Representation

In momentum representation, it is the turn of operator P to be realized as a multi-plication operator onH = L2(R):

P Ψ( p , t) = pΨ( p , t) , QΨ( p , t) = i�∂

∂ pΨ( p , t) . (2.8)

2.2 Four Representations of Quantum States 15

The function Ψ( p , t) (or ψ( p)), its pure momentum part in the stationary case, isthe Fourier transform of Ψ(x , t) at fixed time t (or ψ(x)):

Ψ( p , t) =1√2π�

∫ +∞

–∞e– i

� px Ψ(x , t) dx , (2.9)

Ψ(x , t) =1√2π�

∫ +∞

–∞e

i� px Ψ( p , t) d p . (2.10)

Energy eigenstates ψn( p) in the momentum representation are similar to theirFourier counterpart ψn (x):

ψn( p) = 4

√1

2π p2c

1√2nn!

e– p2

4p2c Hn

(p√2 pc

), (2.11)

where the characteristic momentum pc =√

�mω2 is the standard deviation of the

momentum operator in the ground state.

2.2.3Number or Fock Representation

The space H of states is viewed here on a more abstract level as the closure ofthe linear span of the kets |ψn〉 == |n〉, n ∈ N, that is, the “standard” model of allseparable Hilbert spaces, namely, the space �2(N) of square-summable sequences.

Operators Q and P are now realized via the annihilation operator a and its adjointa†, the creation operator, defined by

a =1√

2�mω(mωQ + iP ) , a† =

1√2�mω

(mωQ – iP ) , (2.12)

that is,

Q = l c (a + a†) , P = –i pc (a – a†) . (2.13)

The respective actions of a and a† on the number basis read as

a|n〉 =√

n|n – 1〉 , a†|n〉 =√

n + 1|n + 1〉 , (2.14)

together with the action of a on the ground or “vacuum” state a|0〉 = 0. In thiscontext, the Hamiltonian takes the simple form

H =12

�ω(a†a + aa†) = �ω(

N +12

), (2.15)

where N = a†a is the “number” operator, diagonal in the basis {|n〉, n ∈ N}, withspectrum N: N |n〉 = n|n〉.


2.2.4A Little (Lie) Algebraic Observation

From the canonical commutation rules

[Q , P ] = i�Id ⇒ [a, a†] = Id , (2.16)

we infer that {Q , P , iId} or alternatively {a, a†, Id} span a Lie algebra (see Ap-pendix B) on the real numbers. It is the Weyl–Heisenberg algebra, denoted w,which plays a central role in the construction of coherent states. Let us adjoin to w

the number operator N that obeys

[a, N ] = a , [a†, N ] = –a† . (2.17)

We obtain a Lie algebra isomorphic to a central extension of the Euclidean group ofthe plane (rotations and translations). Next, the set of generators {Q2, P 2, 1

2 (QP +P Q )} or alternatively {N , a2, a†2} spans the lowest-dimensional symplectic Lie al-gebra, denoted by sp(2, R) � sl(2, R) � su(1, 1). This algebra is involved in theconstruction of the so-called pure squeezed states. Finally, the union of these lin-ear and quadratic generators,

{Q , P , iId , Q2, P 2, 12 (QP + P Q )} ,

or alternatively {a, a†, Id , N , a2, a†2}, spans a six-dimensional Lie algebra, denotedby h6, which is involved in the construction of the general squeezed states (seeChapter 10).

2.2.5Analytical or Fock–Bargmann Representation

Starting from the position representation, let us apply the integral transform

f (z) =∫

R

K(x , z)ψ(x) dx , (2.18)

on ψ ∈ H. In (2.18), z is element of the complex plane C with physical dimensiona square-rooted action, and the integral kernel is defined as a generating functionfor the Hermite polynomials [18]:

K(x , z) =+∞∑n=0

ψn (x)

(z/√

�)n

√n!

= 4

√1

2πl2c

exp

[1�

(z2

2–

(√mω

2x – z

)2)]

. (2.19)

2.2 Four Representations of Quantum States 17

Viewed through the transform (2.18), the eigenstate ψn(x) is simply proportionalto the nth power of z:

f n(z) =∫

R

K(x , z)ψn(x) dx

=1√n!

(z√�

)n

== 〈z|sn〉 . (2.20)

The notation 〈z|swill be explained soon. The inverse transformation for (2.18) reads

as follows:

ψ(x) =∫

C

K(x , z) f (z) μs (dz) . (2.21)

Here μs (dz) is the Gaussian measure on the plane:

μs (dz) =1

π�e– |z|

2

� dx d y =i

2π�e– |z|

2

� dz ∧ dz , (2.22)

with z = x + i y . The transform (2.18) maps the Hilbert space L2(R) onto the spaceFB of entire analytical functions that are square-integrable with respect to μs (dz):

f (z) =+∞∑n=0

αnzn converges absolutely for all z ∈ C ,

that is, its convergence radius is infinite, and

‖ f ‖2FB

def=∫

C

| f (z)|2 μs (dz) <∞ . (2.23)

The Hilbert space FB is known as a Fock–Bargmann Hilbert space. It is equippedwith the scalar product

〈 f 1| f 2〉 =∫

C

f 1(z) f 2(z) μs (dz) = �

+∞∑n=0

n! α1n α2n . (2.24)

From (2.20) a natural orthonormal basis of FB is immediately found to be

f n(z) =1√n!

(z√�

)n

. (2.25)

2.2.6Operators in Fock–Bargmann Representation

The annihilation operator a is represented as a derivation, whereas its adjoint isa multiplication operator:

a f (z) =√

�d

dzf (z) , a† f (z) =

z√�

f (z) . (2.26)


In consequence, the number operator N realizes as a dilatation (Euler), N = z ddz ,

and the Hamiltonian becomes a first order differential operator:

H = �ω(

zd

dz+

12

). (2.27)

Position and momentum then assume a quasi-symmetric form:

Q = l c

(√�

ddz

+1√�

z

), P = –i pc

(√�

ddz

–1√�

z

). (2.28)

2.3Schrödinger Coherent States

Equipped with the basic quantum mechanical material presented in the previ-ous section, we are now in the position to describe the coherent states appear-ing in (2.1). We first note that in position (like in momentum) representation, theground state,

ψ0(x) = 4

√1

2πl2c

e– x2

4l2c , (2.29)

is a Gaussian centered at the origin. Then, let us ask the question: what quantumstates could keep this kind of Gaussian localization in other points of the real line

|ψ(q)(x)|2 ∝ e–const.(x–q)2, q ∈ R ? (2.30)

In our Fock Hilbertian framework, the question amounts to finding the expansioncoefficients bn such that

|ψ(q)〉 =+∞∑n=0

bn |n〉 . (2.31)

The answer is immediate after having a look at the Bergman kernel K(x , z).

2.3.1Bergman Kernel as a Coherent State

Let us first simplify our notation by putting from now on � = 1, m = 1, ω = 1 ⇒l c = 1√

2= pc . Consider again the expansion (2.19) of the kernel K(x , z):

K(x , z) =1

4√

πexp

[(z2

2–

(1√2

x – z

)2)]

=+∞∑n=0

zn√

n!ψn(x) , (2.32)

where we have noted that ψn = ψn . Let us put z = 1√2(q + i p) and adopt the notation

K(x , z) =1

4√

πe|z|2

2

phase︷︸︸︷eix p e–i qp

2 e– 12 (x–q)2 == 〈δx |

sz〉 == 〈δx |

sq, p〉 . (2.33)

2.3 Schrödinger Coherent States 19

These are the Schrödinger or nonnormalized coherent states in position represen-tation (the index “s” is for “Schrödinger”). In Fock representation they read as

|sz〉 = |

sq, p〉 =

+∞∑n=0

zn√

n!|n〉 . (2.34)

We have obtained a continuous family of states, labeled by all points of the complexplane, and elements of the Hilbert spaceHwith as orthonormal basis the set of kets|n〉, n ∈ N.

2.3.2A First Fundamental Property

For a given value of the labeling parameter z, the coherent state |sz〉 is an eigenvector

of the annihilation operator a, with eigenvalue z,

a|sz〉 =

+∞∑n=0

zn√

n!a|n〉 =

+∞∑n=1

zn√

n!

√n|n – 1〉 = z|

sz〉 . (2.35)

It thus follows from this equation that(i) f (a)|

sz〉 = f (z)|

sz〉 for an analytical function of z (with appropriate condi-

tions),

(ii) and 〈z|sa† = z〈z|

s.

2.3.3Schrödinger Coherent States in the Two Other Representations

In momentum representation with variable k, coherent states |sz〉 == |

sq, p〉 are es-

sentially Gaussians centered at k:

〈δk|sz〉 =

14√

πe|z|2

2

phase︷︸︸︷e–ixke–i qp

2 e– 12 (k– p)2

. (2.36)

In Fock–Bargmann representation with variable � = 1√2(x + ik) ∈ C, one gets

〈�|sz〉 =

+∞∑n=0

zn√

n!〈�|n〉 =

+∞∑n=0

zn√

n!

�n

√n!

= ez� = e12 (xq+k p)e

i2 (x p–qk) . (2.37)

One thus obtains a Gaussian depending on z ·� multiplied by a phase factor involv-

ing the form Iz� = 12 (x p – qk) def= � ∧ z. After multiplication by Gaussian factors

present in the measures μs (dz) and μs (d�), one gets a Gaussian localization in thecomplex plane:

e– |�|2

2 〈�|sz〉e– |z|

2

2 = ei(�∧z)e– |z–�|22 . (2.38)


One should notice that the Schrödinger coherent states are not normalized:

〈z|sz〉 = e|z|

2. (2.39)

2.4Glauber–Klauder–Sudarshan or Standard Coherent States

In view of the last remark on the coherent state normalization, we now turn tothe normalized or standard coherent states, those ones which were precisely in-troduced by Glauber1) [5–7], Klauder, [3, 4], and Sudarshan [8]. They are obtainedfrom the Schrödinger coherent states by including in the expression of the latter

the Gaussian factor e– |z|2

2 . They are denoted by

|z〉 = e– |z|2

2

+∞∑n=0

zn√

n!|n〉 . (2.40)

Therefore, the overlap between two such states follows a Gaussian law modulatedby a “symplectic” phase factor:

〈�|z〉 = ei(�∧z)e– |z–�|22 . (2.41)

One should notice that the probability transition

|〈n|z〉|2 = e–|z|2 |z|2n

n!(2.42)

is a Poisson distribution with parameter |z|2. We will come back to this importantpoint in the next chapter.

2.5Why the Adjective Coherent?

Let us compare the two eigenvalue equations:

a|z〉 = z|z〉 , a|n〉 =√

n|n – 1〉 . (2.43)

Hence, an infinite superposition of number states |n〉, each of the latter describing a de-terminate number of elementary quanta, describes a state which is left unmodified (upto a factor) under the action of the operator annihilating an elementary quantum. Thefactor is equal to the parameter z labeling the coherent state considered.

1) In quantum optics the tradition has been touse the first letters of the Greek alphabet todenote the coherent state parameter: |α〉, |�〉,. . .

2.5 Why the Adjective Coherent? 21

More generally, we have f (a)|z〉 = f (z)|z〉 for an analytical function f. This isprecisely the idea developed by Glauber [5, 6]. Indeed, an electromagnetic field ina box can be assimilated to a countably infinite assembly of harmonic oscillators.This results from a simple Fourier analysis of the Maxwell equations. The quan-tization of these classical harmonic oscillators yields a Fock space F spanned byall possible tensor products of number eigenstates

⊗k |nk〉 == |n1, n2, . . . , nk , . . .〉,

where “k” is a shortening for labeling the mode (including the photon polarization)

k ==

⎧⎪⎨⎪⎩�k wave vector ,

ωk = ‖�k‖c frequency ,

λ = 1, 2 helicity ,

(2.44)

and nk is the number of photons in mode “k”. The Fourier expansion of the quan-tum vector potential reads as

�A(�r , t) = c∑

k

√�

2ωk

(ak�uk (�r )e–iωk t + a†k�u

∗k (�r )eiωk t

). (2.45)

As an operator, it acts (up to a gauge) on the Fock space F via ak and a†k defined by

ak0

∏k

|nk〉 =√

nk0 |nk0 – 1〉∏k=/k0

|nk〉 , (2.46)

and obeying the canonical commutation rules

[ak , ak′ ] = 0 = [a†k , a†k′ ] , [ak , a†k′ ] = δkk′ Id . (2.47)

Let us now give more insight into the modes, observables, and Hamiltonian. Onthe level of the mode functions �uk , the Maxwell equations read as

Δ�uk (�r ) +ω2

k

c2�uk(�r ) = �0 . (2.48)

When confined to a cubic box C L with size L, these functions form an orthonormalbasis ∫

C L

�uk (�r ) · �ul (�r ) d3�r = δkl ,

with obvious discretization constraints on “k”. By choosing the gauge∇·�uk (�r ) = 0,their expression is

�uk (�r ) = L–3/2e(λ)ei�k·�r , λ = 1 or 2 , �k · e(λ) = 0 , (2.49)

where the e(λ) stand for polarization vectors. The respective expressions of the elec-tric and magnetic field operators are easily derived from the vector potential:

�E = –1c

∂�A∂t

, �B = �∇ ~ �A .


Finally, the electromagnetic field Hamiltonian is given by

H =12

∫ (‖�E ‖2 + ‖�B‖2

)d3�r =

12

∑k

�ωk(a†kak + aka†k

).

Let us now decompose the electric field operator into positive and negative fre-quencies.

�E = �E (+) + �E (–), �E (–) = �E (+)† ,

�E (+)(�r , t) = i∑

k

√�ωk

2ak�uk(�r )e–iωk t . (2.50)

We then consider the field described by the density (matrix) operator

ρ =∑

nk

cnk

∏k

|nk〉〈nk| , cnk v 0 , tr ρ = 1 , (2.51)

and the derived sequence of correlation functions G (n). The Euclidean tensor com-ponents for the simplest one read as

G (1)i j (�r , t ; �r ′, t ′) = tr

{ρ�E (–)

i (�r , t)�E (+)j (�r ′, t ′)

}, i , j = 1, 2, 3 . (2.52)

They measure the correlation of the field state at different space-time points. A co-herent state or coherent radiation |c.r.〉 for the electromagnetic field is then definedby

|c.r.〉 =∏

k

|αk〉 , (2.53)

where |αk〉 is precisely the Glauber–Klauder–Sudarshan coherent state (2.40) forthe “k” mode:

|αk〉 = e–|αk |

2

2

∑nk

(αk)nk√nk !|nk〉 , ak|αk〉 = αk|αk〉 , (2.54)

with αk ∈ C. The particular status of the state |c.r.〉 is well understood through theaction of the positive frequency electric field operator

�E (+)(�r , t)|c.r.〉 = �E (+)(�r , t)|c.r.〉 . (2.55)

The expression �E (+)(�r , t) which shows up is precisely the classical field expressionsolution to the Maxwell equations.

�E (+)(�r , t) = i∑

k

√�ωk

2αk�uk(�r )e–iωk t . (2.56)

Now, if the density operator is chosen as a pure coherent state, that is,

ρ = |c.r.〉〈c.r.| , (2.57)

2.5 Why the Adjective Coherent? 23

then the components (2.52) of the first-order correlation function factorize intoindependent terms:

G (1)i j (�r , t ; �r ′, t ′) = �E (–)

i (�r , t)�E (+)j (�r ′, t ′) . (2.58)

An electromagnetic field operator is said to be “fully coherent” in the Glauber sense ifall of its correlation functions factorize like in (2.58). Nevertheless, one should noticethat such a definition does not imply monochromaticity.

25

3The Standard Coherent States: the (Elementary) Mathematics

3.1Introduction

In this third chapter we develop, on an elementary level, the mathematical formal-ism of the standard coherent states: Hilbertian properties, resolution of the unity,Weyl–Heisenberg group. We also describe some probabilistic aspects of the co-herent states and their essential role in the existence of four important quantumdistributions, namely, the “R”, “Q”, and “P” distributions and the Wigner distri-bution. Finally, we indicate the way in which coherent states naturally occur in theFeynman path integral formulation of quantum mechanics. In more mathematicallanguage, we tentatively explain in intelligible terms the properties characterizingthe standard coherent states,

|z 〉 = e–|z|2/2∞∑

n=0

zn√

n!|n〉. (3.1)

Let us list these properties that give, on their own, a strong status of uniqueness tothe coherent states.

P0 The map C � z → |z〉 ∈ L2(R) is continuous.P1 |z〉 is an eigenvector of the annihilation operator: a|z〉 = z|z〉.P2 The coherent state family resolves the unity 1

π

∫C|z〉〈z| d2z = I.

P3 The coherent states saturate the Heisenberg inequality: ΔQ ΔP = 1/2.P4 The coherent state family is temporally stable: e–iHt |z〉 = e–i t/2|e–iωt z〉, where H is

the harmonic oscillator Hamiltonian.P5 The mean value (or “lower symbol”) of the Hamiltonian mimics the classical energy-

action relation: H(z) == 〈z|H |z〉 = ω|z|2 + 12 .

P6 The coherent state family is the orbit of the ground state under the action of the Weyl–Heisenberg displacement operator: |z〉 = e(za†–za) |0〉 == D (z)|0〉.

P7 The Weyl–Heisenberg covariance follows from the above:U (s, �)|z〉 = ei(s+I(�z))|z + �〉 .

P8 The coherent states provide a straightforward quantization scheme:Classical state z → |z〉〈z| quantum state .


26 3 The Standard Coherent States: the (Elementary) Mathematics

These properties cover a wide spectrum, starting from the “wave-packet” expres-sion (3.1) together with properties P3 and P4, through an algebraic side (P1),a group representation side (P6 and P7), a functional analysis side (P2), and endingwith the ubiquitous problem of the relationship between classical and quantumworlds (P5 and P8).

3.2Properties in the Hilbertian Framework

3.2.1A “Continuity” from the Classical Complex Plane to Quantum States

With the standard coherent states, we are in the presence of a “continuous” family

of quantum states∣∣∣z = q+i p√

2

⟩or |

sz〉 in the Hilbert space H, labeled by all points of

the complex plane C. We first notice that the scalar product or overlap between twoelements of the coherent state family is given by

〈z1|sz2〉 = ez1z2 , 〈z1|z2〉 = ei(z1∧z2) e– |z1–z2 |2

2 . (3.2)

The latter complex-valued expression never vanishes, and has a Gaussian decreas-ing at infinity. Now, the continuity of the map C � z �→ |z〉 ∈ H, understoodin terms of the respective metric topologies of C and H, results from the aboveoverlap:

‖|z〉 – |z′〉‖2 = 2(1 – R〈z|z′〉)

= 2

(1 – e– |z–z′ |2

2 cos

(q p ′ – pq′

2

))→

z′→z0 . (3.3)

3.2.2“Coherent” Resolution of the Unity

One of the most important features of the coherent states is that not only do theyform a total family in H, that is, H is the closure of the linear span of the family,but they also resolve the identity operator on it.∫

C

|z〉〈z| d2zπ

=∑nn′

|n〉〈n′|∫

C

zn√

n!

zn′

√n′!

e–|z|2 d2zπ︸︷︷︸

δnn′

=∑

n

|n〉〈n| = Id

(=

∫C

|sz〉〈z|

sμs (dz)

). (3.4)

The identity between operators,∫C

|z〉〈z| d2zπ

= Id =∫

C

|sz〉〈z|

sμs (dz) ,

3.2 Properties in the Hilbertian Framework 27

has to be mathematically understood in the so-called weak sense, that is,

for all φ1, φ2 ∈ H ,∫

C

〈φ1|z〉〈z|φ2〉d2zπ

= 〈φ1|φ2〉 . (3.5)

Actually, (3.4) also holds as a strong operator identity.The resolution of the unity by the coherent states will hold as a guideline in the

multiple generalizations presented throughout this book. The abstract nature ofits content should not obscure the great significance of the formula. Before givingmore mathematical insights into the latter, let us present the following elementaryexample that will help the reader grasp (3.4).

3.2.3The Interplay Between the Circle (as a Set of Parameters)and the Plane (as a Euclidean Space)

Everyone is familiar with the orthonormal basis (or frame) of the Euclidean planeR2 defined by the two vectors (in Dirac ket notations) |0〉 and

∣∣ π2

⟩, where |θ〉 denotes

the unit vector with polar angle θ ∈ [0, 2π). This frame is such that

〈0|0〉 = 1 =⟨π

2

∣∣∣ π2

⟩,

⟨0∣∣∣π

2

⟩= 0 ,

and such that the sum of their corresponding orthogonal projectors resolves theunity

Id = |0〉〈0| +∣∣∣π

2

⟩⟨π2

∣∣∣ . (3.6)

This is a trivial reinterpretation of the matrix identity:(1 00 1

)=

(1 00 0

)+

(0 00 1

). (3.7)

To the unit vector

|θ〉 = cos θ|0〉 + sin θ∣∣∣π

2

⟩, (3.8)

there corresponds the orthogonal projector P θ given by

P θ = |θ〉〈θ| =(

cos θsin θ

)(cos θ sin θ

)=

(cos2 θ cos θ sin θ

cos θ sin θ sin2 θ

). (3.9)

The θ-dependent superposition (3.8) can also be viewed as a coherent state superpo-sition. Indeed, integrating the matrix elements of (3.9) over all angles and dividingby π leads to a continuous analog of (3.6)

1π

∫ 2π

0dθ |θ〉〈θ| = Id . (3.10)


Thus, we have obtained a continuous frame for the plane, that is to say the con-tinuous set of unit vectors forming the unit circle, for describing, with an extremeredundancy the Euclidean plane. The operator relation (3.10) is equally understoodthrough its action on a vector |v〉 = ‖v‖ |φ〉 with polar coordinates ‖v‖, φ. By virtueof 〈θ|θ′〉 = cos(θ – θ′), we have

|v〉 =‖v‖π

∫ 2π

0dθ cos (φ – θ)|θ〉, (3.11)

a relation that illustrates the overcompleteness of the family {|θ〉}. The vectors ofthis family are not linearly independent, and their mutual “overlappings” are givenby the scalar products 〈φ|θ〉 = cos (φ – θ). Moreover, the map θ �→ P θ furnishesa noncommutative version of the unit circle since

P θ P θ′ – P θ′ P θ = sin (θ – θ′)(

0 –11 0

). (3.12)

More generally, we find in this example the notion of a positive-operator-valued mea-sure (POVM) on the unit circle, which means that to any measurable set Δ ⊂ [0, 2π)there corresponds the positive 2 ~ 2 matrix:

Δ �→ P (Δ) def=1π

∫Δ

dθ |θ〉〈θ|

=

(1π

∫Δ dθ cos2 θ 1

π

∫Δ dθ cos θ sin θ

1π

∫Δ dθ cos θ sin θ 1

π

∫Δ dθ sin2 θ

). (3.13)

This matrix is obviously positive since, for any nonzero vector |v〉 in the plane, wehave 〈v |P (Δ)|v〉 = 1

π

∫Δ dθ |〈v |θ〉|2 > 0. POVMs are important partners of resolu-

tions of the unity, as will be seen at different places in this book.

3.2.4Analytical Bridge

The resolution of the identity (3.4) provided by the coherent states is the key fortransforming any abstract or concrete realization of the quantum states into theFock–Bargmann analytical one. Let

|φ〉 =∑

n

ϕn |n〉 (3.14)

be a vector in the Hilbert space H of quantum states in some of its realizations. Itsscalar product with a Schrödinger coherent state (chosen preferably to be a stan-dard coherent state to avoid the Gaussian weights) reads as the power series

〈z|sφ〉 =

+∞∑n=0

ϕnzn√

n!def= Φs (z) , (3.15)

3.2 Properties in the Hilbertian Framework 29

with infinite convergence radius. The series (3.15) defines an entire analytical func-tion, Φs (z), which is square-integrable with respect to the Fock–Bargmann measureμs (dz). Hence, this function is an element of the Fock–Bargmann space FB andwe have established a linear map fromH into FB. This map is an isometry:

‖Φs‖2FB =

∫C

|Φs (z)|2 μs (dz)

=∫

C

|〈z|sφ〉|2 μs (dz)

=∫

C

〈φ|sz〉〈z|

sφ〉 μs (dz) = ‖φ‖2

H , (3.16)

where we have used the fact that μs (dz) = μs (dz) and the resolution of the iden-tity (3.4).

3.2.5Overcompleteness and Reproducing Properties

The coherent states form an overcomplete family of states in the sense that(i) it is total, which is equivalent to stating that if there exists φ ∈ H such that〈φ|

sz〉 = 0 for all z ∈ C, then φ = 0. Now 〈φ|

sz〉 = 0 implies Φs (z) = 0 for all

z ∈ C. So, by analyticity, ϕn = 0 for all n and finally φ = 0,

(ii) at least two of them in the family are not linearly independent.

Thus, the coherent states do not form a Hilbertian basis ofH, but they form a densefamily in the Hilbert space H and they resolve the unity.An immediate consequence of the latter is their reproducing action on the ele-ments of FB that emerge from the map H � φ↔ Φs ∈ FB:

Φs (z) = 〈z|sφ〉 = 〈z|

s

∫C

|sz′〉〈z′|

sφ〉 μs (dz′)

=∫

C

〈z|sz′〉Φs (z′) μs (dz′) . (3.17)

Hence, the scalar product 〈z|sz′〉 = ezz′ = 〈z′|

sz〉 plays the role of the reproducing

kernel in the Fock–Bargmann space FB. The latter is a reproducing kernel space. Notethat this object indicates to what extent the coherent states are linearly dependent:

|sz〉 =

∫C

ezz′ |sz′〉μs (dz′) , (3.18)

|z〉 =∫

C

e–iz∧z′e– 12 |z–z′|2 |z′〉 d2z′

π. (3.19)

This precisely shows that a coherent state is, by itself, a kind of average over all thecoherent state family weighted with a Gaussian distribution (up to a phase).


3.3Coherent States in the Quantum Mechanical Context

3.3.1Symbols

As shown in 3.2.5, the coherent states form an overcomplete system (often abusive-ly termed overcomplete “basis”) or frame that allows one to analyze quantum statesor observables from a “coherent state” point of view. Indeed, when we decomposea state |φ〉 inH as

|φ〉 =∫

C

〈z|sφ〉 |

sz〉 μs (dz) =

∫C

Φs (z) |sz〉μs (dz) , (3.20)

the continuous expansion component Φs (z) is an entire (anti-) holomorphic func-tion called symbol. On the same footing, the nonanalytical function

Φ(z) = e– |z|2

2 Φs (z) = 〈z|φ〉 (3.21)

is the symbol of |φ〉 in its decomposition over the family of standard coherentstates:

|φ〉 =∫

C

〈z|φ〉 |z〉 d2zπ

=∫

C

Φ(z) |z〉 d2zπ

. (3.22)

Note that the following upper bound, readily derived from the Cauchy–Schwarzinequality,

|Φ(z)| = |〈z|φ〉| u√〈z|z〉‖φ‖ = ‖φ‖ , (3.23)

implies the continuity of the map φ �→ Φ for the uniform convergence topology inthe target space.

3.3.2Lower Symbols

The mean value of a quantum observable in standard coherent states is called the“lower” [19] or “contravariant” [20] symbol of this operator. For instance, for thecreation and annihilation operators (“ladder operators”), we have

〈z|a|z〉 = z , 〈z|a†|z〉 = z . (3.24)

For the number operator N = a†a, we have

〈z|N |z〉 = |z|2 . (3.25)

Hence, although a coherent state is an infinite superposition of Fock states |n〉, onegets a finite mean value of N that can be arbitrarily small at z → 0.

3.3 Coherent States in the Quantum Mechanical Context 31

For the canonical operator position Q = 1√2(a+a†) and momentum P = 1√

2i(a–a†)

and with the classical parameterization z = 1√2(q + i p), we have

〈z|Q |z〉 = q , 〈z|P |z〉 = p . (3.26)

These formulas give coherent states a quite “classical face”, although they are rigor-ously quantal! The physical meaning of the variable x in the Schrödinger positionrepresentation of the wave is that of a sharp position. Unlike x in the Schrödingerrepresentation, the variables p and q represent mean values in the coherent states.It is for this reason that we can specify both values simultaneously at the sametime, something that could not be done if instead they both had represented sharpeigenvalues.

3.3.3Heisenberg Inequalities

Let us now calculate the mean quadratic values of the position and of the mo-mentum.

〈z|Q2|z〉 = q2 + 12 , 〈z|P 2|z〉 = p2 + 1

2 . (3.27)

It follows for the quantum harmonic oscillator Hamiltonian H = 12

(P 2 + Q2

)that

〈z|H |z〉 = 12 (q2 + p2) + 1

2 . (3.28)

The “absolutely quantal face” of the coherent states reappears here. Indeed, theadditional term 1

2 , called “symplectic correction” or “vacuum energy” according tothe context, has no classical counterpart.

Finally, from the respective variances of position and momentum,

(ΔQ )2 == 〈z|Q2 – 〈Q〉2z|z〉 = 12 = (ΔP )2 ,

we infer the saturation of the uncertainty relation or, more properly, Heisenberginequality:

ΔQ ΔP =12

(==

�

2

). (3.29)

This is one of the most important features exhibited by coherent states: their quan-tal face is the closest possible to its classical counterpart.


3.3.4Time Evolution and Phase Space

From the eigenvalue equation H |n〉 =(n + 1

2

)|n〉 for the Hamiltonian of the oscil-

lator, one deduces the time evolution of a coherent state:

e–iHt |z〉 =e– |z|2

2

+∞∑n=0

zn√

n!e–iHt |n〉

=e–i t2 e– |z|

2

2

+∞∑n=0

(e–i t z

)n

√n!|n〉

= e–i t2︸︷︷︸

phase

| e–i t z︸︷︷︸rotation of z

〉 . (3.30)

We thus ascertain the temporal stability of the family {|z〉, z ∈ C} or {|sz〉, z ∈ C}

since a quantum state is defined up to a phase factor. There follows the illuminatingform of this time evolution in position representation:

〈δx |z〉 =1

4√

πeix pe–i qp

2 e– 12 (x–q)2

,

〈δx |e–iHt |z〉 =1

4√

πeix p ′e–i q′ p ′

2 e– 12 (x–q′)2

,(3.31)

where

q′ = q cos t + p sin t , p ′ = –q sin t + p cos t .

Hence, at given z = (q, p), the time evolution of the Gaussian localization ofthe “particle” is a harmonic motion taking place between –|z| and |z|. We exactlyrecover the classical behavior of the harmonic oscillator such as it is described byits phase diagram in which, at constant energy,

2E = p2 + q2 = 2|z|2 = p ′2 + q′2 ,

the phase trajectory is a circle (or, depending on the units utilized, an ellipse). Thecomplex number z = q+i p√

2can unambiguously be identified as a phase space point

(a “classical state”) and the Liouville measure or 2-form �ω = –idz∧dz = dq∧d p onthe phase space is naturally present in the definition of the Fock–Bargmann space.Given an initial state z = z0, say, at t = 0, its time evolution is simply given byz = z(t) = z0 eit . The alternative phase space formalism “angle–action” becomesapparent here. The angle is precisely the argument θ = t + θ0 of z, whereas itscanonical conjugate is the action

I0 =1

2π

∫|z′|u|z0|

�ω′ = |z0|2 . (3.32)

3.3 Coherent States in the Quantum Mechanical Context 33

Fig. 3.1 This is a java animation devel-oped by L. Kocbach from an original ap-plet written by O. Psencik. It is found athttp://web.ift.uib.no/AMOS/MOV/HO/. Thisinteractive toy shows the shape of Glauber(i.e. standard) coherent states and their timeevolution. It offers the possibility to select theenergy ~ |z|2 : the vacuum z = 0 is shown on

the top, and an example of CS with intermedi-ate z =/ 0 is shown in the middle figure. Theshape of a perfect Poisson law is easily recog-nized for these two cases. One can see in thethird figure (bottom) to what extent a smallpertubation of the Poisson distribution de-stroys the “coherent” Gaussian shape of CSstates.


One notices that this physical quantity can be viewed as the average value of thenumber operator in the coherent state |z0〉, I0 = 〈z0|N |z0〉 (lower symbol).

At this point it is worth observing, as Schrödinger did, the close relationshipbetween classical and quantum solutions to the oscillator problem. To be moreconcrete, let us reintroduce physical dimensions, namely, mass m and frequency ω.The solution to the Newton equation x + ω2x = 0 is given by x(t) = |A| cos (ω t – ϕ)with amplitude = |A| ==

√2I0 and phase ϕ = θ0 as initial conditions. Let us rewrite

this solution as the real part of the complex Ae–iωt , with A = |A|eiϕ ==√

2I0eiθ0 :

x(t) =12

(Ae–iωt + Aeiωt ) . (3.33)

Let us now go to the quantum side by adopting the time-dependent (Heisenberg)point of view for position and momentum operators. Their respective time evolu-tions

Q (t) = ei� Ht Q e– i

� Ht , P (t) = ei� Ht P e– i

� Ht , (3.34)

produced by the Hamiltonian H = P 2

2m + 12 mω2Q2, obey

Q(t) = –i�

[Q (t), H ] = P (t)/m , (3.35)

P (t) = –mω2Q (t) . (3.36)

From these two equations one easily derives the analog of the classical Newtonequation:

Q(t) + ω2Q (t) = 0 . (3.37)

The same holds for P (t), and, as well, for the time-dependent lowering and raisingoperators defined consistently as (2.12)

a(t) =1√

2�mω(mωQ (t)+iP (t)) , a†(t) =

1√2�mω

(mωQ (t)–iP (t)) . (3.38)

Note that, in the Heisenberg representation, these operators act on quantum statesat the initial time, since we have for any operator, say, O, and any states, say, ψ1,ψ2, the identity

〈ψ1(t)| O︸︷︷︸Schrödinger

|ψ2(t)〉 = 〈ψ1(0)| ei� HtOe– i

� Ht︸︷︷︸HeisenbergO(t)

|ψ2(0)〉 . (3.39)

Now, from (3.35), (3.36) and (3.38) we derive the time-evolution equation for a(t)and a†(t):

a(t) = –iωa(t) , a†(t) = iωa†(t) . (3.40)

They are easily solved with respective initial conditions a(0) = a, a†(0) = a†:

a(t) = a e–iωt , a†(0) = a† eiωt . (3.41)

3.4 Properties in the Group-Theoretical Context 35

Hence, the solution to (3.37) parallels exactly the classical solution (3.33) as

Q (t) = l c(a e–iωt + a† eiωt

), (3.42)

where the quantum characteristic length l c =√

�2mω was introduced in Section 2.1.

Since the mean value of a(t) in a coherent state |z〉 is given by 〈z|a(t)|z〉 =e–iωt〈z|a|z〉 = ze–iωt , we can now understand Schrödinger’s interest in these statesas pointed out at the beginning of this book:

〈z|Q (t)|z〉 = 2l c |z| cos (ωt – ϕ) . (3.43)

3.4Properties in the Group-Theoretical Context

3.4.1The Vacuum as a Transported Probe . . .

From

|z〉 = e– |z|2

2

+∞∑n=0

zn√

n!|n〉

and

|n〉 =(a†)n√

n!|0〉 ,

we obtain the alternative expression for the coherent states

|z〉 = e– |z|2

2

+∞∑n=0

(za†)n√

n!|0〉 = e– |z|

2

2 eza† |0〉 . (3.44)

One should notice here the presence of the “exponentiated” action of an alternativeversion, denoted by wm , of the Weyl–Heisenberg Lie algebra

wm = linear span of {iQ , iP , iId} . (3.45)

A generic element of wm is written as

wm � X = i sId + i ( pQ – qP ) = i sId + (za† – za) , s ∈ R , (3.46)

with the notation z = q+i p√2

, Q = a+ia†√2

, and P = a–ia†√2i

. The operator X is anti-self-adjoint inH and is the infinitesimal generator of the unitary operator:

eX = eis ei( pQ–qP ) = eis eza†–za def= eisD (z) . (3.47)


3.4.2Under the Action of . . .

The operator-valued map z �→ D (z) = eza†–za = ei( pQ–qP ) that appears in (3.47) isa unitary representation, up to a (crucial!) phase factor, of the group of translationsof the complex plane. Indeed, let us apply to the product D (z1)D (z2) == eA1 eA2 ,where Ai = zi a† – zi a and [A1, A2] = 2iI(z1z2) = –2iz1 ∧ z2, the Weyl formula,eA eB = e( 1

2 [A,B ]) e(A+B ), which is valid for any pair of operators that commute withtheir commutator, [A, [A, B ]] = 0 = [B , [A, B ]]. One gets the composition rule

D (z1)D (z2) = e–iz1∧z2 D (z1 + z2) . (3.48)

The unitarity of the operator D (z), D (z)–1 = D (z)† = eza–za† = D (–z) is then easy tocheck:

D (z)D (–z) = e–iz∧zD (z – z) = Id . (3.49)

More generally,

D (zn)D (zn–1) · · ·D (z1) = eiδD (z1 + z2 + · · · + zn) . (3.50)

The phase δ = –∑

j<k z j ∧ zk that appears in this expression has a topological (orsymplectic) meaning: it is equal to the oriented area of the polygon AΓ delimitedby the path Γ with vertices z1, z1 +z2, . . . , z1 +z2 + · · ·+zn , as shown in Figure 3.2.In canonical coordinates zi = 1√

2(qi + pi ),

δ =∑

j<k

12

(q j pk – qk p j ) , (3.51)

and this represents a discrete version of the Stokes formula∫

Γ p dq =∫AΓ

dq ∧ d p .

��

��

��

�

��

�

�

�

�

�

�

�

�

�

O

z1

z2

z3

z4

zn

Fig. 3.2 An example of piecewise linear path in phase space.


3.4.3. . . the D-Function

The operators X, through the map X �→ eX = eisD (z), thus define an irreducibleunitary representation of the Weyl–Heisenberg algebra w. All other representations,except an unimportant case, are of the same type: there is uniqueness of the real-ization of the canonical commutation rules (Stone–von Neumann theorem [21]).The phase factor eis that appears here commutes with all other operators in therepresentation but nevertheless plays a crucial role in quantum mechanics, sinceit encodes its noncommutativity, as will be seen below.

The D-“function” (it is actually an operator, also named displacement operator)D (z) = exp (za† – za) is essential in determining the properties and applications ofthe coherent states. Using the Weyl formula, one can also write

D (z) = e– |z|2

2 eza† e–za = e– i2 q pei pQ e–iqP . (3.52)

With this formulation and from a|0〉 = 0 ⇒ e–za|0〉 = |0〉, one gets a new defi-nition of the standard coherent states, precisely the one that emerges from grouptheory [10, 11]:

|z〉 = D (z)|0〉 . (3.53)

Hence, the family of coherent states is to be viewed as the orbit of the Fock vacuumunder the action of operators D (z), z ∈ C.

Let us show how we derive from (3.53) the coherent states in the position repre-sentation. From

|z〉 = D (z)|0〉 , �z(x) def= 〈δx |z〉 = 〈D (–z)δx |0〉 , (3.54)

and from

D (–z)δx (y ) = e– i2 q pe–i pQ eiqP δx (y ) = e– i

2 q pe–i pQ δx (y + q)

= ei2 q pe–i px δx–q(y ) , (3.55)

one finds the following expression for the coherent states:

�z(x) =1

4√

πe– i

2 q pei px e– (x–q)2

2 ,

to be compared with (2.33).

3.4.4Symplectic Phase and the Weyl–Heisenberg Group

In fact, the coherent state family should be viewed as the orbit of any particularcoherent state since

D (z′)|z〉 = D (z′)D (z)|0〉 = eiz∧z′D (z′ + z)|0〉 = eiz∧z′ |z + z′〉 . (3.56)


One notices that the presence of the phase factor in (3.56) prevents us from view-ing the displacement operator D (z) as the representation of a simple translation.Indeed, it appears as the subtle mark of noncommutativity of two successive dis-placements:

D (z1)D (z2) = e2iz2∧z1 D (z2)D (z1) . (3.57)

This equation is the integrated version of the canonical commutation rules [Q , P ] =iId . In this regard, one speaks of projective representation z �→ D (z) of the Abeliangroup C, since the composition of two operators D (z1), D (z2) produces a phasefactor, with phase Iz1z2 = –z1 ∧ z2. In other words and to insist on this importantfeature of the formalism, the unavoidable appearance of this phase compels us towork with a wider set than the complex numbers. This set is precisely the (Lie)Weyl–Heisenberg group W � R ~ C, the Lie algebra of which is wm

W � g = (s, z) = (s, q, p)

�Wm � X = i sId + za† – za .

(3.58)

The group law is given by

(s1, z1)(s2, z2) = (s1 + s2 – z1 ∧ z2, z1 + z2)⇔(s1, q1, p1)(s2, q2, p2) =

(s1 + s2 – 1

2 (q1 p2 – q2 p1), q1 + q2, p1 + p2)

.(3.59)

The neutral element is (0, 0, 0) and the inverse is (s, z)–1 = (–s, –z).

3.4.5Coherent States as Tools in Signal Analysis

An intriguing question arises from the group-theoretical interpretation of the co-herent states: what about transporting a state different from the vacuum? Concrete-ly, let us make D (z) act on an arbitrarily chosen state |ψ〉 ∈ H:

|z〉ψ def= D (z)|ψ〉 . (3.60)

For instance, in position representation, these states read as

〈δx |z〉ψ def= ψz(x) = e– i2 q pei px ψ(x – q) . (3.61)

A question naturally arises: which genuine coherent state properties are still valid?

(i) The states |z〉ψ are normalizable:

ψ〈z|z〉ψ = 〈ψ|D (z)†D (z)|ψ〉 = ‖ψ‖2 . (3.62)

(ii) They solve the identity. Indeed, consider the operator

A =∫

C

|z〉ψψ〈z|d2zπ

=∫

C

D (z)|ψ〉〈ψ|D (–z)d2zπ

. (3.63)


This operator commutes with all operators eis D (z′) of the unitary irreduciblerepresentation of the Weyl–Heisenberg group (exercise!). Hence, by apply-ing the Schur lemma,2)A = constant Id , with consequently the same repro-ducing properties. The computation of the constant == cψ is straightforward:

cψ ‖ψ‖2 = 〈ψ|A|ψ〉 =∫

C

|〈ψ|D (z)|ψ〉|2 d2zπ

.

(iii) These coherent states enjoy the same covariance properties with respect tothe action of W:

D (z′)|z〉ψ = eiz∧z′ |z + z′〉ψ . (3.64)

But then what have we lost?(iv) We have lost an important property:

a|z〉ψ =/ z|z〉ψ , (3.65)

since the equality only holds true for |ψ〉 ∝ |0〉.

Nevertheless, we get a serious improvement with regard to the freedom in thechoice of ψ ! This is essentially the main interest in these states from the point ofview of signal analysis. The so-called Gabor (or “windowed Fourier”) transform ofsignal analysis [22–24] precisely rests upon the resolution of the identity

Id =∫

C

|z〉ψψ〈z|d2zcψπ

. (3.66)

This identity allows one to implement a Hilbertian analysis of any state |φ〉 fromthe point of view of the continuous frame of coherent states |z〉ψ:

|φ〉 =∫

Cψ〈z|φ〉 |z〉ψ

d2zcψπ

. (3.67)

The projection ψ〈z|φ〉 of the state |φ〉 onto the state

ψz(x) = e– i2 q pei px ψ(x – q) , (3.68)

which is the “window” or “Gaboret” or even “wavelet”, |ψ〉, translated and modu-lated, is called the Gabor transform or the windowed Fourier transform or the time–frequency representation of the “signal” φ. This transform reads as

ψ〈z|φ〉 =∫ +∞

–∞e

i2 q pe–i px ψ(x – q)φ(x) dx def= Gφ(q, p) . (3.69)

Note that, in practice, one ignores the phase factor ei2 q p in the definition of the win-

dow. We give in Figure 3.3 an example of such a “time–frequency” representationof a signal.

2) Schur lemma: If T on vector space E andT ′ on E ′ are irreducible representations ofa group G and L : E �→ E ′ is a linear mapsuch that T ′(g ) L = L T (g ) for all g ∈ G,

then L = 0 or L is invertible. Furthermore,if E = E ′ is a vector space over complexnumbers, then L is a scalar.


500 1000 1500 2000−2

−1

0

1

2

0.02

0.04

0.06

0.08

0.1

0.12

0246

x 104

500 1000 1500 2000

20

40

60

80

100

120

Fig. 3.3 Windowed Fourier (or Gabor, or time–frequency)transform of a signal (top). On the left, the Fourier transform,with two visible frequency peaks corresponding to the dis-cernible regimes in the signal. On the right, the time–frequencyrepresentation of the signal. The two regimes clearly manifestthemselves with their respective durations and frequencies (bycourtesy of Pierre Vandergheynst, EPFL).

3.5Quantum Distributions and Coherent States

In quantum mechanics, the existence, due to some classical statistical estimate,of a probability distribution on the states accessible to a system is described bya statistical operator called its density matrix. This operator is viewed as a “mixedstate” superposition of pure state projectors |ψ〉〈ψ| with ‖ψ‖ = 1.

“mixed” state: ρ =∑

ψ

pψ |ψ〉〈ψ|︸︷︷︸“pure” state with norm 1

, (3.70)

where 0 u pψ u 1,∑

ψ pψ = 1. In the case of an orthonormal basis |ψi 〉, ρ =∑i pi |ψi 〉〈ψi |, pi is the probability (estimated from a classical point of view) for

the system to be in state |ψi 〉. So, a quantum mixed state is therefore a convex linearsuperposition of pure states (projectors).

Let {|n〉, n ∈ N} be an orthonormal basis (e.g., Fock number states) of theHilbert space of states of the system. The mean value of a quantum observable Oin a unit-norm state ψ is 〈O〉ψ =

∑n〈n|ψ〉〈ψ|O|n〉. Hence, the mean value ofO in

3.5 Quantum Distributions and Coherent States 41

the state ρ is

〈O〉ρ =∑

ψ

pψ〈O〉ψ =∑

n

〈n|ρO|n〉 = tr(Oρ) . (3.71)

In this section we give a short account of different types of phase space repre-sentations of the density matrix of a system, mainly pertaining to quantum op-tics [25, 26].

3.5.1The Density Matrix and the Representation “R”

In the Fock representation, the density matrix is just defined by its matrix elementsρm,n :

ρ =∑m,n

ρm,n︸︷︷︸〈m|ρ|n〉

|m〉〈n| .

In the “full” coherent state representation, the density matrix is determined by the(generally complex-valued) function named distribution “R” defined as

R(α, �) def= 〈α|sρ|

s�〉 = 〈α|ρ|�〉e 1

2 (|α|2+�|2) . (3.72)

This distribution appears in the Fock–Bargmann representation of ρ:

ρ =∫∫

C2

R(α, �) e– 12 (|α|2+�|2) |α〉〈�| d2α

πd2�π

=∫∫

C2

R(α, �) |sα〉〈�|

sμs (dα) μs (d�) . (3.73)

In the case of a pure number state, ρ = |n〉〈n|, this distribution reads as

R(α, �) =(α�)n

n!. (3.74)

In the case of a coherent state ρ = |z〉〈z|,

R(α, �) = e–|z|2 ezα+z� . (3.75)

3.5.2The Density Matrix and the Representation “Q”

The distribution “Q”, or “lower symbol” of ρ, or Husimi function of the state ρ, isthe set of expected values of ρ in coherent state representation:

Q (α, α) def= 〈α|ρ|α〉 . (3.76)


It is a positive function bounded by 1:

0 u Q (α, α) =∑

ψ

pψ|〈ψ|α〉|2︸︷︷︸u1

u 1 .

Moreover, it is a true probability density on the phase space since we have∫C

Q (α, α)d2απ

= 1 . (3.77)

The simple choice of the pure state ρ = |n〉〈n| illustrates its probabilistic meaningin terms of a Poisson distribution:

Q (α, α) = |〈n|α〉|2 = e–|α|2 |α|2n!

. (3.78)

The Husimi function has a remarkable property (we will come back to this point):an operator OA(a, a†) =

∑m,n dmn ama†n , in its “antinormal order” expansion, that

is, all creation operators are placed on the right in monomials appearing in theseries expansion of O(a, a†), has the following expectation value in state ρ:⟨

OA(a, a†)⟩

ρ=

∫C

OA(α, α) Q (α, α)d2απ

. (3.79)

3.5.3The Density Matrix and the Representation “P”

The distribution “P”, or “upper symbol” of ρ, is defined by the components of ρ inits diagonal representation in terms of coherent state projectors:

ρ =∫

C

P (α, α) |α〉〈α| d2απ

. (3.80)

It is a bounded function but has indeterminate sign. It is a pseudodensity of prob-ability on the phase space in the sense that∫

C

P (α, α)d2απ

= 1 . (3.81)

There exists an inversion formula to (3.80)

P (α, α) = e|α|2

∫C

〈–�|ρ|�〉 e|�|2 e–2iα∧� d2�π

. (3.82)

This formula is derived from the so-called symplectic Fourier transform of generalfunctions of the complex variable α and its conjugate:

f s (α, α) =1

2π

∫C

eiα∧� f (�, �) d2� ,

f (�, �) =1

2π

∫C

ei�∧α f s (α, α) d2α,

with1

4π2

∫C

eiα∧� d2� = δ(2)(α) .

(3.83)

3.5 Quantum Distributions and Coherent States 43

Let us determine the distribution “P” for a coherent state itself, ρ = |�〉〈�|. Onefinds, as expected and up to the factor π, a Dirac distribution on C centered at �:

P (α, α) = e–|α|2–|�|2∫

C

e2i(α–�)∧� d2�π

= πδ2(α – �) . (3.84)

Another interesting example [25] concerns a thermal radiation field, emitted bya source in thermal equilibrium at temperature T:

ρ =e– H

kB T

tr e– HkB T

=∑

n

〈n〉n(1 + 〈n〉)n+1 |n〉〈n| ,

where 〈n〉 == tr(a†aρ). One then gets the Gaussian distribution

P (α, α) =1〈n〉 e

– |α|2

〈n〉 .

Similarly to the Husimi function Q, a remarkable property holds concerning meanvalues of operators: an operator ON (a, a†) =

∑m,n cmn a†n am , in its “normal or-

der” expansion, that is, all creation operators are placed on the left in monomi-als appearing in the series expansion of O(a, a†), has as the expected value in thestate ρ ⟨

ON (a, a†)⟩

ρ=

∫C

ON (α, α) P (α, α)d2απ

. (3.85)

3.5.4The Density Matrix and the Wigner(–Weyl–Ville) Distribution

The Wigner distribution W c(α, α) associated with a matrix density ρ is the symplec-tic Fourier transform of the mean value of the displacement operator or function Din the state ρ:

W c(α, α) def=1

2π

∫C

tr(D (�)ρ) e2iα∧� d2�π

=1

2π

∫C

tr(e�a†–�aρ) e–(�α–�α) d2�π

. (3.86)

Similarly to the distribution “P”, it is a pseudodensity of probability on the phasespace ∫

C

W c

(1√2

(q + i p),1√2

(q – i p)

)dq d p = 1 .

A yet more remarkable property than those encountered for Q and P holds formean values of operators. An operatorO(a, a†) has as the mean value in the state ρ⟨

O(a, a†)⟩

ρ= tr(Oρ) =

∫C

OS (α, α) W c(α, α) d2α , (3.87)

where OS (α, α) is obtained from O(a, a†) by just substituting a �→ α, a† �→ α after


symmetrization of the monomials in a and a† appearing in the series expansion ofO(a, a†). For instance,

O(a, a†) = a†a = 12 (a†a + aa†) – 1

2 → OS (α, α) = |α|2 – 12 .

Let us establish the link between (3.86) and a more traditional expression of theWigner(–Weyl–Ville) distribution. Putting α = 1√

2(q + i p) and � = –i√

2(σ + iτ) in the

integral

W c(α, α) =1

2π

∫C

tr(e�a†–�aρ) e–(�α–�α) d2�π

,

one gets the Fourier transform (actually with permuted variables) in the phasespace:

W c(α, α) def= W (q, p) =1

2π

∫C

tr(e–i(τP +σQ)ρ) ei(τ p+σq) dτ dσ2π

=1

2π

∫ +∞

–∞

⟨q –

y2

∣∣∣ ρ∣∣∣q +

y2

⟩ei y p d y . (3.88)

For example, the Wigner function for a pure state ρ = |ψ〉〈ψ| reads as

W (q, p) =1

2π

∫ +∞

–∞ψ

(q +

y2

)ψ

(q –

y2

)ei y p d y . (3.89)

Within a signal processing framework, this expression is precisely the Wigner–Villetransform of the “signal” ψ. The real-valued function W, with indeterminate sign,is a phase space representation of the state ρ. In the case of a pure state, marginalintegrations restore a true quantum probabilistic content:∫ +∞

–∞W (q, p) d p = |ψ(q)|2 ,

∫ +∞

–∞W (q, p) dq = |ψ( p)|2 .

3.6The Feynman Path Integral and Coherent States

The path integral was introduced by Feynman in 1948 as an alternative formula-tion of (nonrelativistic) quantum mechanics [27]. Starting from the Schrödingerequation (in which and in the sequel we put � = 1)

i∂Ψ(x , t)

∂t= –

12 m

∂2Ψ(x , t)∂x2 + V (x) Ψ(x , t) (3.90)

for a particle of mass m moving in a potential V (x), a solution can be written as anintegral,

Ψ(x , t) =∫

K (x , t ; x ′, t ′) Ψ(x ′, t ′) dx ′ , (3.91)

3.6 The Feynman Path Integral and Coherent States 45

which represents the wave function Ψ(x , t) at time t as a linear superposition overthe wave function Ψ(x ′, t ′) at the initial time t ′, t ′ < t . The integral kernel or propa-gator K (x , t ; x ′, t ′) can be formally expressed as an integral running over all contin-uous paths x ′′(u), t ′ u u u t ′′, where x ′′(t) = x and x ′′(t ′) = x ′ are fixed end pointsfor all paths.

K (x , t ; x ′, t ′) = N∫

ei∫

[(m/2) x ′′2(u) – V (x ′′(u))] du Dx ′′ . (3.92)

Note that the integrand involves the classical Lagrangian for the system.To give some meaning to this mathematically ill-defined object, Feynman adopt-

ed the following lattice regularization with spacing ε:

K (x , t ; x ′, t ′) = limε→0

(m/2πiε)(N +1)/2∫

. . .

∫~ exp

{i

N∑l=0

[(m/2ε)(xl+1 – xl )2 – ε V (xl )

]} N∏l=1

dxl , (3.93)

where xN +1 = x , x0 = x ′, and ε == (t – t ′)/(N + 1), N ∈ {1, 2, 3, . . . }. This procedurethat yields well-defined integrals has to be validated by the existence of a “contin-uum limit as ε → 0. Following the original Feynman approach, various authors,such as Feynman himself, Kac, Gel’fand, Yaglom, Cameron, and Itô, attempted tofind a suitable continuous-time regularization procedure along with a subsequentlimit to remove that regularization that ultimately should yield the correct propa-gator (see the illuminating review by Klauder [28], from which a large part of thissection is borrowed).

Now, it appears that a phase space formulation of path integrals is more natural,as was also suggested by Feynman [29], and later successfully carried further byDaubechies and Klauder [2, 30]. Feynman (1951) proposed for the propagator thefollowing integral on paths in the phase space

K (q, t ; q′, t ′) =M∫

exp{

i∫ [

p ′′q′′ – H(q′′, p ′′)]

du}D p ′′ Dq′′ . (3.94)

Here one integrates over all paths q′′(u), t ′ u u u t , with q′′(t) == q and q′′(t ′) == q′

held fixed, as well as over all paths p ′′(u), t ′ u u u t , without restriction.A lattice space version of this expression is commonly given by

K (q, t ; q′, t ′) = limε→0

∫· · ·

∫exp

{i

N∑l=0

[12 pl+1/2(ql+1 – ql )

–ε H( pl+1/2, 12 (ql+1 + ql ))

] } N∏l=0

d pl+1/2/(2π) ΠNl=1 dql . (3.95)

Like before ε = (t – t ′)/(N + 1). The integration is performed over all p and q vari-ables except for qN +1 == q and q0 == q′. It is important to observe the presence of re-strictions due to the canonical formalism of quantum mechanics: since ql implies


a sharp q value at time t ′ + lε, the conjugate variable has been denoted by pl+1/2 toemphasize that a sharp p value must occur at a different time, here at t ′ + (l + 1/2)ε,since it is not possible to have sharp p and q values at the same time. Note that thereis one more p integration than q integration in this formulation. This discrepancybecomes clear when one imposes the composition law that requires that

K (q, t ; q′, t ′) =∫

K (q, t ; q′′, t ′′) K (q′′, t ′′; q′, t ′) dq′′ , (3.96)

a relation that implies, just on dimensional grounds, that there must be one morep integration than q integration in the definition of each K expression.

The contribution of Daubechies and Klauder (1985) was to reexamine (3.94) byusing a complete phase space formalism combined with coherent states throughthe Fock–Bargmann representation of wave functions and operators. They intro-duced coherent states, denoted here by |

fq, p〉 and defined as

|fq, p〉 def= e–iqP ei p Q |0〉 = e–i p q/2 ei( pQ–qP )|0〉 . (3.97)

Since they are the standard ones |z〉, z = q+i p√2

times the (important here!) phase

factor e–i p q/2, they have unit norm and they resolve the unity as well:

〈q, p|fq, p〉 = 1 ,

∫R2

|fq, p〉〈q, p|

fdq d p/2π = Id . (3.98)

Let us introduce, with the phase space variables (q, p), the (nonanalytical) symbolof the wave function like was defined in (3.21):

�Ψ(q, p , t) def= 〈q, p|fΨ(·, t)〉 . (3.99)

At this point, we should make clear the probabilistic interpretation of this objectcompared with Ψ(x , t). We know that |ψ(x , t)|2 is the probability density of findingthe particle at position x (at time t). On the other hand, the quantity |�Ψ(q, p , t)|2 isthe probability that the state |Ψ〉 can be found in the state |

fq, p〉. What is important

is to be aware that in coherent state representation we can specify both values ofthe variables simultaneously at the same time.

In this Fock–Bargmann representation, position and momentum operators aregiven by

Q f = q + i∂

∂ p, P f = –i

∂

∂q, (3.100)

and thus the coherent state representation of the Schrödinger equation with classi-cal Hamiltonian H(q, p) is given by

i ∂ �Ψ(q, p , t)/∂t = H( –i ∂/∂q, q + i ∂/∂ p ) �Ψ(q, p , t) . (3.101)

3.6 The Feynman Path Integral and Coherent States 47

The solution to this form of Schrödinger’s equation can be expressed in the form

�Ψ(q, p , t) =∫

K (q, p , t ; q′, p ′, t ′) �Ψ(q′, p ′, t ′) dq′ d p ′/2π , (3.102)

where K (q, p , t ; q′, p ′, t ′) denotes the propagator in the coherent state representa-tion.

The propagator for the coherent state representation of Schrödinger’s equationcan also be given a formal phase space path integral form, namely,

K (q, p , t ; q′, p ′, t ′) =M∫

exp{i∫

[ p ′′ q′′ – H(q′′, p ′′)] dt}Dq′′ D p ′′ . (3.103)

Despite the fact that this expression looks the same as (3.94), the pinned values andthe lattice space formulations are different. In the coherent state case, we have

K (q, p , t ; q′, p ′, t ′) =

limε→0

∫. . .

∫exp

{i

N∑l=0

[12 ( pl+1 + pl )(ql+1 – ql )

–εH(

12 ( pl+1 + pl ) + i 1

2 (ql+1 – ql ), 12 (ql+1 + ql ) – i 1

2 ( pl+1 – pl )) ] }

~ exp

{–(1/4)

N∑l=0

[( pl+1 – pl )2 + (ql+1 – ql )2 ]

}N∏l=1

dql d pl/(2π) . (3.104)

Observe that there are now the same number of p and q integrations in this expres-sion. Such a conclusion is fully in accord with the combination law as expressed inthe coherent state representation, namely,

K (q, p , t ; q′, p ′, t ′) =∫K (q, p , t ; q′′, p ′′, t ′′) K (q′′, p ′′, t ′′; q′, p ′, t ′) dq′′ d p ′′/2π . (3.105)

Daubechies and Klauder [30] gave a mathematical rigor to the expressions (3.103)and (3.104) by introducing a Brownian type regularization term. They eventuallyproved the existence of the following limit:

limν→∞Mν

∫exp

{i∫ [

p q′′ – H(q′′, p ′′)]

du

}~ exp

{–

12ν

∫ [p ′′

2+ q′′

2]du

}Dq′′ D p ′′

= limν→∞

2π eνT/2∫

ei∫

[ p ′′ dq′′ – H(q′′, p ′′) du] dμνW (q′′, p ′′)

==〈q, p| e–i(t–t ′)H |q′, p ′〉 == K (q, p , t ; q′, p ′, t ′) , (3.106)

where the second line of (3.106) is a mathematically rigorous formulation of theheuristic and formal first line, and μν

W denotes the measure on continuous path


q′′(u), p ′′(u), t ′ u u u t , said to be a pinned Wiener measure. Such a choice ofregularization also justifies the choice of coherent states (3.97) and imposes thecondition that H(q, p) be precisely the upper symbol of the quantum Hamiltonian

H =∫

H(q, p) |fq, p〉〈q, p|

fdq d p/(2π) . (3.107)

A sufficient set of technical assumptions ensuring the validity of this representa-tion is given by

(a)∫

H(q, p)2 e–α( p2+q2) dq d p <∞, for all α > 0 ,

(b)∫

H(q, p)4e–�( p2+q2) dq d p <∞, for some � < 1/2� ,

(c) The quantum HamiltonianH is essentially self-adjoint on the span of finite-ly many number eigenstates.

As a matter of fact, Hamiltonians that are semibounded, symmetric (Hermitian)polynomials of Q f and P f are admissible.One can conclude that a continuous-time, Brownian motion regularization of thephase space path integral can be rigorously established. It applies to a wide classof Hamiltonians. It can also be proved that the formulation is fully covariant undergeneral canonical coordinate transformations.

Recently, dos Santos and Aguiar [31] constructed a representation of the coherentstate path integral using the Weyl symbol of the Hamiltonian operator. Their coher-ent state propagator provides an explicit connection between the Wigner and theHusimi representations of the evolution operator. The dos Santos–Aguiar repre-sentation is different from the usual path integral forms suggested by Klauder andSkagerstam presented in this section. These different representations, althoughequivalent quantum mechanically, lead to different semiclassical limits.

49

4Coherent States in Quantum Information:an Example of Experimental Manipulation

4.1Quantum States for Information

Quantum information processing is about exploiting quantum mechanical fea-tures in all facets of information processing (data communication, computing).Excellent textbooks, monographs, and reviews exist which give all the material nec-essary to understand this fast-developping field [32–34].

The states act as information carriers, while the communication is processedthrough a sequence of quantum operations constituting the channel. The senderencodes information by preparing the channel into a well-defined quantum state ρbelonging to an alphabet A = {ρ0, ρ1, . . . , ρM}. The receiver, following any rele-vant signal propagation, performs a measurement on the transmission channel toascertain which state was transmitted by the sender. Quantum information theoryis mainly based on superposition-basis and entanglement measurements. This re-quires high-fidelity implementation to be effective in the laboratory. Unfortunately,quantum measurements are “invasive” in the sense that little or no refinement isachieved by further observation of an already measured system. Some of the diffi-culties in implementing communication in quantum information stems from thefragility of Schrödinger-cat-like superpositions. Even with transmission of orthog-onal codewords, decoherence, energy dissipation, and other imperfections deterio-rate orthogonality.

If the states in the sender’s alphabet are not orthogonal, no measurement can dis-tinguish between overlapping quantum states without some ambiguity [21, 35–38].Then errors seem unavoidable: there exists a nonzero probability that the receiverwill misinterpret the transmitted codeword.

However, this impossibility of discriminating between nonorthogonal quantumstates presents an advantage for quantum key distribution [39]. Indeed, nonorthog-onality prevents an eavesdropper from acquiring information without disturbingthe state. Also, in some cases it has been shown by Fuchs [36] that the classicalinformation capacity of a noisy channel is actually maximized by a nonorthogonalalphabet.

Mathematically, the question of distinguishing between nonorthogonal states[36, 37] is addressed by optimizing a state-determining measurement over all pos-


50 4 Coherent States in Quantum Information: an Example of Experimental Manipulation

itive-operator-valued measures (POVM) [38]. But arbitrary POVMs are not easy tomanipulate!

In a recent work, Cook, Martin, and Geremia [40] demonstrated that real-timequantum feedback can be used in place of a quantum superposition of the type“Schrödinger cat state” to implement an optimal quantum measurement for dis-criminating between optical coherent states. This work gives us an excellent op-portunity of presenting standard coherent states as they are produced and used inrealistic conditions. As a preliminary to the description of the experiment, we willalso give an account of the theoretical background needed.

4.2Optical Coherent States in Quantum Information

The optical field produced by a laser provides a convenient quantum system for car-rying information. Since optical coherent states |α〉 are not orthogonal, one wouldattempt to minimize the overlapping 〈α′|α〉 = eiI(α′α)e–|α–α′|2/2 by using large-ampli-tude regimes. However, one faces power limitations and the appearance of nonlin-ear effects. So one is more inclined to develop optimization methods for commu-nication processes based on small-amplitude optical coherent states and photode-tection. When one tries to distinguish between two nonorthogonal states throughsome receiver device, there exists a quantum error probability. The latter is bound-ed below by some minimum, named the quantum limit or Helstrom bound [38] inthis context.

Three types of receivers were described by Geremia [42]. Kennedy [44] proposedin 1972 a receiver based on simple photon counting to distinguish between twodifferent coherent states. However, the Kennedy receiver error probability liesabove the quantum mechanics minimum, that is, the Helstrom bound. Then,Dolinar [41] proposed in 1973 a measurement scheme capable of achieving thequantum limit. Dolinar’s receiver, while still based on photon counting, approxi-mates an optimal POVM by superposing a local feedback signal on the channel.A serious experimental drawback was that real-time adjustment of the local signalfollowing each photon was considered as quite impracticable. As a result, Sasakiand Hirota [45] later proposed an alternative receiver that applies an open-loop uni-tary transformation to the incoming coherent state signals to render them moredistinguishable by simple photon counting.

Geremia [42] compared, theoretically and numerically, the relative performanceof the Kennedy, Dolinar, and Sasaki–Hirota receivers under realistic experimentalconditions, insisting on the following aspects:

(i) subunity quantum efficiency, where it is possible for the detector to mis-count incoming photons,

(ii) nonzero dark counts, where the detector can register photons even in theabsence of a signal,

4.3 Binary Coherent State Communication 51

(iii) nonzero dead-time, or finite detector recovery time after registering the ar-rival of a photon,

(iv) finite bandwidth of any signal processing necessary to implement the de-tector,

(v) fluctuations in the phase of the incoming optical signal.

4.3Binary Coherent State Communication

4.3.1Binary Logic with Two Coherent States

Let us consider an alphabet consisting of two pure coherent states,

ρ0 = |Ψ0〉〈Ψ0| , ρ1 = |Ψ1〉〈Ψ1| ,

corresponding to the logic states “0” and “1,” respectively. Without loss of general-ity, Ψ0(t) can be chosen as the vacuum, Ψ0(t) = 0, that is, |Ψ0〉 = |0〉, while

Ψ1(t) = ψ1(t) exp[–i (ωt + ϕ)

]+ c.c. , (4.1)

where ω is the frequency of the optical carrier and ϕ is (ideally) a fixed phase.The envelope function, ψ1(t), is normalized such that∫ τ

0|ψ1(t)|2 dt = n , (4.2)

where n is the mean number of photons arriving at the receiver during the mea-surement interval, 0 u t u τ. That is, �ω|ψ1(t)|2 is the instantaneous average powerof the optical signal for logic “1.”

By combining the incoming signal with an appropriate local oscillator, one canalways transform the amplitude keying with the alphabet of two coherent statesA = {|0〉〈0|, |α〉〈α|}, with |α〉 = |Ψ1〉, to the phase-shift keyed alphabet,{∣∣∣∣– 1

2α⟩⟨

–12

α

∣∣∣∣ ,

∣∣∣∣12

α⟩⟨

12

α

∣∣∣∣} ,

via the unitary displacement, D(

– 12 α

)= exp(– 1

2 (αa† – αa)). Similarly, if |Ψ0〉 =/ |0〉,a simple displacement can be used to restore |Ψ0〉 to the vacuum state.

4.3.2Uncertainties on POVMs

In the case of nonorthogonal quantum states as codewords, the receiver attemptsto ascertain which state was transmitted by performing a quantum measurement,


say, Π, on the channel. The operator Π is described by an appropriate POVM rep-resented by a complete (here countable) set of positive operators [37] resolving theidentity,∑

n

Πi = Id , Πi v 0 , (4.3)

where n indexes the possible measurement outcomes.We already gave a simple example of a continuous POVM in Section 3.2.3. In

the same vein, an example of a finite POVM in the Euclidean plane is given by thefollowing cyclotomic polygonal resolution of the unity:

2n

n–1∑q=0

Π 2πqn

= Id , Πθ = |θ〉〈θ| =(

cos2 θ cos θ sin θcos θ sin θ sin2 θ

).

For binary communication, for which the POVM resolution of the unity reads Π0 +Π1 = Id , the measurement by the receiver amounts to a decision between twohypotheses: H0, that the transmitted state is ρ0, selected when the measurementoutcome corresponds to Π0, and H1, that the transmitted state is ρ1, selected whenthe measurement outcome corresponds to Π1.

4.3.3The Quantum Error Probability or Helstrom Bound

Now, possibilities of errors mean that there is some chance that the receiver willselect the null hypothesis, H0 (or H1), when ρ1 (or ρ0) is actually present. Thus, wehave in terms of conditional probabilities

p (H0|ρ1) = tr[Π0ρ1] = tr[(Id – Π1)ρ1] , p (H1|ρ0) = tr[Π1ρ0] . (4.4)

The total receiver error probability, say, p [Π0, Π1], is then given by

p [Π0, Π1] = �0 p (H1|ρ0) + �1 p (H0|ρ1) , �0 + �1 = 1 , (4.5)

where �0 = p0(ρ0) and �1 = p0(ρ1) are the probabilities that the sender will transmitρ0 and ρ1, respectively; they reflect the prior knowledge that enters into the hypoth-esis testing process implemented by the receiver, and, in many cases �0 = �1 = 1/2.

Minimizing the error in receiver measurement over all possible POVMs (Π0, Π1)leads to the so-called quantum error probability or Helstrom bound,

P H == minΠ0,Π1

p [Π0, Π1] , (4.6)

P H is the smallest physically allowable error probability, given the overlap betweenρ0 and ρ1.

4.3 Binary Coherent State Communication 53

4.3.4The Helstrom Bound in Binary Communication

The receiver error probability

p [Π0, Π1] = �0 tr [Π1ρ0] + �1 tr[(Id – Π1)ρ1

]= �1 + tr

[Π1(�0ρ0 – �1ρ1)

]is minimized by optimizing minΠ1 tr[Π1Γ ], Γ def= �0ρ0 – �1ρ1 , over Π1 subject to0 u Π1 u Id .

Let Γ =∑

n λn |γn〉〈γn| be the spectral decomposition of the operator Γ . One canwrite tr[Π1Γ ] =

∑n λn〈γn|Π1|γn〉. Then the Helstrom bound can be expressed as

P H = �1 +∑

λn <0 λn , which corresponds to the case in which Π1 is the projector onall eigenstates |γn〉 with negative λn .

For pure states, where ρ0 = |Ψ0〉〈Ψ0| and ρ1 = |Ψ1〉〈Ψ1|, Γ has two eigenvalues,of which only one is negative,

λ– =12

(1 –

√1 – 4�0�1|〈Ψ0|Ψ1〉|2

)– �1 < 0 . (4.7)

Then the quantum error probability is [38]

P H = �1 + λ– =12

(1 –

√1 – 4�0�1|〈Ψ1|Ψ0〉|2

). (4.8)

To prove this formula, let us consider an orthonormal basis {|e0〉 == |Ψ0〉, |e1〉}and the corresponding decomposition of |Ψ1〉 : |Ψ1〉 = μ0|e0〉 + μ1|e1〉 with μ0 =〈Ψ0|Ψ1〉. Thus, the operator Γ reads as the 2 ~ 2 matrix:

Γ = �0ρ0 – �1ρ1 =

(�0 – �1|μ0|2 –�1μ0μ1

–�1μ0μ1 –�1|μ1|2

).

By taking into account �0 + �1 = 1 (completeness of the probabilities) and |μ0|2 +|μ1|2 = 1 (normalization of |Ψ1〉), we get the eigenvalues of Γ as equal to

λ± =12

(1±

√1 – 4�0�1|〈Ψ0|Ψ1〉|2

)– �1 ,

from which we derive (4.8).

4.3.5Helstrom Bound for Coherent States

From the expansion of coherent states over the number states,

|α〉 = e–|α|2/2∞∑

n=0

αn√

n!|n〉,

the overlap between |Ψ1〉 = |α〉 and |Ψ0〉 = |0〉 is just given in terms of the expected


number of photons or the average value or lower symbol of the number operatorin the coherent state |α〉: nα == 〈α|N |α〉:

〈Ψ1|Ψ0〉 = 〈α|0〉 = e–|α|2/2 = e–nα/2 . (4.9)

So, the Helstrom bound is given by

P H =12

(1 –

√1 – 4�0�1e–nα

). (4.10)

4.3.6Helstrom Bound with Imperfect Detection

It is further possible to evaluate the Helstrom bound for imperfect detection.Nonunit efficiency of a photodetector leads to a photon count which is related tothe ideal (efficiency η = 1) premeasured photon distribution by a Bernoulli trans-formation [43]. Accordingly, the probability pn(η) of detecting n photons usinga nonideal photodetector (η < 1) is given in terms of the probability pm(η = 1)(using an ideal one) by

pn(η) =∞∑

m=n

(mn

)ηn (1 – η)m–n pm (η = 1) . (4.11)

Coherent states have the convenient property that subunity quantum efficien-cy is equivalent to an ideal detector masked by a beam splitter with transmissioncoefficient, η u 1. Indeed, in the case of coherent states, we have the Poisson dis-tribution pm (η = 1) = e–|α|2 |α|2m/m!, and so changing m into s = m – n in thesummation (4.11) gives

pn(η) =ηn|α|2n

n!e–η|α|2 , (4.12)

which amounts to replacing α by√

ηα in the expression of coherent states.Accordingly, the Helstrom bound becomes

P H(η) =12

(1 –

√1 – 4�0�1e–nαη

). (4.13)

This result and (4.8) indicate that there is a finite quantum error probability for allchoices of |Ψ1〉, even when an optimal measurement is performed.

4.4The Kennedy Receiver

4.4.1The Principle

The Kennedy receiver is based on the following principle. A near-optimal receiversimply counts the number of photon arrivals registered by the detector between

4.4 The Kennedy Receiver 55

t = 0 and T. The receiver decides in favor of H0 when the number of clicks is zero,otherwise H1 is chosen.

This hypothesis testing procedure corresponds to the measurement operators

Π0 = |0〉〈0| , Π1 =∞∑

n=1

|n〉〈n| . (4.14)

The receiver always correctly selects H0 when the channel is in ρ0, since no photoncan be registered when the vacuum state is present (ignoring background light anddetector dark counts for now). Therefore, p(H1|ρ0) = 0.

On the other hand, the Poisson statistics of coherent state photon numbers al-lows for the possibility that zero photons will be recorded even when ρ1 is present.So

p(H0|ρ1) == tr[Π0ρ1] = |〈0|Ψ1〉|2 (4.15)

is nonzero owing to the finite overlap of all coherent states with the vacuum.

4.4.2Kennedy Receiver Error

Now, an imperfect detector, although able to count photons, can misdiagnose ρ1 ifit fails to generate clicks for photons that do arrive at the detector.

The probability for successfully choosing H1 when ρ1 is present is given by

pη(H1|ρ1) =∞∑

n=1

∞∑k=1

p(n, k)|〈n|α〉|2 , (4.16)

where the Bernoulli distribution,

p(n, k) =n!

k!(n – k)!ηk(1 – η)n–k , (4.17)

gives the probability that a detector with quantum efficiency, η will register k clickswhen the actual number of photons is n.

The resulting Kennedy receiver error,

P K(η) = 1 – pη(H1|ρ1) = �1e–nαη , (4.18)

asymptotically gets closer to the Helstrom bound for large signal amplitudes, butis larger for small photon numbers.


4.5The Sasaki–Hirota Receiver

4.5.1The Principle

Still in the simple photon counting implementation, a unitary transformation tothe incoming signal states prior to detection could help to approach better the Hel-strom bound. With Sasaki and Hirota [45] let us consider the rotation

U (θ) = exp[θ(|Ψ ′0〉〈Ψ ′1| – |Ψ ′1〉〈Ψ ′0|)

], (4.19)

generated by the transformed alphabet, A′ = {|Ψ ′0〉〈Ψ ′0| , |Ψ ′1〉〈Ψ ′1|}, where thestates are obtained from Gram–Schmidt orthogonalization of those of A:

|Ψ ′0〉 = |Ψ0〉, |Ψ ′1〉 =|Ψ1〉 – c0|Ψ0〉√

1 – c20

, c0 = 〈Ψ1|Ψ0〉 = e–|α|2/2 == e–nα/2 .

(4.20)

The angle θ must be optimized to achieve the Helstrom bound.The action of U (θ) on the incoming signal states is given by

U (θ)|Ψ0〉 =

(cos θ +

c0 sin θ√1 – c2

0

)|Ψ0〉 –

sin θ√1 – c2

0

|Ψ1〉

U (θ)|Ψ1〉 =sin θ√1 – c2

0

|Ψ0〉 +cos θ

√1 – c2

0 – c0 sin θ√1 – c2

0

|Ψ1〉 .

4.5.2Sasaki–Hirota Receiver Error

Since |Ψ ′0〉 is the vacuum state, hypothesis testing can still be performed by simplephoton counting. However, unlike the Kennedy receiver, it is possible to misdiag-nose ρ0 since U (θ)|Ψ0〉 contains a nonzero contribution from |Ψ1〉.

The probability for a false-positive detection by a photon counter with efficiency ηis given by

pθη(H1|ρ0) =

∞∑n=1

∞∑k=1

p(n, k)|〈n|U (θ)|Ψ0〉|2 =c2η

0 – 1c2

0 – 1sin2 θ , (4.21)

with

〈n|U (θ)|Ψ0〉 =

[cos θ +

c0 sin θ√1 – c2

0

]δn,0 –

c0αn sin θ√n!(1 – c2

0), (4.22)

where α is the (complex) amplitude of |Ψ1〉.

4.6 The Dolinar Receiver 57

Similarly, the probability for correct detection is given by

pθη(H1|ρ1) =

∞∑n=1

∞∑k=1

p(n, k)|〈n|U (θ)|Ψ1〉

=c2η

0 – 1c2

0 – 1

[c0 sin θ –

√1 – c2

0 cos θ]2

, (4.23)

with

〈n|U [θ]|Ψ1〉 =

[c0 cos θ –

c20αn sin θ√n!(1 – c2

0)

]+

sin θ√1 – c2

0

δn,0 . (4.24)

The total Sasaki–Hirota receiver error is then given by the weighted sum

P SH(η, θ) = �0 pθη(H1|ρ0) + �1

[1 – pθ

η(H1|ρ1)]

(4.25)

and can be minimized over all possible values of θ to give

θ = – tan–1

√√1 – 4�0�1c2

0 – 1 + 2�1c20√

1 – 4�0�1c20 + 1 – 2�1c2

0

. (4.26)

For perfect detection efficiency, η = 1, (4.25) is equivalent to the Helstrom bound;however, for η < 1, it is larger.

4.6The Dolinar Receiver

4.6.1The Principle

The Dolinar receiver utilizes an adaptive strategy to implement a feedback approxi-mation to the Helstrom POVM [41]. It operates by combining the incoming signal,Ψ(t), with a separate local signal,

U(t) = u(t) exp[–i (ωt + φ)

]+ c.c . (4.27)

Here u(t) is the “displacement” or “feedback” amplitude.The detector counts photons with total instantaneous mean rate

Φ(t) = |ψ(t) + u(t)|2 , (4.28)

where ψ(t) = 0 (for logic “0”) when the channel is in the state ρ0, and ψ(t) = ψ1(t)(for logic “1”) when the channel is in ρ1.


4.6.2Photon Counting Distributions

Given the alphabetA = (ρ0, ρ1), the feedback amplitude u(t), a transmission coeffi-cient η, and some subdivision (t0 == 0, t1, . . . , tn , tn+1 == τ) of the measurement timeinterval (or “counting interval”) [0, τ], the conditional probability w

[tk|ρi , u(t)

]that

a photon will arrive at time tk and that it will be the only click during the half-closed interval (tk–1, tk ] [7] is called the exponential waiting time distribution foroptical coherent states. It is defined as

wη[tk|ρi , u(t)

]= η Φ(tk) exp

(–η

∫ tk

tk–1

Φ(t ′) dt ′)

. (4.29)

The corresponding exclusive counting densities for the measurement interval arethen given by

pη[t1, . . . , tn |ρi , u(t)

]=

n+1∏k=1

wη[tk|ρi , u(t)

]. (4.30)

They allow one to evaluate, using the Bayes rule, the conditional arrival time proba-bilities pη

[ρi |t1, . . . , tn , u(t)

]= pη

[t1, . . . , tn|ρi , u(t)

]p0(ρi ). The latter reflect the

likelihood that n photon arrivals occur precisely at the times t1, . . . , tn ,3) given thatthe channel is in the state ρi , the feedback amplitude is u(t), and the detector quan-tum efficiency is η.

4.6.3Decision Criterion of the Dolinar Receiver

The receiver decides between hypotheses H0 and H1 by selecting the one that ismore consistent with the record of photon arrival times observed by the detectorgiven the choice of u(t). H1 is selected when the ratio of conditional arrival timeprobabilities,

Λ =pη

[ρ1|t1, . . . , tn , u(t)

]pη

[ρ0|t1, . . . , tn , u(t)

] , (4.31)

is greater than one; otherwise it is assumed that ρ0 was transmitted.By employing the Bayes rule, one can reexpress Λ in terms of the photon count-

ing distributions

Λ =pη

[t1, . . . , tn|ρ1, u(t)

]p0(ρ1)

pη[t1, . . . , tn|ρ0, u(t)

]p0(ρ0)

=�1

�0

pη[t1, . . . , tn |ρ1, u(t)

]pη

[t1, . . . , tn |ρ0, u(t)

] , (4.32)

3) Even though the term “arrival time” is notappropriate from an experimental point ofview. Time interval is more appropriate.

4.6 The Dolinar Receiver 59

In terms of error probabilities, the likelihood ratio is given by

Λ =pη

[H1|ρ1, u(t)

]pη

[H1ρ0, u(t)

] =1 – pη

[H0|ρ1, u(t)

]pη

[H1|ρ0, u(t)

] , for Λ > 1 (4.33)

(i.e., the receiver definitely selects H1), and

Λ =pη

[H0|ρ1, u(t)

]pη

[H0|ρ0, u(t)

] =pη

[H0|ρ1, u(t)

]1 – pη

[H1|ρ0, u(t)

] , for Λ < 1 (4.34)

(i.e., the receiver definitely selects H0).

4.6.4Optimal Control

The minimization over u(t) of the Dolinar receiver error probability,

P D[u(t)] = �0 pη[H1|ρ0, u(t)

]+ �1 pη

[H0|ρ1, u(t)

], (4.35)

can be accomplished by employing the technique of dynamical programming [47].The optimal control policy, u∗(t), is identified by solving the Hamilton–Jacobi–

Bellman equation,

minu(t)

[∂

∂tJ [u(t)] +∇pJ [u(t)]T

∂

∂tp(t)

]= 0 , (4.36)

where the “control cost” J [u(t)] == P D[u(t)] = �Tp in an effective state-space picturegiven by the conditional error probabilities,

p(t) =

(pη

[H1|ρ0, u(t)

](t)

pη[H0|ρ1, u(t)

](t)

). (4.37)

The partial differential equation for J is based on the requirement that p(t) andu(t) are smooth (continuous and differentiable) throughout the entire receiver op-eration. However, like all quantum point processes, our conditional knowledge ofthe system state evolves smoothly only between photon arrivals. Fortunately, the dy-namical programming optimality principle allows us to optimize u(t) in a piecewisemanner [47]. Performing the piecewise minimization leads to the control policy

u∗1(t) = –ψ1(t)

(1 +

J [u∗1(t)]1 – 2J [u∗1(t)]

)(4.38)

for Λ > 1 (see [42] for the proof), where pη[H0|ρ1, u∗1(t)] = 0 and

J [u∗1(t)] = �1 pη[H1|ρ0, u∗1(t)] =12

(1 –

√1 – 4�0�1e–ηn(t)

).

Here, n(t) =∫ t

0 |ψ1(t ′)|2 dt ′ is the average number of photons expected to arrive atthe detector by time t when the channel is in the state ρ1.


Conversely, the optimal control takes the form

u∗0(t) = ψ1(t)

(J [u∗0(t)]

1 – 2J [u∗0(t)]

)(4.39)

for Λ < 1, where pη[H1|ρ0, u∗0(t)] = 0 and

J [u∗0(t)] = �1 pη[H0|ρ1, u∗0(t)] =12

(1 –

√1 – 4�0�1e–ηn(t)

).

4.6.5Dolinar Hypothesis Testing Procedure

The Hamilton–Jacobi–Bellman solution leads to a conceptually simple procedurefor estimating the state of the channel. The receiver begins at t = 0 by favoringthe hypothesis that is more likely based on the prior probabilities, p0(0) = �0 andp1(0) = �1. Note that if �0 = �1, then neither hypothesis is a priori favored andthe Dolinar receiver is singular with P D = 1

2 . Assuming that �1 v �0 (for �0 > �1,the opposite reasoning applies), the Dolinar receiver always selects H1 during theinitial measurement segment. The probability of deciding on H0 is exactly zeroprior to the first photon arrival such that an error only occurs when the channel isactually in ρ0.

To see what happens when a photon does arrive at the detector, it is necessary toinvestigate the behavior of Λ(t) at the boundary between two measurement seg-ments. Substituting the optimal control policy, u∗(t), which alternates betweenu∗1(t) and u∗0(t), into the photon counting distribution leads to

p(t1, . . . , tn |ρi ) = ηnn+1∏k=0

Φi [uk|2(tk–1, tk ]]

~ exp

(–η

∫ tk

tk–1

Φi[uk|2(t ′k–1, t ′k ]

]dt ′

). (4.40)

Here, the notation k|2 stands for k mod 2. This expression can be used to showthat the limit of Λ(t) approaching a photon arrival time, tk , from the left is thereciprocal of the limit approaching from the right:

limt→t–

k

Λ(t) =

[limt→t+

k

Λ(t)

]–1

. (4.41)

That is, if Λ > 1 such that H1 is favored during the measurement interval ending attk , the receiver immediately swaps its decision to favor H0 when the photon arrives.Each photon arrival invalidates the current hypothesis and the receiver completelyreverses its decision on every click. This result implies that H1 is selected when thenumber of photons, n, is even (or zero) and that H0 when the number of photonsis odd.

4.7 The Cook–Martin–Geremia Closed-Loop Experiment 61

Despite the discontinuities in the conditional probabilities, pη[H1|ρ0, u∗(t)] andpη[H0|ρ1, u∗(t)], at the measurement segment boundaries, the total Dolinar re-

ceiver error probability, P D(η, t) = 12

(1 –

√1 – �0�1e–ηn(t)

), evolves smoothly since

limt→t–kJ [u∗(t)] = limt→t+

kJ [u∗(t)] at the boundaries.

Recognizing that n(τ) = nα leads to the final Dolinar receiver error,

P D(η) =12

(1 –

√1 – 4�0�1c2η

0

), (4.42)

which is equal to the Helstrom bound for all values of the detector efficiency,0 < η u 1.

4.7The Cook–Martin–Geremia Closed-Loop Experiment

4.7.1A Theoretical Preliminary

Cook, Martin, and Geremia [40] demonstrated that shot noise can be surpassed andeven the quantum limit can be approached by using the Dolinar real-time quantumfeedback in place of the cat-state measurement. They exploited the finite durationof any real measurement, and quantum states |0〉 and |α〉 are realized as opticalwave packets with spatiotemporal extent.

Measurements on an optical pulse inherently persist for a time set by the pulselength τ. Photon counting generates a measurement record �[0,τ] == (t1, t2, . . . , tn),t0 = 0, tn+1 = τ, consisting of the observed photon arrival times. The total numberof photon arrivals in the counting interval [0, τ] is viewed as one aggregate “instan-taneous” measurement of the number operator.

In the closed-loop measurement as sketched in Figure 4.1a, photon counting iscombined with feedback-mediated optical displacements applied during the pho-ton counting interval. The amplitude of the displacement u(t), here denoted by ut ,applied at each time t during the measurement is conditional on the accumulatedmeasurement record �[0,τ] and is based on an evolving Bayesian estimate of theincoming wave-packet state.

Discrimination is performed by selecting the state |ψ〉 ∈ {|0〉, |α〉} that maxi-mizes the conditional probability P

(�[0,τ]|ψ, u [0,t ]

)that the measurement record

�[0,τ] would be observed given the state ψ and the history of applied displacementsdenoted by u [0,t ] .

The feedback controller determines which state is most consistent with the accu-mulating record �[0,τ] and chooses the feedback amplitude at each point in time tominimize the probability of error over the remainder of the measurement interval(t , τ].

The policy for determining the optimal displacement amplitude u∗t is based onthe optimal control theory presented in Section 4.6.4: it engineers the feedback


Fig. 4.1 A measurement that combines pho-ton counting with feedback-mediated opticaldisplacements to enact quantum-limited statediscrimination between the coherent states|0〉 and |α〉 is considered in (a). A diagram of

the laboratory implementation of (a) is shownin (b). Source Cook et al. [40] (reprinted bypermission from Macmillan Publishers Lim-ited: [Nature] (Cook, R.L., Martin, P.J., andGeremia, J.M. 446, p. 774, 2007)).

such that the photon counter is least likely to observe additional clicks if it is cor-rectly based on its best knowledge of the channel state at that time. In the presentexperiment, it is conveniently summarized as the minimization of the time-addi-tive extension of total receiver error probabilities like (4.5), with �0 = �1 = 1/2and where, at time t, the displacement history u [0,t ] is also taken conditionally intoaccount:

P E [ut ] =12

∫ τ

0dt

[P

(α|0, u [0,t ]

)+ P

(0|α, u [0,t ]

)]. (4.43)

The functional minimization of this expression leads to feedback policies (4.38)and (4.39) with ψ1(t) = α. They can be written as the unique formula

u∗t(n[0,t ]

)=

α2

(eiπ(n [0,t]+1)√

1 – e|α|2t/τ– 1

)(4.44)


with the decision procedure that |α〉 (or |0〉) is chosen when the number of photoncounts n[0,t ] in the measurement interval [0, t ] is even (or odd), that is, the ratio Λis greater than 1 (or the ratio Λ is less than 1). We recall that this expression ana-lytically achieves the fundamental quantum limit or Helstrom bound.

4.7.2Closed-Loop Experiment: the Apparatus

As shown in Figure 4.1b the laboratory implementation of the closed-loop mea-surement consists in the following. Light from an external-cavity grating-stabilizeddiode laser4) operating at 852nm is coupled into a polarization-maintaining fiber-optic Mach–Zehnder interferometer. A Mach–Zehnder interferometer is a deviceused to determine the phase shift caused by a small sample which is placed inthe path of one of two collimated beams (thus having plane wave fronts) froma coherent light source. The input beamsplitter (FBS1) provides two optical fieldswith a well-defined relative phase: the upper arm of the interferometer acts as thetarget quantum system for state discrimination and the lower arm provides an aux-iliary field used to perform closed-loop displacements at the second beamsplitter(FBS2). Photon counting on the outcoupled field is implemented using a gatedsilicon avalanche photodiode (APD). APDs are photodetectors that can be regard-ed as the semiconductor analog of photomultipliers. By applying a high reversebias voltage (typically 100–200 V in silicon), APDs show an internal current gaineffect (around 100) due to impact ionization (avalanche effect). The feedback con-troller is constructed from a combination of programmable waveform generatorsand high-speed digital signal processing electronics (feedback bandwidth 30 MHz).A digital counter records the number of photon counter clicks generated in eachmeasurement interval [0, τ], during which time the feedback controller determinesthe feedback amplitude u∗t

(n[0,t ]

)in (4.44) via the accumulating count record n[0,t) .

Coherent states for discrimination are realized as τ = 20 μs optical pulses pro-duced by a computer-controlled polarization-maintaining fiber-optic intensity mod-ulator (FIM1) in the upper arm of the interferometer. The calibration parity be-tween desired and observed values of α in Figure 4.2a (squares) highlights theability to prepare arbitrary optical coherent states with amplitudes 0.1 u |α| u 1.Counting statistics for one such preparation with the mean photon number nα =〈α|N |α〉 = |α|2 are shown in the inset. The circles in the inset are a Poisson fit tothe counting data, for which �2 – 1 has been computed below 1ppm, showing thequantum noise-limitation of the state preparation.

The phase of the prepared coherent states φ(α) and the phase of the feedbackdisplacements φ(ut ) are implemented by the modulator (FPM) in the upper armof the interferometer. Without loss of generality, φ(α) = 0 was always chosen tosimplify the interpretation of the displacements.

4) From Encyclopedia of Laser Physics andTechnology, http://www.rp-photonics.com/:An external-cavity diode laser isa semiconductor laser based on a laser

diode chip which typically has one end anti-reflection coated, and the laser resonator iscompleted with, e.g., a collimating lens andan external mirror.


Fig. 4.2 (a) Quality of the preparation of theoptical coherent states (circles) and the con-trol of the displacements (squares). The insetshows a Poisson fit of the photon countingstatistics for α W 1, illustrating the quantumlimitation of the state preparation. (b) Qual-ity of the implementation of the controlledphase displacements (φut ). The calibration in(b) illustrates the low control voltages (V π)required to drive the apparatus, which allows

for high-bandwidth application of the mea-surement feedback control. Each data pointin (a) and (b) reflects a statistical ensembleof 100 000 replicate measurements, with errorbars given by the estimated sample standarddeviation. Source Cook et al. [40] (reprintedby permission from Macmillan PublishersLimited: [Nature] (Cook, R.L., Martin, P.J., andGeremia, J.M. 446, p. 774, 2007)).


Residual technical imperfections in the experiment result primarily from detec-tor dark counts (nd = 0.0078), interferometer phase noise (δφ W 8 mrad), and finiteextinction of the modulators.

4.7.3Closed-Loop Experiment: the Results

The measured probability of error versus mean photon number is shown in Fig-ure 4.3a. The squares correspond to the case in which the feedback is disabled. Thecircles correspond to the closed-loop measurement.

4.7.3.1Disabled FeedbackThe measurement reduces to direct photon counting. The observed probability oferror for discriminating between |0〉 and |α〉 faithfully reproduces shot noise (linewith squares) as a function of nα. To get a precise view of the experimental chal-lenge, data points were calculated using 100 000 optical pulses sampled randomlyfrom {|0〉, |α〉} with equal probability. The label |0〉 has been used to signify thedarkest field n0 W 0.008. The residual field appears to have a negligible effect, witha discrepancy of �2 – 1 = 1.13 ~ 10–5 between the photon counting data in Fig-ure 4.3a and the shot-noise error probability defined as P SN = e–nα/2.

Fig. 4.3 (a) The measured probability of errorversus mean photon number for both directphoton counting (squares) and the Cook–Martin–Geremia closed-loop measurementinterpreted using a Bayesian estimator thatassumes application of the optimal closed-loop control policy (circles) and that accountsfor experimental imperfections (triangles).All data points were obtained fromensembles of 100 000 measurementtrajectories, with error bars that reflect thesample standard deviation. The four tracesin (b) (1–4) depict a single-shot closed-loop

measurement trajectory. Attention shouldbe paid to several technical issues: first, thefinite dynamical range of the displacements(point A on the graph); second, the initialavalanche photodiode click is a timing sig-nal, not a real detection event (B); and, third,the apparent rise time is that of the monitorphotodiode not the feedback (C). SourceCook et al. [40] (reprinted by permissionfrom Macmillan Publishers Limited: [Nature](Cook, R.L., Martin, P.J., and Geremia, J.M.446, p. 774, 2007)).


4.7.3.2Closed-Loop MeasurementThe premise behind the closed-loop measurement is to displace the field to thevacuum in each shot and decide which state is present on the basis of the displace-ment applied to cancel the field. As the controller gains increased confidence in itsguess, it is better able to perform the correct nulling displacement. From (4.44), thedisplacement magnitude |u∗t | is inversely proportional to the time-dependent deci-sion uncertainty

√1 – e–nαt/τ. Performing the optimization of (4.43) reveals that it is

statistically optimal for the closed-loop measurement to reverse its state hypothesiswith each detector click during the counting interval [41, 42].

Many aspects of this closed-loop measurement are evident from the single-shottrajectory in Figure 4.3b. At t = 0 there is no reason to prefer one state, |0〉 or |α〉,over the other. But as more data become available, the controller refines its Bayesianestimate of the incoming optical state by updating the conditional probabilitiesP (ψ|�[0,t ] , u [0,t ]). The sequence of hypothesis reversals in the example closed-looptrajectory is denoted along with the measurement record in Figure 4.3b, trace 4. Asthe measurement record accumulates, the controller eventually settles on its final(correct) decision, which in this case is |α〉.

The data (circles) in Figure 4.3a demonstrate that the closed-loop state discrimi-nation procedure (alternating guesses between |0〉 and |α〉with each photon arrival)surpasses the shot-noise error probability for amplitudes |α| less than about one.The fundamental quantum limit is essentially saturated over a nontrivial region ofparameter space nα.

The data (circles) in Figure 4.3a were determined assuming that the optimalfeedback control policy in (4.44) was implemented perfectly by selecting the state|ψ〉 ∈ {|0〉, |α〉} to maximize the conditional probability P

(�[0,t ]|ψ, u∗[0,τ]

). This ap-

proach is clearly suboptimal owing to the technical imperfections in the controldisplacements just described. Actually, the raw measurement data should be rein-terpreted to account for deviations between the feedback actually performed in theexperiment u [0,τ] and the optimum policy u∗[0,τ] .

Owing to the nature of coherent states, the detection efficiency η (resulting fromthe combination of detector quantum efficiency ηd and optical efficiency ηe) fac-tors out of a comparison between the shot-noise and quantum limits [42]. For com-parison, the shot-noise error and quantum limits that would correspond to idealdetection (η = 1) have been plotted in Figure 4.3a. The intrinsic efficiency of theapparatus has been independently determined to be approximately η W 0.35.

The data (triangles) in Figure 4.3a reflect an analysis based on the true condi-tional probability P

(�[0,t ]|ψ, u∗[0,τ]

). It can be seen that the Cook–Martin–Geremia

procedure nearly achieves the quantum limit (for the actual detection efficiency)over the full range of coherent states investigated. It can be observed that even withdetection efficiency η = 0.35 the closed-loop measurement slightly outperforms theideal shot-noise error that would be achieved in a technically lossless experimentη = 1 for photon numbers n < 0.2.

4.8 Conclusion 67

4.8Conclusion

Quantum feedback can be viewed as manipulating the outcome statistics of thenumber operator N. In the absence of feedback, the detailed measurement recordconsisting of photon arrival times �[0,τ] = (t1, t2, . . . , tn) provides no more informa-tion than the total number n: Poisson processes are stationary in time, but withfeedback the significance of each click depends on when it occurs, even though thefield is described by some coherent state at each point in time. The optimal feed-back policy applies displacements in a manner that extracts as much informationout of each photon arrival as possible. It is in this manner that shot noise can besurpassed to achieve the fundamental quantum limit over a nontrivial range of |α|.

Furthermore, the Cook–Martin–Geremia procedure that has been described hereappears as less demanding on the measurement resources needed to achieve op-timal statistics than a direct implementation of a cat state: at no point in time hasa superposition between optical coherent states been generated.

It should be added to these comments that very recently Wittman et al. [48] ex-perimentally realized a new quantum measurement that detects binary optical co-herent states with fewer errors than the homodyne and the Kennedy receiver for allamplitudes of the coherent states. Although the scheme is not capable of achievingthe Helstrom bound, the implementation discriminates between binary coherentstates with an error probability lower than the optimal Gaussian receiver, namely,the homodyne receiver. For more details on the theoretical background, see [49].

69

5Coherent States: a General Construction

5.1Introduction

We now depart from the standard situation and present a general method of con-struction of coherent states, starting from a few observations on the structure ofthese objects as superpositions of eigenstates of some self-adjoint operator, as wasthe harmonic oscillator Hamiltonian for the standard coherent states. It is theessence of quantum mechanics that this superposition has a probabilistic flavor.As a matter of fact, we notice that the probabilistic structure of the standard co-herent states involves two probability distributions that underlie their construction.There are, in a sort of duality, a Poisson distribution ruling the probability of detect-ing n excitations when the quantum system is in a coherent state |z〉, and a gammadistribution on the set C of complex parameters, more exactly on the range R+

of the square of the radial variable. The generalization follows that duality scheme.Given a set X, equipped with a measure ν and the resulting Hilbert space L2(X , ν) ofsquare-integrable functions on X, we explain how the choice of an orthonormal sys-tem of functions in L2(X , ν), precisely {φ j (x) | j ∈ J },

∫X φ j (x)φ j ′ (x) ν(dx) = δ j j ′ ,

carrying a probabilistic content,∑

j∈J |φ j (x)|2 = 1, determines the family of co-herent states |x〉 =

∑j φ j (x)|φ j 〉. The relation to the underlying existence of a re-

producing kernel space will be briefly explained. This will be the guideline rulingthe content of the subsequent chapters concerning each family of coherent statesexamined (in a generalized sense).


70 5 Coherent States: a General Construction

5.2A Bayesian Probabilistic Duality in Standard Coherent States

5.2.1Poisson and Gamma Distributions

In the first chapters we reviewed the standard coherent states,

|z〉 =∞∑

n=0

e–|z|2/2 zn√

n!|n〉 , (5.1)

in their principal physical and mathematical aspects. The key ingredients of theseobjects are

(i) The original set of parameters, namely, the complex plane C, equipped withits Lebesgue measure d2z/π. This set may or not be given a phase spacestatus. The latter takes place within the framework of the classical motionof a particle on the real line. Otherwise, we could think about the set oftime–frequency parameters in signal analysis.

(ii) Another set of parameters, namely, the natural numbers N == J . Withinthe context of quantum physics, this set labels the possible issues of a cer-tain experiment, such as counting the number of elementary excitations or“quanta” of a certain physical entity.

In relation with these two sets, the quantity

pn(|z|2) def= |〈n|z〉|2 = e–|z|2 |z|2n

n!(5.2)

leads to the following two interpretations.

(i) The discrete probability distribution n �→ pn(|z|2) is a Poisson distribu-tion on N with parameter u == |z|2 equal to the average number of oc-currences. Clearly, this probability concerns experiments performed on thesystem within some experimental protocol, say, E , and might be viewed asa stochastic model. Note that in the Poisson experiment, state preparationis determined only up to the unknown parameter |z|2.

(ii) The continuous distribution z �→ pn(|z|2) on C with measure d2z/π isa gamma distribution when it is considered with respect to the u = |z|2variable, with the Lebesgue measure du on R+, and with n as a shape pa-rameter.

The duality of interpretations that appears here is reminiscent of a similar duali-ty observed in the theory of Bayesian statistical inference [50, 51]. In that context,the Poisson experiment would be performed and the Bayesian method used to ob-tain information about the behavior of the unknown parameter |z|2 in the form ofa conditional probability distribution on the parameter space R+ given an observedPoisson experimental value [52–55].

5.2 A Bayesian Probabilistic Duality in Standard Coherent States 71

5.2.2Bayesian Duality

We have already encountered Bayesian probabilities in the previous chapter. Let ustry to become more familiar with this Bayesian context [55] (see Appendix A).

Suppose we have an experiment for which we postulate an experimental mod-el in the form of a (one-parameter) family n �→ P (n, u) of discrete probabilitydistributions, where the unknown parameter u takes values in a measure space(U , m(du)). Suppose that the experiment has been performed, producing the re-sult k. In Bayesian parlance, P (k, u) as a function of u is called the “likelihoodfunction” (see Appendix A) and m(du) is called the “prior” measure on the pa-rameter space U. Then we have a conditional probability density function f on theparameter space via an “inverse probability” formula where f (u; k)du is propor-tional to P (k, u) m(du). In Bayesian language, this final probability distribution onU is called the “posterior” probability distribution.

Thus, we have a duality of two probability distributions. We have the original dis-crete family indexed by parameter u, wherein, if the “true” value of u were known,it would serve as a predictive model for experimentally obtained data. Then wehave the Bayesian posterior probability distribution on the (continuous) parameterspace, which, if an experimental value were known, would serve as an “inferred” or“retrodictive” probability distribution for the unknown parameter.

As can be seen from the Poisson–gamma duality described above, the choice ofmeasure space (U , m(du)) and coherent states (5.1) along with the expression (5.2)leads to a similar duality of the two probability distributions.

5.2.3The Fock–Bargmann Option

Now, the exclusive character of the possible outcomes n ∈ N or f (n) in the mea-surement of some quantum observable f (N ), N being the number operator, isencoded by the orthogonality between elements of the set of functions

en(z) = e–|z|2/2 zn√

n!.

These functions are complex square roots of the probability distributions in bothsenses above: |en (z)|2 = pn(|z|2).

They are in one-to-one correspondence with the Fock or number states |n〉. Theclosure of their linear span within the Hilbert space L2(C, d2z/π) is a sub-Hilbertspace, say,FBe . The latter is reproducing, isomorphic to the Fock–Bergmann spaceintroduced in Chapter 2, and also to the Fock space H generated by the numberstates.


5.2.4A Scheme of Construction

In summary, what do we have?

(i) A set of parameters or data (in a classical sense), C, for example, the set ofinitial conditions for the motion of a particle on the line, equipped by theLebesgue measure d2z/π.

(ii) The “large” Hilbertian arena, L2(C, d2z/π), which could be viewed as thespace of images (with finite energy) in a signal analysis framework.

(iii) An orthonormal set of functions, {en(z) ∈ L2(C, d2z/π), n ∈ N}, whichobeys the probabilistic identity

∞∑n=0

|en(z)|2 = 1 . (5.3)

(iv) A resulting family of states, the standard coherent states |z〉, in FBe (or inH), with en(z) as the orthogonal projection on the basis element en (or |n〉).

(v) The identity (5.3) entails the normalization 〈z|z〉 = 1, and the orthonormal-ity of the en(z)’s entails the resolution of the identity in FBe (or inH).

In the next section, we will extend this scheme of construction to any measureset X.

5.3General Setting: “Quantum” Processing of a Measure Space

In a first approach, one notices that quantum mechanics and signal analysis havemany aspects in common. As a departure point of their respective formalism, onefinds a raw set X of basic parameters, which we denote generically by X = {x ∈ X }.This set may be a classical phase space in the former case, like the complex planefor the particle motion on the line, whereas it may be a time–frequency plane (forGabor analysis) or a time-scale half-plane (for wavelet analysis) in the latter one.Actually, it can be any set of data accessible to observation. For instance, it mightbe a temporal line or the circle or some interval. The minimal significant struc-ture one requires so far is the existence of a measure μ(dx) on X. As a measurespace, (X , μ), or simply X, could be given the name of an observation set, and theexistence of a measure provides us with a statistical reading of the set of all mea-surable real- or complex-valued functions f (x) on X: it allows us to compute, forinstance, average values on subsets with bounded measure. Actually, both theo-ries deal with quadratic mean values, and the natural framework of study is theHilbert space L2(X , μ) of all square-integrable functions f (x) on the observationset X:

∫X | f (x)|2 μ(dx) < ∞. The function f is referred to as a finite-energy signal

in signal analysis and might be referred to as a (pure) quantum state in quantum

5.3 General Setting: “Quantum” Processing of a Measure Space 73

mechanics. However, it is precisely at this stage that the “quantum processing” ofX differs from signal processing in at least three points:

(i) not all square-integrable functions are eligible as quantum states,

(ii) a quantum state is defined up to a nonzero factor,

(iii) among the functions f (x), those that are eligible as quantum states andthat are of unit norm,

∫X | f (x)|2 μ(dx) = 1, give rise to a probabilistic in-

terpretation: the correspondence X ⊃ Δ �→∫

Δ | f (x)|2μ(dx) is a probabilitymeasure, which is interpreted in terms of localization in the measurableset Δ and which allows one to determine mean values of quantum observ-ables, which are (essentially) self-adjoint operators defined in a domain thatis included in the set of quantum states.

The first point lies at the heart of the quantization problem (to which we devotethe second part of the book): what is the more or less canonical procedure allowingus to select quantum states among simple signals? In other words, how should weselect the true (projective) Hilbert space of quantum states, denoted by K, that is,a closed subspace of L2(X , μ), or equivalently the corresponding orthogonal projec-tor IK?This problem can be solved if one finds a map from X to the Hilbert space K,x �→ |x〉 ∈ K, defining a family of states {|x〉}x∈X obeying the following twoconditions:

• normalization

〈x |x〉 = 1 , (5.4)

• resolution of the unity in K∫X|x〉〈x | ν(dx) = IK , (5.5)

where ν(dx) is another measure on X, usually absolutely continuous with re-spect to μ(dx): this means that there exists a positive measurable function h(x)such that ν(dx) = h(x)μ(dx).

The explicit construction of such a set of vectors as well as its physical relevance areclearly crucial. It is remarkable that signal and quantum formalisms meet again onthis level, since the family of states is called, in a wide sense, a wavelet family [56]or a coherent state family [11] according to the practitioner’s field of interest. Twomethods for constructing such families are generally in use. The first one restsupon group representation theory: a specific state or probe, say, |x0〉, is transport-ed along the orbit {|g · x0 == x〉 , g ∈ G} by the action of a group G for whichX is a homogeneous space. Irreducibility (Schur lemma) and unitarity conditions,combined with square integrability of the representation in some restricted sense,automatically lead to properties (5.4) and (5.5). Various examples of such group-the-oretical constructions are given in [10, 11]. The second method has a wave-packet


flavor in the sense that the state |x〉 is obtained from the superposition of elementsin a fixed family of states {|λ〉}λ∈Λ that is total inH:

|x〉 =∫

Λ|λ〉 σ(x , dλ) . (5.6)

Here, the complex-valued x-dependent measure σ has its support Λ contained inthe support of the spectral resolution E (dλ) of a certain self-adjoint operator A, andthe |λ 〉’s are precisely eigenstates of A: A|λ 〉 = λ|λ 〉. The choice of the operator Ais ruled by the existence of the experimental device that allows us to measure allpossible and exclusive issues λ ∈ Sp(A) of the physical quantity precisely encodedby A. The eigenstates can be understood in a distributional sense so as to put intothe game of the construction portions belonging to the possible continuous part ofthe spectrum of A. Examples of such wave-packet constructions are given in [57–59], and here we will follow a similar procedure.

For pedagogical purposes, we now suppose that A is a self-adjoint operator ina Hermitian space (with finite dimension, say, N + 1) or a separable Hilbert space(with infinite dimension N = ∞), say, H, of quantum states or of something else,it does not matter. Let us assume that the spectrum of A has only a discrete com-ponent, say, {an , 0 u n u N}. Normalized eigenstates of A are denoted by |en〉 andthey form an orthonormal basis ofH. Next, suppose that the basis {|en〉}0unuN is inone-to-one correspondence with an orthonormal set {φn (x)}0unuN of elements ofL2(X , μ). The generic H could be the Hilbert space K, subspace of L2(X , μ), but wekeep our freedom in the choice of realization ofH. Furthermore, and this a decisivestep in the wave-packet construction, we assume, in the case N =∞, that

0 < N (x) ==∑

n

|φn(x)|2 <∞ almost everywhere on X . (5.7)

Then, the states inH,

|x〉 ==1√N (x)

∑n

φn(x)|en〉 , (5.8)

satisfy both of our requirements (5.4) and (5.5). Indeed, the normalization is en-sured because of the orthonormality of the set {|en〉} and the presence of the nor-malization factor (5.7). The resolution of the unity IH in H holds by virtue of theorthonormality of the set {φn(x)} if ν(dx) is related to μ(dx) by

ν(dx) = N (x)μ(dx) . (5.9)

Indeed, we have∫X|x〉〈x | ν(dx) =

∑n,n′

|en〉〈en′ |∫

Xφn(x) φn′ (x) μ(dx)

=∑

n

|en〉〈en| = IH ,

where we have used the orthonormality of the φn ’s and the resolution of the unity

5.3 General Setting: “Quantum” Processing of a Measure Space 75

inH obeyed by the orthonormal basis |en〉. Note that Hilbertian superposition (5.8)makes sense provided that the set X is equipped with a mild topological structurefor which this map is continuous.

A direct and important consequence of the resolution of the unity is the existenceof a positive-operator-valued measure [11, 35, 60, 61] on the measure space (X ,F ) fora σ-algebra F of subsets of X,

F � Δ �→∫

Δ|x〉〈x | N (x) μ(dx) ∈ L(H)+ . (5.10)

The resolution of the unity in H can alternatively be understood in terms of thescalar product 〈x |x ′〉 of two states of the family. Indeed, (5.5) implies that, with anyvector |φ〉 inH, one can isometrically associate the function

φ(x) ==√N (x)〈x |φ〉 (5.11)

in L2(X , μ). In particular,

φn(x) =√N (x)〈x |en〉 , (5.12)

and this justifies our hypothesis that the basis {|en〉} is in one-to-one correspon-dence with the orthonormal set {φn(x)}. Hence, H is isometric to the Hilbert sub-space K of L2(X , μ), closure of the linear span of the φn(x)’s defined by (5.12).

Now, by direct application of the resolution of the unity, the function φ(x)in (5.11), as an element of K, obeys

φ(x) =∫

X

√N (x)N (x ′)〈x |x ′〉φ(x ′) μ(dx ′) . (5.13)

Hence, K is a reproducing Hilbert space with kernel

K(x , x ′) =√N (x)N (x ′)〈x |x ′〉 , (5.14)

and the latter assumes finite diagonal values (almost everywhere), K(x , x) = N (x),by construction.

A last point of this construction of the space of quantum states concerns its sta-tistical aspects, already pointed out in Section 5.2. There is indeed an interplaybetween two probability distributions:

• For almost each x, a discrete distribution,

n �→ |φn (x)|2N (x)

. (5.15)

Like for the standard coherent states, this probability could be consideredas concerning experiments performed on the system within some exper-imental protocol, say, E , to measure the spectral values of the “quantumobservable” A.


• For each n, a “continuous” distribution on (X , μ),

X � x �→ |φn (x)|2 . (5.16)

Here, we observe the previously encountered Bayesian duality. There are two in-terpretations. The resolution of the unity verified by the “coherent” states |x〉 in-troduces a preferred prior measure on the observation set X, which is the set of pa-rameters of the discrete distribution, with this distribution itself playing the role ofthe likelihood function. The associated discretely indexed continuous distributionsbecome the related conditional posterior distribution.

Hence, a probabilistic approach to experimental observations concerning Ashould serve as a guideline in choosing the set of the φn(x)’s.

Coming back to the standard coherent states, one briefly states the way in whichtheir construction fits perfectly the above procedure:

• the observation set X is the classical phase space R2 � C = {x == z =1√2(q + i p)} of a particle with one degree of freedom, more exactly the set of

initial conditions (position and velocity) for motions of the particle on theline,

• the measure on X is Lebesgue, μ(dx) = 1π d2z,

• the Hilbert spaceH is the Fock space with orthonormal basis the eigenstates|n〉 of the number operator A == N ,

• the functions φn(x) are the normalized powers of the complex variable zweighted by a Gaussian factor

φn(x) == e– 12 |z|2

zn√

n!,

• The Hilbert subspace K is the Fock–Bargmann space of all square-inte-

grable functions that are of the form φ(z) = e– |z|2

2 g (z), where g (z) is ana-lytically entire,

• the coherent states read

|z〉 = e– |z|2

2

∑n

zn√

n!|n〉 .

5.4Coherent States for the Motion of a Particle on the Circle

We now illustrate the general construction given in the previous section byconsidering an elementary although nonstandard example of coherent states.These states were proposed independently by De Bièvre–González [62], Kowalski–Rembielinski–Papaloucas [63–66], and González–Del Olmo [67] through approach-es that substantially differ from the method illustrated in this chapter.

5.4 Coherent States for the Motion of a Particle on the Circle 77

The observation set X is the phase space of a particle moving on the circle. Itis the Cartesian product of the set of angular positions with the set of all possiblevelocities. It is naturally described by the cylinder

S1 ~ R = {x == (�, J) | 0 u � < 2π, J ∈ R} . (5.17)

The velocity or momentum or “angular momentum” J (depending on the choice ofunits for the physical constants involved in the description of the motion) and theangle � are canonically conjugate variables and dJ d� is the invariant measure onthe phase space. So, we choose μ(dx) = 1

2π dJ d� as a measure on X.The functions φn(x), for n ∈ Z, are suitably weighted Fourier exponentials:

φn(x) =( ε

π

)1/4e– ε

2 ( J–n)2ein� , n ∈ Z , (5.18)

where ε > 0 can be arbitrarily small. This parameter could be viewed as the analogof the inverse of the square of the Planck constant, since J, as conjugate to an angle,could be considered as an angular momentum or an action. Actually ε representsa regularization. Notice that the continuous distribution x �→ |φn(x)|2 is the normallaw centered at n (for the momentum variable J).

The normalization factor

N (x) == N ( J) =

√επ

∑n∈Z

e–ε( J–n)2<∞ (5.19)

is a periodic train of normalized Gaussians, and is proportional to an elliptic thetafunction. Note that the Poisson summation formula leads to the alternative form

N ( J) =∑n∈Z

e2πin J e– π2ε n2

. (5.20)

Hence, we derive its regular behavior at ε = 0: limε→0N ( J) = 1.Choosing a separable Hilbert spaceH with orthonormal basis {|en〉 , n ∈ Z}, we

now have all the ingredients to define the coherent states on the circle:

| J, �〉 =1√N ( J)

( επ

)1/4 ∑n∈Z

e– ε2 ( J–n)2

e–in�|en〉 . (5.21)

For instance, the states |en〉 can be considered as Fourier exponentials ein� form-ing the orthonormal basis of the Hilbert space L2(S1). They are the spatial modesin this representation, and are eigenstates of the (angular momentum) operatorJ = –i∂/∂� with eigenvalues n ∈ Z.

As already mentioned, they could also be defined as elements of the reproduc-ing kernel Hilbert space K, subspace of L2

(R ~ S1, 1

2π d J d�)

with the set {φn ==|φn〉 , n ∈ Z} as an orthonormal basis,

| J, �〉 =1√N ( J)

( επ

)1/4 ∑n∈Z

e– ε2 ( J–n)2

e–in�|φn〉 , (5.22)


The coherent states (5.21) or (5.22) are, as expected, normalized and resolve theunity in the Hilbert spaceH (or K ⊂ L2(X , 1

2π d J d�)):

〈 J , �| J , �〉 = 1 ,∫

X

d J d�2π

| J, �〉〈 J , �| = Id . (5.23)

The “quantum” processing of the observation set R ~ S1 is hence achieved byselecting in the (modified) Hilbert space L2(S1 ~ R,

√επ

12π e–ε J2

d J d�) all Laurentseries in the complex variable z = eε J–i�. The overlap of two coherent states canalso be expressed in terms of an elliptic theta function:

〈 J , �| J ′, �′〉 =e– ε( J– J ′)2

4√N ( J)N ( J ′)

( επ

)1/2 ∑n∈Z

e–ε(

J+ J ′2 –n

)2

ein(�′–�) . (5.24)

Again the Poisson summation formula leads to the other form:

〈 J , �| J ′, �′〉 =e– ε( J– J ′)2

4√N ( J)N ( J ′)

ei( J+ J ′2 )(�′–�)

∑n∈Z

e– 14ε (�′–�+2πn)2

eπin( J+ J ′) . (5.25)

It is easily proven that as ε→ 0 this expression goes to zero if � =/ �′ and goes to 1if � = �′.

5.5More Coherent States for the Motion of a Particle on the Circle

Our choice of the Gaussian distribution for the J variable in the construction ofthe coherent states for the motion on the circle was essentially determined by theexisting literature on the subject [63, 68]. Now, it is interesting to note that we canreplace this Gaussian distribution by any (possibly even) probability distributionR ∈ J �→ π( J ) such thatN ( J) =

∑n π( J – n) <∞. Let us be more precise. For the

measure on the cylinder X, we still choose μ(dx) = 12π d J d�. The functions φn(x),

for n ∈ Z, are now given by

φn(x) =1√ν

√π( J – n) ein� , n ∈ Z , (5.26)

with ∫ +∞

–∞π( J) d J = ν .

The corresponding family of coherent states on the circle reads as

| J , �〉 =1√N ( J)

1√ν

∑n∈Z

√π( J – n) e–in�|φn〉 . (5.27)

They are normalized and resolve the unity.

79

6The Spin Coherent States

6.1Introduction

The spin or SU (2) coherent states form the second most known family of coherentstates. They were introduced in the early 1970s by Radcliffe [69], Gilmore[70, 71],and Perelomov [72]. They also bear the name atomic or Bloch coherent states. Thisdiversity of appellations reflects the range of domains in quantum physics wherethese objects play some role. The way in which we will introduce these states fol-lows the probabilistic and Hilbertian scheme explained in the previous chapter.The central object or observation set is now the two-dimensional unit sphere S2,equipped with the usual rotationally invariant measure. As orthonormal systemsnecessary for the construction of coherent states, sets of special functions calledspin spherical harmonics will be selected. These functions are closely related to theunitary irreducible representations of the rotation group SO (3) and its coveringSU (2) and imply the rich set of mathematical properties described in the presentchapter.

6.2Preliminary Material

Within the classical mechanics framework, the free rotation of a rigid body is char-acterized by its angular momentum JCl, where the superscript “Cl” stands for “clas-sical.” If the norm ‖JCl‖ or the rotational kinetic energy ‖JCl‖2/2I is conserved, the(pseudo)-vector JCl describes a sphere of radius JCl = ‖JCl‖.

The quantum version of this angular momentum or classical spin is well known.At given j = 0, 1

2 , 1, 32 , . . . let us consider the Hermitian (== finite-dimensional

Hilbert) space

H j def= C2 j+1 , (6.1)

with the set of kets {| j , m〉 , m = – j , – j + 1, . . . , j – 1, j}, 〈 j , m′| j , m〉 = δmm′ , asan orthonormal basis.


80 6 The Spin Coherent States

Writing J = ( Jx , J y , Jz ) for the angular momentum operator, we have, with� = 1, the commutation rules

[ Jx , J y ] = i Jz , [ Jz , Jx ] = i J y , [ J y , Jz ] = i Jx . (6.2)

The basis vectors | j , m〉 are selected as common eigenvectors of the commuting

pair of operators Jz , J2 def= J · J:

J2 | j , m〉 = j ( j + 1) | j , m〉 , Jz | j , m〉 = m | j , m〉 . (6.3)

The ladder operators J±def= Jx ± i J y act on the basis considered as

J+| j , m〉 =√

( j – m)( j + m + 1) | j , m + 1〉 , (6.4)

J–| j , m〉 =√

( j + m)( j – m + 1) | j , m – 1〉 , (6.5)

J+| j , j〉 = 0 , J–| j , – j〉 = 0 .

All elements of the basis of H j are readily derived from those “extremal” states| j , j〉 and | j , – j〉.

| j , m〉 =

√( j – m)!

( j + m)!2 j !( J+) j+m | j , – j〉 =

√( j + m)!

( j – m)!2 j !( J–) j–m | j , j〉 .

(6.6)

6.3The Construction of Spin Coherent States

Following the guideline explained in the previous chapter, we start from the sphere

as the observation set: X = S2 = {x def= r def= (θ, φ) ∈ S2} in spherical coordi-

nates. This set is equipped with the rotationally invariant measure μ(dx) def= dr =sin θdθdφ.5) We then proceed to the selection of the following orthonormal set inL2(S2, dr ):

φ j ,m (r) =1√4π

√(2 j

j + m

)(cos

θ2

) j+m (sin

θ2

) j–m

e–i( j–m)φ . (6.7)

We immediately check by using a simple binomial expansion that

N (r) =j∑

m=– j

|φ j,m (x)|2 =1

4π. (6.8)

5) In the original definitions of coherentstates, the measure is normalized,

μ(dx) def= dr4π = sin θdθdφ/4π, and so our

definitions of spin coherent states differ bya factor 1/

√4π.

6.3 The Construction of Spin Coherent States 81

We naturally define the Hilbert space H as H = H j . Then, the “spin” or “Bloch”or “atomic” coherent states |r〉 ∈ H6) are given by

|r〉 = |θ, φ〉 =j∑

m=– j

√(2 j

j + m

)(cos

θ2

) j+m (sin

θ2

) j–m

ei( j–m)φ| jm〉 .

(6.9)

By construction, they are normalized and solve the identity Id == IH

〈r |r〉 = 1 ,∫

S2

dr4π|r〉〈r | = Id . (6.10)

There exists another useful parameter for labeling the elements of the familyof spin coherent states. The number � = tan θ

2 eiφ parameterizes the (Riemann)sphere through a stereographic projection onto C of unit vectors (θ/2, φ) with re-spect to the south pole:

S2 � r = (x , y , z) �→ � =x + i y1 + z

∈ C ⇔

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩x =

2R(�)1 + |�|2

y =2I(�)

1 + |�|2

z =1 – |�|21 + |�|2 .

(6.11)

In terms of it, spin coherent states read as

C � � �→ |r〉 == |�〉 =j∑

m=– j

√(2 j )!

( j – m)!( j + m)!� j–m

(1 + |�|2) j | j , m〉 , (6.12)

and they resolve the unity inH j in the following way:

2 j + 1π

∫C

d2�(1 + |�|2)2 |�〉〈�| = Id . (6.13)

One should notice here the similarity with the standard coherent states

C � z �→ |z〉 =∞∑

n=0

e– |z|2

2zn√

n!|n〉 , (6.14)

which are easily obtained from the spin coherent state at the limit of high spinsthrough a contraction process. The latter is carried out through a scaling of thecomplex variable �, namely, z =

√N �, with N = 2 j and n = j – m, | j , m〉 == |n〉,

and the limit N →∞:∣∣∣∣� =z√N

⟩spin

→N→∞

|z〉 . (6.15)

6) Gilmore, Radcliffe, Perelomov


6.4The Binomial Probabilistic Content of Spin Coherent States

For a given polar angle θ, consider the Bernoulli process, that is, a sequence ofn = 2 j independent trials, with two possible outcomes for each trial:

1. “+” (win) with probability p = cos2 θ2

,

2. “–” (loss) with probability 1 – p = sin2 θ2

.

Thus, we are certain to always win at the north pole of the sphere and to alwayslose at the south pole! At the equator the chances are equal, p = 1/2. Then theprobability of winning after k = j + m trials is the discrete binomial distributionwith parameter p = cos2 θ

2 :

k → p (n)k =

(nk

)pk(1 – p)n–k = 4π|φ j ,m (θ, φ)|2 = |〈 j , m|θ, φ〉|2 . (6.16)

In duality, we have the continuous binomial distribution with parameters k = j +mand n = 2 j

p = cos2 θ2→ p (n)

k =

(nk

)pk (1 – p)n–k = 4π|φ j ,m (θ, φ)|2 = |〈 j , m|θ, φ〉|2 .

(6.17)

We recover here the existence of the Bayesian duality of interpretations. The res-olution of the unity verified by the spin coherent states introduces a preferred priormeasure on the parameter space of polar angles θ ∈ [0, π] of the discrete distri-bution, with this distribution itself playing the role of the likelihood function [55].The associated discretely indexed continuous distributions become the related con-ditional posterior distributions. We will illustrate this probabilistic content in thenext chapter with the example of a quantum spin in a magnetic field.

6.5Spin Coherent States: Group-Theoretical Context

The way the operators J± and Jz act on the basis elements of the Hermitian spaceH j is the infinitesimal version of the action of a unitary irreducible representationof the rotation group SO (3) or of its covering SU (2).

At this point, we recall that any proper rotation in space is determined by a unitvector n defining the rotation axis and a rotation angle 0 u ω < 2π about the axis.

6.5 Spin Coherent States: Group-Theoretical Context 83

��

O

r′��

�ω

n

��

r

The action of such a rotation,R(ω, n), on a vector�r is given by

�r ′ def= R(ω, n) ·�r = �r · n n + cos ω n ~ (�r ~ n) + sin ω (n ~ �r ) , (6.18)(0, �r ′

)=

(cos

ω2

, sinω2

n)

(0,�r )(

cosω2

, – sinω2

n)

, (6.19)

the latter being expressed in scalar–vector quaternionic form. Let us give here theminimal material necessary to understand the quaternionic formalism (Hamil-ton [73], 1843). We recall that the quaternion field as a multiplicative group isH � R+ ~ SU (2). The correspondence between the canonical basis of H � R4 (1 ==e0, e1, e2, e3) and the Pauli matrices is ei ↔ (–1)i+1σi , with i = 1, 2, 3. Hence, the2 ~ 2 matrix representation of these basis elements is the following:(

1 00 1

)↔ e0 ,

(0 ii 0

)↔ e1 == ı ,(

0 –11 0

)↔ e2 == j ,

(i 00 –i

)↔ e3 == k .

(6.20)

Any quaternion decomposes as q = (q0,�q) (or qaea , a = 0, 1, 2, 3) in scalar–vectornotation (or in Euclidean metric notation). We also recall that the multiplication lawexplicitly reads in scalar–vector notation as qq′ = (q0q′0 –�q ·�q′, q′0�q+q0�q′+�q ~ �q′). The(quaternionic) conjugate of q = (q0,�q) is q = (q0, –�q), the squared norm is ‖q‖2 = qq,and the inverse of a nonzero quaternion is q–1 = q/‖q‖2.

The rotations R(ω, n), direct about their axis n, are represented in H j by theunitary operator:

R(ω, n) –→ exp (–iω n · J) . (6.21)

In particular, for a given unit vector

r = (sin θ cos φ, sin θ sin φ, sin θ cos θ) def= (θ, φ) ,

0 u θ u π , 0 u φ < 2π , (6.22)

one considers the specific rotation Rr that brings the unit vector pointing to thenorth pole, k = (0, 0, 1), to r ,

r = Rr (θ, uφ) , uφdef= (– sin φ, cos φ, 0) , (6.23)


and the resulting unitary operator inH j :

exp (–iθ uφ · J) == D j (Rr ) = e(� J+–� J–) , � def= –θ2

e–iφ . (6.24)

��

��

��

��

O

r

uφ

� �

θ

k�

Spin coherent states then result from the “rotational” transport by D j (Rr ) of theextremal state | j , j〉.

|x == r〉 = D j (Rr ) | j , j〉 . (6.25)

As is indicated in (6.24), they are labeled as well by the complex numbers � in theclosed disk centered at the origin and with radius π/2. The proof of (6.25) restsupon the application of the Baker–Campbell–Hausdorff or Zassenhaus formula,7)

which allows us to “disentangle” the exponential of a sum of operators:

e(� J+–� J–) = e–� J+ eln(1+|�|2) Jz e� J– = e� J– e– ln(1+|�|2) Jz e–� J+ , (6.26)

where � is the alternative parameter introduced in (6.11):

� = –�|�|

sin |�|cos |�| = tan

θ2

eiφ .

Hence, |r〉 is equal to

|r〉 = e� J– e– ln(1+|�|2) Jz e–� J+ | j , j〉︸︷︷︸=| j , j〉︸︷︷︸

=(cos θ2 )2 j | j , j〉

=

(cos

θ2

)2 j

e� J– | j , j〉

=j∑

m=– j

√(2 j )!

( j – m)!( j + m)!

(cos

θ2

) j+m (sin

θ2

) j–m

ei( j–m)φ | j , m〉 .

(6.27)

Let us now describe the covariance property of the spin coherent states. Undera space rotation R(ω, n) represented by a (2 j + 1) ~ (2 j + 1) unitary matrix acting

7) Let X and Y be two matrices and[X , Y ] = Z be their commutator. Then,e(X +Y ) = eX eY e–Z/2 e([X ,Z ]+2[Y ,Z ])/6 · · ·

6.5 Spin Coherent States: Group-Theoretical Context 85

onH j , R(ω, n) –→ e–iω n·J == D j (R(ω, n)), spin coherent states transform as

e–iω n·J |r〉 = ei j A(k,r ,r ′ ) |r ′〉 , r ′ = R(ω, n) · r , (6.28)

where A(k, r , r ′) denotes the (symplectic) oriented area of the geodesic sphericaltriangle with vertices k, r , r ′.

This is a manifestation of the “quasi-classical” behavior of the spin coherentstates, “quasi” since there appears a phase factor with topological interpretationin terms of U (1) fiber, originating in the Hopf fibration of the three-dimension-al sphere S3 → S2. We recall here that the Lie group SU (2) is, as a manifold,identical to S3. Precisely, the rotation Rr can be associated with the element � =(�0, �1, �2, �3) of S3, which, viewed as a unit quaternion, acts on the north pole ofthe 2-sphere S2 through formula (6.24):

� (0, 0, 0, 1) � = (0, x1, x2, x3) = r . (6.29)

This gives rise to three quadratic relations,

x1 = 2(�0�2 + �1�3) ,

x2 = 2(�2�3 – �0�1) ,

x3 = 2(�20 + �2

3 – �21 – �2

2) ,

(6.30)

which exemplifies the so-called Hopf map � �→ r . The Hopf inverse of a pointof S2 is a circle ~= S1 ⊂ S3. The Hopf inverse of a circle ~= S1 ⊂ S2 is a torus~= S1 ~ S1 ⊂ S3, and so on.

The topological factor in (6.28) appears as well in the overlap of two spin coherentstates

〈r |r ′〉 =

(1 + r · r ′

2

) j (ei A(k,r ,r ′ )

) j. (6.31)

This overlap reads in terms of complex parameters �, �′, as

〈�|�′〉 =(1 + ��′)2 j

(1 + |�|2) j (1 + |�′|2) j . (6.32)

Note that the orthogonality holds for antipodal points: 〈r | – r〉 = 0.The spin coherent states saturate appropriate inequalities like standard coherent

states do for Heisenberg inequalities. Those inequalities concern the triplet of op-erators Jx , J y , Jz . From [ Jx , J y ] = i Jz , one derives the inequalities for the productof variances calculated in an arbitrary state |ψ〉 ∈ H j :

Δ Jx Δ J y v12|〈 Jz〉| . (6.33)

Precisely, the equality is achieved when |ψ〉 is chosen to be a spin coherent state|r〉 , r ∈ S2. The proof consists in checking first the equality in (6.33) for |k〉 ==| j , j〉 and next using the unitarity of (6.28) of the operator representing the rotationbringing k to an arbitrary r .


6.6Spin Coherent States: Fock–Bargmann Aspects

Like for the standard coherent states, the spin coherent states formalism allowsa Fock–Bargmann realization (up to a nonanalytical factor) of the Hermitian spaceH j . The coherent state parameter used here is naturally the complex � parameter-izing the Riemann sphere through (6.11)

H j � |ψ〉 �→ Ψ(�) def=1√4π〈�|ψ〉 . (6.34)

From the expression of |�〉, the function Ψ(�) assumes the form

Ψ(�) = (1 + |�|2)– j P (�) , (6.35)

where P (�) is an analytical polynomial of degree u 2 j . The space of such functionsΨ(�) is the finite-dimensional Hilbert space, denoted byK j , of dimension 2 j +1, ofall functions of the type (6.35) that are square-integrable with respect to the scalarproduct

〈Ψ1|Ψ2〉 def=2 j + 1

π

∫C

d2�(1 + |�|2)2 Ψ1(�) Ψ2(�) . (6.36)

The mapH j � |ψ〉 �→ Ψ(�) ∈ K j is an isometry. It provides an “analytical” readingthrough a reproducing Hilbert space, of the quantum spin states inH j = C2 j+1 (wecould of course forget the quantum spin context and apply this isometry to anyHermitian space). Under this isometry, the orthonormal basis elements | j , m〉 inH j are in one-to-one correspondence with the functions

u jm (�) =

√2 j !

( j – m)!( j + m)!� j–m (1 + |�|2)– j (6.37)

that form an orthonormal basis of K j .Note that we could also consider the holomorphic purely polynomial realization

of the Hermitian space K j . This finite-dimensional Fock–Bargmann space is de-noted by FB j , consistent with the notation in Section 2.2.5, and is the space of allholomorphic polynomials P (�), of degree u 2 j equipped with the scalar product

〈P 1|P 2〉 def=2 j + 1

π

∫C

d2�(1 + |�|2)2 j+1 P 1(�) P 2(�) . (6.38)

6.7Spin Coherent States: Spherical Harmonics Aspects

Another realization makes use of the well-known material of the functions f (ν),ν == (θ, φ), on the sphere, square-integrable with respect to the scalar product

〈 f 1| f 2〉 def=∫

S2

dν f 1(ν) f 2(ν) , (6.39)

and the orthonormal basis of spherical harmonics Y lm at fixed integer l v 0. The

6.8 Other Spin Coherent States from Spin Spherical Harmonics 87

isometry between the Fock–Bargmann realization FBl and the Hermitian spaceY l generated by these Y lm ’s at fixed l is based on the following generating functionfor the spherical harmonics:

K (�, ν == (θ, φ)) =1

l ! 2l

√(2l + 1) 2l !

4π

~(– sin θ eiφ + 2� cos θ + �2 sin θ e–iφ

)l(1 + |�|2)–l . (6.40)

We will see in the next sections the SU (2) group representation origin of this ex-pansion. The expression (6.40) is a spin coherent state in spherical representation.On the same footing, it is the kernel providing the link between the two spaces

K (�, ν) =l∑

m=–l

ulm (�) Y lm (ν) , (6.41)

where the functions ulm (�) are given in (6.37). Precisely, the isomorphism betweenthe two Hermitian spaces is given by

Y l � f (ν) �→ Ψ(�) =∫

S2

K (�, ν) f (ν) dν , (6.42)

FBl � Ψ(�) �→ f (ν) =∫

C

K (�, ν) Ψ(�)2l + 1

πd2�

(1 + |�|2)2 . (6.43)

6.8Other Spin Coherent States from Spin Spherical Harmonics

A whole set of families of spin coherent states actually exists, as they were intro-duced by Perelomov within a group-theoretical context [10, 72]. Here, we will recov-er these Perelomov states by picking in our coherent state construction procedurethe so-called spin spherical harmonics, of which the orthonormal system (6.7) rep-resents a particular case. First, it is necessary to recall some technical facts aboutthe group SU (2) and its unitary irreducible representations. More details are givenin Appendix C.

6.8.1Matrix Elements of the SU(2) Unitary Irreducible Representations

Let � = (�0, �1, �2, �3) = �aea, �20 + �2

1 + �22 + �2

3 = 1, be a vector pointing to theunit sphere S3 in the Euclidean space R4 equipped with the orthonormal basis{ea , a = 0, 1, 2, 3} introduced in Section 6.5. With the correspondence establishedin (6.20), the group SU (2) can be defined as the set of 2 ~ 2 matrices in one-to-onecorrespondence with such unit-norm vectors:

SU (2) � � =

(�0 + i�3 –�2 + i�1

�2 + i�1 �0 – i�3

), (6.44)

where we use the same notation for a unit 4-vector as for the corresponding SU (2)


matrix. In bicomplex angular coordinates,

�0 + i�3 = cos ωeiψ1 , �1 + i�2 = sin ωeiψ2 , (6.45)

0 � ω � π2

, 0 � ψ1, ψ2 < 2π , (6.46)

such SU (2) matrices read as

SU (2) � � =

(cos ωeiψ1 i sin ωeiψ2

i sin ωe–iψ2 cos ωe–iψ1

), (6.47)

in agreement with the notation of Talman [74].Let us choose j ∈ N/2 and m ∈ Z/2 such that – j u m u j and j – m ∈ Z. With

any vector z =( z1

z2

)in C2 let us associate the monomial function

emj (z) =

z j+m1 z j–m

2√( j + m)!( j – m)!

. (6.48)

At fixed j these monomials span a (2 j + 1)-dimensional vector space of polynomialsp j (z). The group SU (2) acts on this space through the following action:

p j (z)�∈SU (2)

–→ p j (�† z) . (6.49)

The corresponding matrix elements of this (2 j +1)-dimensional unitary irreduciblerepresentation of SU (2) are defined through the action (6.49) on the monomialbasis elements:

em2j (�† z) =

(cos ωe–iψ1 z1 – i sin ωeiψ2 z2) j+m2 (–i sin ωe–iψ2 z1 + cos ωeiψ1 z2) j–m2√( j + m2)!( j – m2)!

=∑m1

D jm1m2 (�) em1

j (z) . (6.50)

This expression provides a generating function for the matrix elements. The latterare given by [74]

D jm1m2 (�) = (–1)m1–m2

[( j + m1)!( j – m1)!( j + m2)!( j – m2)!

]1/2

~∑

t

(�0 + i�3) j–m2–t

( j – m2 – t)!(�0 – i�3) j+m1–t

( j + m1 – t)!(–�2 + i�1)t+m2–m1

(t + m2 – m1)!(�2 + i�1)t

t !.

(6.51)

With angular variables the matrix elements of the unitary irreducible represen-tation of SU (2) are given in terms of Jacobi polynomials [18] by

D jm1m2 (�) = e–im1(ψ1+ψ2)e–im2(ψ1–ψ2)im2–m1

√( j – m1)!( j + m1)!( j – m2)!( j + m2)!

~1

2m1(1 + cos 2ω)

m1 +m22 (1 – cos 2ω)

m1 –m22 P (m1–m2,m1+m2)

j–m1(cos 2ω) , (6.52)

in agreement with Edmonds [75] (up to an irrelevant phase factor).


6.8.2Orthogonality Relations

Let us equip the SU (2) group with its invariant (Haar) measure:

μ(d�) = sin 2ω dω dψ1 dψ2 , (6.53)

in terms of the bicomplex angular parameterization. Note that the volume of SU (2)with this choice of normalization is 8π2. The orthogonality relations satisfied by thematrix elements D j

m1m2 (�) read as∫SU (2)

D jm1m2 (�) D j ′

m′1m′2(�) μ(d�) =

8π2

2 j + 1δ j j ′δm1m′1

δm2m′2. (6.54)

6.8.3Spin Spherical Harmonics

The spin spherical harmonics, as functions on the 2-sphere S2, are defined as fol-lows:

σY jμ(r) =

√2 j + 1

4πD j

μσ (� (Rr )) = (–1)μ–σ

√2 j + 1

4πD j

–μ–σ (� (Rr ))

=

√2 j + 1

4πD j

σμ(�† (Rr )

), (6.55)

where � (Rr ) is a (nonunique) element of SU (2) that corresponds to the space rota-tionRr introduced in (6.23) and that brings the unit vector e3 == k to the unit vectorr with spherical coordinates (θ, φ):

We immediately infer from the definition (6.55) the following properties:

σY jμ(r) = (–1)σ–μ–σY j–μ(r) , (6.56)

μ= j∑μ=– j

∣∣σY jμ(r)∣∣2

=2 j + 1

4π. (6.57)

Now, from the quaternionic description of 3-space rotations, we have the grouphomomorphism � = �(R) ∈ SU (2)↔R ∈ SO (3) ~= SU (2)/Z2:

r ′ =

(ix ′3 –x ′2 + ix ′1

x ′2 + ix ′1 –ix ′3

)= R · r = �

(ix3 –x2 + ix1

x2 + ix1 –ix3

)�† . (6.58)

In the particular case of (6.55) the angular coordinates ω, ψ1, ψ2 of the SU (2) ele-ment �(Rr ) are constrained by

cos 2ω = cos θ, sin 2ω = sin θ , so 2ω = θ , (6.59)

ei(ψ1+ψ2) = ieiφ, so ψ1 + ψ2 = φ +π2

. (6.60)


Here we should pay special attention to the range of values for the angle φ, de-pending on whether j and consequently σ and m are half-integer or not. If j is half-integer, then the angle φ should be defined mod (4π) whereas if j is integer, itshould be defined mod (2π).

We still have one degree of freedom concerning the pair of angles ψ1, ψ2. Weleave open the option concerning the σ-dependent phase factor by putting

i–σeiσ(ψ1–ψ2) def= eiσψ , (6.61)

where ψ is arbitrary. With this choice and considering (6.51) and (6.52) we get theexpression for the spin spherical harmonics in terms of φ, θ/2, and ψ, and of Jacobipolynomials, valid in the case in which μ± σ > –18):

σY jμ(r) = (–1)μeiσψ

√2 j + 1

4π

√( j – μ)!( j + μ)!( j – σ)!( j + σ)!

~12μ (1 + cos θ)

μ+σ2 (1 – cos θ)

μ–σ2 P (μ–σ,μ+σ)

j–μ (cos θ) eiμφ . (6.62)

For other cases, it is necessary to use alternative expressions based on the rela-tions [18]

P (–l,�)n (x) =

(n+�l

)(nl

) (x – 1

2

)l

P (l,�)n–l (x) , P (α,�)

0 (x) = 1 . (6.63)

Note that with σ = 0 we recover the expression for the normalized sphericalharmonics (see Appendix C).

Finally, introducing the single complex variable � = z2/z1, one should retain thefollowing expression, directly emerging from (6.51), for the generating function ofthe spin spherical harmonics:√

2 j + 14π

(cos

θ2

+ sinθ2

e–iφ �) j+σ (

– sinθ2

eiφ + cosθ2

�) j–σ

=j∑

m=– j

√( j + σ)!( j – σ)!

( j + m)!( j – m)!� j–m (–1)–σ e–iσφ

σY jm (r) . (6.64)

In particularizing the above equation to the case σ = 0, one recovers the ker-nel (6.40).

8) This expression is not exactly in agreementwith the definitions of Newman andPenrose [77], Campbell [76] (note that thereis a mistake in the expression given byCampbell, in which a cos θ

2 should read

cot θ2 ), and Hu and White [78]. Besides the

presence of different phase factors, thedisagreement is certainly due to a differentrelation between the polar angle θ and theEuler angle.


6.8.4Spin Spherical Harmonics as an Orthonormal Basis

Specifying (6.54) to the spin spherical harmonics leads to the following orthogo-nality relations that are valid for integer j (and consequently integer σ):∫

S2σY jμ(r)

(σY j ′ν(r )

)∗μ(dr) = δ j j ′δμν . (6.65)

We recall that in the integer case, the range of values assumed by the angle φ is0 � φ < 2π. Now, if we consider half-integer j (and consequently σ), the range ofvalues assumed by the angle φ becomes 0 � φ < 4π. The integration above has tobe carried out on the “doubled” sphere S2 and an extra normalization factor equalto 1√

2is needed in the expression of the spin spherical harmonics.

For a given integer σ the set{

σY jμ, –∞ � μ �∞, j � max (0, σ, m)}

forms anorthonormal basis of the Hilbert space L2(S2). Indeed, at μ fixed so that μ± σ � 0,the set {√

2 j + 14π

√( j – μ)!( j + μ)!( j – σ)!( j + σ)!

12μ (1 + cos θ)

μ+σ2 (1 – cos θ)

μ–σ2

~ P (μ–σ,μ+σ)j–μ (cos θ), j � μ

}is an orthonormal basis of the Hilbert space L2([–π, π], sin θ dθ). The same holdsfor other ranges of values of μ by using alternative expressions such as (6.63)for Jacobi polynomials. Then it suffices to view L2(S2) as the tensor productL2([–π, π], sin θ dθ)

⊗L2(S1). Similar reasoning is valid for half-integer σ. Then,

the Hilbert space to be considered is the space of “fermionic” functions on thedoubled sphere S2, that is, such that f (θ, φ + 2π) = – f (θ, φ).

6.8.5The Important Case: σ = j

For σ = j , owing to the relations (6.63), the spin spherical harmonics reduce totheir simplest expressions:

j Y jμ(r ) = (–1) j ei jψ

√2 j + 1

4π

√(2 j

j + μ

)(cos

θ2

) j+μ (sin

θ2

) j–μ

eiμφ .

(6.66)

They are precisely the functions that appear, up to the factor√

2 j + 1 and a phasefactor also, in the construction of the spin coherent states described in Section 6.3.


6.8.6Transformation Laws

Let Hσ j be the (2 j + 1)-dimensional Hermitian space, subspace of L2(S2, μ(dr)),spanned by the orthonormal set

{σY jμ(r) , – j u m u j

}of spin spherical harmon-

ics at fixed σ and j. Given a function f (x) on the sphere S2 belonging to Hσ j anda rotation R ∈ SO (3), we define the rotation operator Dσ j (R) for that representa-tion by(

Dσ j (R) f)

(x) = f (R–1 · x) = f (tR · x) . (6.67)

We now consider the transformation law of the spin spherical harmonics underthis representation of the rotation group. From the relation

RRtRr = Rr (6.68)

for any R ∈ SO (3), and from the homomorphism �(RR′) = �(R)�(R′) betweenSO (3) and SU (2), we deduce from the definition (6.55) of the spin spherical har-monics the transformation law(

Dσ j (R)σY jμ)

(r) =σ Y jμ(tR · r) =

√2 j + 1

4πD j

σμ(�† (RtR·r )

)=

√2 j + 1

4πD j

σμ(�†

(tRRr

))=

√2 j + 1

4πD j

σμ(�† (Rr ) � (R)

)=

√2 j + 1

4π

∑ν

D jσν

(�† (Rr )

)D j

νμ (� (R))

=∑

ν

σY jν(r)D jνμ (� (R)) , (6.69)

as expected if we think of the special case (σ = 0) of the spherical harmonics [74].

6.8.7Infinitesimal Transformation Laws

The generators of the representative of the three rotationsR(a) , a = 1, 2, 3, aroundthe three Cartesian axes, are the components of the angular momentum operatorin the representation Dσ j . When σ = 0, these generators are the usual angularmomentum operators Ja = –iεabc xb∂c (short notation for J ( j)

a ), which, in sphericalcoordinates, are given by

J3 = –i∂φ ,

J+ = J1 + i J2 = eiφ(∂θ + i cot θ∂φ

),

J– = J1 – i J2 = –e–iφ(∂θ – i cot θ∂φ

).

(6.70)


In the general case σ =/ 0, we denote the generators by Λ(σ j)a . These “spin” angular

momentum operators are given by

Λσ j3 = J3 = –i∂φ, (6.71)

Λσ j+ = Λσ j

1 + iΛσ j2 = J+ + σ csc θeiφ, (6.72)

Λσ j– = Λσ j

1 – iΛσ j2 = J– + σ csc θe–iφ. (6.73)

They obey the expected commutation rules,

[Λσ j3 , Λσ j

± ] = ±Λσ j± , [Λσ j

+ , Λσ j– ] = 2Λσ j

3 . (6.74)

Their actions on the spin spherical harmonics are similar to the case σ = 0.

Λσ j3 σY jμ = μ σY jμ (6.75a)

Λσ j+ σY jμ =

√( j – μ)( j + μ + 1) σY jμ+1 (6.75b)

Λσ j– σY jμ =

√( j + μ)( j – μ + 1)σY jμ–1. (6.75c)

6.8.8“Sigma-Spin” Coherent States

For a given pair ( j , σ), we now define the family of coherent states in the (2 j + 1)-dimensional Hilbert space Hσ j by following our method of construction. We willcall them “sigma-spin” coherent states because of the context, although they are,up to a constant, identical to the coherent states constructed by Perelomov [72] onpurely group-theoretical arguments, as will be shown later.

Here, we just pick the orthonormal set {σY ∗jμ(r), – j u μ u j}, in one-to-one cor-respondence with an orthonormal basis {|σ jμ〉, – j u μ u j} of Hσ j , and considerthe following superposition inHσ j :

|r ; σ〉 = |θ, φ; σ〉 =1√N (r )

j∑μ=– j

σY jμ(r)|σ jμ〉 ; |r〉 ∈ Hσ j , (6.76)

with

N (r ) =j∑

μ=– j

|σY jμ(r )|2 =2 j + 1

4π.

They are the sigma-spin coherent states. In particular, at the north pole of thesphere, they reduce to the state

|e3; σ〉 = |σ jσ〉 . (6.77)


For σ = j , they are equal to the spin coherent states, |r ; 0〉 == |r〉. But, for a given jand two different σ =/ σ′, the corresponding families are distinct because they livein different Hermitian spaces of the same dimension 2 j + 1. This is due to the factthat the map between the two orthonormal sets is not unitary, since we should dealwith expansions such as

σY jμ =∑

j ′μ′

M j ′μ′, jμ(σ′, σ)σ′Y j ′μ′ , (6.78)

where

M j ′μ′ , jμ(σ′, σ) =∫

S2σ′Y j ′μ′ (r)σY jμ(r) μ(dr) = [ j ′ jσ′σμ] δμμ′ , (6.79)

the (nontrivial!) coefficient [ j ′ jσ′σμ] to be determined and forcing the sum to runon values of j ′ different from j.

The sigma-spin coherent states, by construction, are normalized and solve theidentity inHσ j .

〈r ; σ|r ; σ〉 = 1 ,2 j + 1

4π

∫S2

dr |r ; σ〉〈r ; σ| = Id . (6.80)

From the probabilistic point of view at the basis of the construction, the discretedistribution μ �→ 4π

2 j+1 |σY jμ(r )|2 has an interesting meaning in terms of the tran-sition probability for a quantum spin interacting with a transient magnetic field.This point will be developed in the next chapter.

Their overlap 〈r ; σ|r ′; σ〉 is given, up to a phase, in terms of Jacobi polynomialsand the dot product r · r ′:

〈r ; σ|r ′; σ〉 = e–2iσψ

(1 + r · r ′

2

)σ

P (0,2σ)j–σ (r · r ′) , (6.81)

with

tan ψ = –sin (φ – φ′)

cos (φ – φ′) + tan θ2 tan θ′

2

.

The proof is based on the definition of the spin spherical harmonics (6.55) as par-ticular cases of representation matrix elements of SU (2), the group compositionrule for the latter, and the expression (6.52)

〈r ; σ|r ′; σ〉 =4π

2 j + 1

j∑μ=– j

σY jμ(r ′)σY jμ(r)

=j∑

μ=– j

D jμσ (� (Rr ′ )) D j

σμ(�† (Rr )

)= D j

σσ (� (Rr Rr ′ )) .

It is then necessary to identify the angular parameters of the matrix elements ofRr Rr ′ ∈ SU (2) by using (6.59) and (6.45):

Rr Rr ′ ==

(cos ω eiψ1 i sin ω eiψ2

i sin ω e–iψ2 cos ω e–iψ1

).

We find that ψ1 = ψ and cos 2ω = r · r ′.


6.8.9Covariance Properties of Sigma-Spin Coherent States

The definition of the rotation operator Dσ j (R) was given in (6.67). Starting froma coherent state |r ; σ〉, let us consider the coherent state with rotated parameterR · r . Owing to the transformation property (6.69), the invariance of N (r ), and theunitarity of D j , we find

|R · r ; σ〉 =1√N (r )

j∑μ=– j

σY jμ(tR · r)|σ jμ〉

=1√N (r )

j∑μ,μ′=– j

σY jμ′ (r) D jμ′μ

(�(R–1

))|σ jμ〉

=1√N (r )

j∑μ′=– j

σY jμ′ (r)j∑

μ=– j

D jμμ′ (� (R)) |σ jμ〉

= Dσ j (R)|r ; σ〉 . (6.82)

Hence, we get the (standard) covariance property of the sigma-spin coherentstate:

Dσ j (R)|R–1 · r ; σ〉 = |r ; σ〉 . (6.83)

From this relation and (6.77) we derive the SU (2) theoretical content of the sigma-spin coherent states as the element of the orbit of the state |σ jσ〉 under the actionof the representation Dσ j of SO (3) (or rather SU (2) in the case of half-integer j):

|r ; σ〉 = Dσ j (Rr )|σ jσ〉 . (6.84)

This equality generalizes (6.25) and can also be viewed as a definition of the sigma-spin coherent states. It is actually the Perelomov genuine definition.

97

7Selected Pieces of Applications of Standardand Spin Coherent States

7.1Introduction

We now proceed with the presentation of a first (small, but instructive) panoramaof applications of the standard coherent state and spin coherent state to some prob-lems encountered in physics, quantum physics, statistical physics, and so on. Theselected models that are illustrated as examples in the present chapter were chosenbecause of their high pedagogical and illustrative content.

Coherent States and the Driven Oscillator The driven oscillator is a pedagog-ical model for presenting the S matrix as a unitary operator transforming an ini-tial state into a final state. By introducing coherent states into the formalism aswas done by Carruthers and Nieto [79], we show this operator to be nothing oth-er than the displacement operator D (z). Also, there will appear some interestingdiscrete probability distributions that generalize the Poisson distribution and forwhich there exists a limpid physical interpretation.

A Nice Application of Standard or Spin Coherent States in Statistical Physics:Superradiance This second part of this chapter is devoted to a nice example of theapplication of the coherent state formalism. The object pertains to atomic physics:two-level atoms in resonant interaction with a radiation field (Dicke model andsuperradiance). The content is based on two selected ancient papers, the first oneby Wang and Hioe [80], and the second one, more mathematically oriented, byHepp and Lieb [81].

Application to Quantum Magnetism Inspired by the Perelomov monograph[10], we explain how the spin coherent states can be used to solve exactly or ap-proximately the Schrödinger equation for certain systems, such as a spin inter-acting with a variable magnetic field. Again, there will appear discrete probabilitydistributions involving spin spherical harmonics and that generalize the binomialdistribution.

Classical and Thermodynamical Limits Coherent states are useful in thermo-dynamics. Following a paper by Lieb [19], we establish a representation of the par-tition function for systems of quantum spins in terms of coherent states. Afterintroducing the so-called Berezin–Lieb inequalities, we show how that coherent


98 7 Selected Pieces of Applications of Standard and Spin Coherent States

state representation makes crossed studies of classical and thermodynamical lim-its easier.

7.2Coherent States and the Driven Oscillator

The Newton equation for the classical version of the driven oscillators is given by

mx + kx = F (t) , (7.1)

where F (t) represents the time-dependent driving force. Hence, we have to add tothe free oscillator energy 1

2m p2 + 12 kx2 the interaction potential energy V int(x , t) =

–x F (t). The Hamiltonian of the quantum version of this model is just obtainedby replacing x by Q = l c

(a + a†

), l c = 1/

√2�mω and p by P = –i pc

(a – a†

),

pc =√

�mω/2, while we still consider the driving force as classical:

H =1

2mP 2 +

12

kQ2 – Q F (t) = �ω(

aa† +12

)– l c(a + a†) F (t) . (7.2)

In a first approach, let us suppose that there exists just a constant driving: F (t) =F 0. By shifting the lowering operator to

b = a –l cF 0

�ωId , [b , b†] = Id , (7.3)

one gets the Hamiltonian

H = �ω(

bb† +12

)–

(l cF 0)2

�ωId . (7.4)

The energy levels of the oscillator are, as expected, just translated by(l cF 0)2

�ω. The

new ground state, say, |0〉b , vanishes under the action of b:

b |0〉b = 0 ⇔ a |0〉b =l cF 0

�ω|0〉b . (7.5)

This means that |0〉b is an element of the family of coherent states for the value of

the parameter α =l cF 0

�ω:

|0〉b =

∣∣∣∣α =l cF 0

�ω

⟩= e

lc F 0�ω (a†–a) |0〉 . (7.6)

Let us now examine the general situation for which we just suppose the asymp-totic vanishing of the driving force,

F (±∞) = 0 , (7.7)

7.2 Coherent States and the Driven Oscillator 99

a hypothesis that pertains to scattering theory. Otherwise said, the question is todetermine the “S matrix” or unitary operator that transforms, from the Heisenbergviewpoint, the “ingoing” state |Ψin〉 into the “outgoing” state |Ψout〉:

|Ψin〉 �→ |Ψout〉 = S† |Ψin〉 . (7.8)

The method for finding S consists in first solving the classical version of this scat-tering process, and next translating the solution into the quantum language byusing coherent states.

The treatment of the classical scattering described by

x(t) + ω2x(t) = F (t)/m (7.9)

consists in finding the corresponding Green function G or elementary solutionwithin the framework of distribution theory and establishing the solution througha convolution with G. The Green function is a solution to(

d2

dt2 + ω2

)G(t) = ω δ(t) , (7.10)

and is completely determined by stating initial conditions. Introducing the Heavi-side function

H(t) =

{0 t < 0 ,

1 t > 0 ,(7.11)

the retarded Green function GR(t) def= H(t) sin ωt vanishes for t < 0, whereas the

advanced Green function GA(t) def= –H(–t) sin ωt vanishes for t > 0. The solutionx(t) of (7.9) is then expressed in terms of initial condition x in or final condition xout

defined as

x(t) ~

{x in(t) t → –∞ ,

xout(t) t →∞ ,(7.12)

both being solutions to the free oscillator equation

x inout

+ ω2x inout

= 0 .

So, one obtains

x(t) = x in(t) +1

mω

∫ +∞

–∞GR (t – t ′) F (t ′) dt ′ (7.13)

= xout(t) +1

mω

∫ +∞

–∞GA(t – t ′) F (t ′) dt ′ . (7.14)

By eliminating x(t) from the above equations, we obtain the classical counterpartof the S matrix:

xout(t) = x in(t) +1

mω

∫ +∞

–∞G(t – t ′) F (t ′) dt ′ , (7.15)


with

G(t) = GA(t) – GR (t) = sin ωt , t =/ 0 . (7.16)

The scattering relation (7.15) is easily expressed in terms of the (actually inverse)Fourier transform of F (t) defined here as

F (ω) def=∫ +∞

–∞eiωt F (t) dt .

xout(t) = x in(t) –

(1

2imω

∫ +∞

–∞eiωt ′ F (t ′) dt ′

)e–iωt

+

(1

2imω

∫ +∞

–∞e–iωt ′ F (t ′) dt ′

)eiωt

= x in(t) +i F (ω)2mω

e–iωt –i F (ω)2mω

eiωt . (7.17)

By writing the asymptotic free oscillator solutions as

x in(t) =12

(Ae–iωt + Aeiωt

), xout(t) =

12

(Be–iωt + Beiωt

), (7.18)

one gets from (7.17)

B = A + iF (ω)mω

. (7.19)

Let us now transpose this into the quantum side through the correspondences

12 A �→ l ca , 1

2 B �→ l cb (7.20)

so that the quantum counterpart of (7.18) reads as

Q in(t) = l c(a e–iωt + a† eiωt

), Qout(t) = l c

(b e–iωt + b† eiωt

), (7.21)

with the relation between a and b being

a S�→ b = a + iF (ω)√2mω�

Iddef= S†aS . (7.22)

Hence, the effect of the interaction is to translate the ingoing normal mode a by anamount equal (up to a factor) to the Fourier component of F (t) corresponding tothe proper frequency of the free oscillator. The unitary operator that executes thistask is precisely the displacement operator D (z). Indeed, the following propertyholds true.

[a, D (α)] = α D (α) ⇔ D †(α)aD (α) = a + α Id . (7.23)

To prove (7.23), it is enough to check it on a coherent state, say, |�〉 with arbitrary�. We have

aD (α)|�〉 = eiI(α�)a|α + �〉 = (α + �)eiI(α�)|α + �〉 = (α + �)D (α)|�〉 ,

7.2 Coherent States and the Driven Oscillator 101

whereas

D (α)a|�〉 = �D (α)|�〉 ,

and so aD (α) – D (α)a = αD (α).It follows that b = S†aS , where

S = D (α) , with α = iF (ω)√2m�ω

. (7.24)

An alternative form of this S matrix is given by

S = D (α) = e(αa†–αa)

= expi l c

�

(F (ω)a† + F (ω)a

)= exp

i�

∫ +∞

–∞Q in(t) F (t) dt . (7.25)

To understand better the physical significance of this S matrix, let us examineits matrix elements and the resulting transition probabilities when ingoing andoutgoing states are energy eigenstates of the oscillator:

|Ψin〉 == |Ψin,m〉 , |Ψout〉 == |Ψout,n〉 .

Let us expand the ingoing eigenstate in terms of the complete orthonormal set ofoutgoing eigenstates:

|Ψin,m〉 =+∞∑n=0

〈Ψout,n |Ψin,m〉 |Ψout,n〉

=+∞∑n=0

〈Ψout,n |S|Ψout,m〉︸︷︷︸Snm

|Ψout,n〉 . (7.26)

The matrix element Snm , which is also equal to 〈Ψin,n |S|Ψin,m〉, is the matrixelement of the displacement operator in the Fock basis:

Snm = Snm(α) = 〈n|D (α)|m〉 , α = iF (ω)√2m�ω

. (7.27)

The calculation of this matrix element can be carried out by using the resolution,by coherent states, of the identity and the subsequent reproducing property (3.17)of the kernel:

〈n|D (α)|m〉 =∫∫

C2

d2zπ

d2z′

π〈n|z〉〈z|D (α)|z′〉〈z′|m〉

=1√

n!m!

∫∫C2

d2zπ

d2z′

πe– |z|

2

2 e– |z′ |22 zn z′m eiI(αz′) 〈z|z′ + α〉

=e– |α|

2

2

√n!m!

∫C

d2z′

πe–|z′|2 e– ¯αz′ z′m (z′ + α)n .


After binomial and exponential expansions and integration, one ends up with thefollowing expression:

Snm(α) =

√m!n!

e– |α|2

2 αn–m L(n–m)m

(|α|2

)for m u n ,

=

√n!m!

e– |α|2

2 (–α)m–n L(m–n)n

(|α|2

)for m > n , (7.28)

where

L(μ)n (x) =

n∑k=0

(–1)k Γ(n + μ + 1)Γ(μ + k + 1)(n – k)!

xk

k!(7.29)

is a generalized Laguerre polynomial.Hence, the transition probability, say, in the case m → n v m, is given by

P nm(|α|2) = |Snm (α)|2 =m!n!

e–|α|2 |α|2(n–m)(

L(n–m)m

(|α|2

))2. (7.30)

This is a generalization of the discrete Poisson distribution in the sense that onerecovers the latter for m = 0:

P n0(|α|2) =|α|2n!

e–|α|2 .

One notices that, in the spirit of Chapter 5, the interpretation of the matrix ele-ments (7.27) as “square roots” of a discrete probability distribution sheds interest-ing light on the following family of normalized coherent states that emerges fromthe unitary transport by D (z) of the excited state |m〉:

|z; m〉 def= D (z)|m〉 =∞∑

n=0

Snm (z) |n〉 , (7.31)

with a normalization factor equal to 1 because of the unitarity of the matrix S.As is well known, the Poisson distribution, as a function of its parameter, here|α|2 = |F (ω)|2

2m�ω , reaches its maximum at |α|2 = n. Inverting this yields the most prob-

able transition oscillator level ν = |F (ω)|22m�ω for a given driving force F (t). Otherwise

said, the most probable energy transferred to the oscillator is ΔE = ν�ω = |F (ω)|22m ,

and this coincides with the mean transfer of energy, namely,+∞∑n=0

P n0n�ω = ν�ω . (7.32)

It should be noted that the most probable energy transferred to the oscillator wouldbe the same whatever the initial state |m〉. Indeed, for all Ψin, and with H in =�ω(a†a + 1/2), Hout = �ω(b†b + 1/2), b = a + α,

ΔE = 〈Ψin|Hout – H in|Ψin〉= �ω

[|α|2 + i (〈Ψin|a†|Ψin〉α – c.c.)

], (7.33)

and the factor of i vanishes for |Ψin〉 = |m〉.

7.3 An Application of Standard or Spin Coherent States in Statistical Physics: Superradiance 103

7.3An Application of Standard or Spin Coherent States in Statistical Physics:Superradiance

7.3.1The Dicke Model

In quantum optics, superradiance [85] is a phenomenon of collective emission ofan ensemble of excited atoms or ions, first considered by Dicke [86–89]. It is similarto superfluorescence, but it starts with the coherent excitation of the ensemble,usually with an optical pulse. This coherence (i.e., a well-defined phase relationshipbetween the excitation amplitudes of lower and upper electronic states) leads toa macroscopic dipole moment. The maximum intensity of the emitted light scaleswith the square of the number of atoms, because each atom contributes a certainamount to the emission amplitude, and the intensity is proportional to the squareof the amplitude.

In the Dicke model [86] for the interaction between matter and radiation, oneconsiders a system of N two-level atoms interacting with a radiation field. Theatoms have fixed positions in a one-dimensional box of length L, and they are sup-posed to be far enough from each other so that their mutual interaction is negli-gible. Nevertheless, the assembly has to be thought of as a single quantum entity,because of its collective interaction with the radiation field. Hepp and Lieb [81] haveobtained some exact results for the thermodynamical properties of this system inthe limit N → +∞, L → +∞, N/L = const. In particular, they have shown theexistence of a second-order phase transition, radiance → superradiance, at a cer-tain critical temperature T c , when the atom-field coupling is strong enough. Thiscoupling is supposed to hold in the so-called dipolar approximation. On the oth-er hand, the box of size L is supposed to be sufficiently small compared with theradiation wavelength so that all the atoms experience the same field. The latter isunderstood as collective with the hypothesis L << λray. In the rotating-wave approxi-mation, for which more details will be given below, the Hamiltonian of the systemreads as (in units � = c = 1)

H = H0 + HI , (7.34)

where H0 and HI are the free and interaction Hamiltonians, respectively:

H0 = Hrad + Hat =∑

k

νk a†k ak +12

ωN∑j=1

σzj , (7.35)

HI =1

2√

L

⎡⎣(∑k

λ′k ak

)⎛⎝ N∑j=1

σ+j

⎞⎠ +

(∑k

λ′k a†k

)⎛⎝ N∑j=1

σ–j

⎞⎠⎤⎦ .

(7.36)

Here, a†k and ak are the creation and annihilation operators, respectively, for thekth radiation mode with frequency νk , ω is the two-level energy shift, and λ′ is the


atom-field coupling strength. In the dipolar approximation, the Hamiltonian HI

is supposed to be linear in a†k and ak . Concerning the two-level atoms, they aredescribed by Pauli matrices:

σ±j = σxj ± iσ y

j ,

σxj =

(0 11 0

)j

, σ yj =

(0 –ii 0

)j

, σzj =

(1 00 –1

)j

.

7.3.1.1Rotating-Wave ApproximationThe rotating-wave approximation consists in neglecting the terms akσ–

j (energy loss

W �(νk +ω j )) and a†kσ+j (energy gain). Let us present here the elementary example of

a two-level atom subjected to an oscillating electric field E = E cos νt . On a quasi-classical level [25], the time evolution of this system is described by the Schrödingerequation:

|Ψ(t)〉 = –i (H0 + HI )|Ψ(t)〉 , with |Ψ(t)〉 = C +(t)|+〉 + C –(t)|–〉 , (7.37)

where H0 = 12 � ω σz , HI = –e Q E cos νt , where Q is the position operator. This

results in the differential system:

C + = – i2 ω C + + I ΩRe–iφ cos νt C – ,

C – = i2 ω C – + I ΩReiφ cos νt C + ,

(7.38)

where φ is the argument of the complex 〈–|Q |+〉, 〈–|Q |+〉 = |〈–|Q |+〉|eiφ , and

ΩRdef= |〈+|Q |–〉| E/� (Rabi frequency). Putting c± = C± e±i ω

2 t , we get for the timeevolution of the latter

c+ � iΩR

2e–iφ c– ei(ω–ν)t , c– � i

ΩR

2eiφ c+ e–i(ω–ν)t , (7.39)

where terms proportional to e±i(ω+ν)t have been neglected.

7.3.1.2Dicke Hamiltonian with a Single ModeIn the Dicke model dealing with a single mode of frequency ν only, the correspond-ing Hamiltonian reads as

H = a† a +N∑j=1

[12

ε σzj +

λ2√

N

(aσ+

j + a†σ–j

)], (7.40)

where we have introduced the modified parameters

ε = ω/ν, λ = λ′√

ρ/ν, ρ = N/L ,

which are better suited to thermodynamical considerations.


7.3.2The Partition Function

The thermodynamical properties of the system are encoded by the partition func-tion

Z (N , T ) = Tr e–�H , � =1

kBT, (7.41)

where kB is the Boltzmann constant and T is the absolute temperature. It is pre-cisely at this point in the explicit computation of the partition function that thestandard coherent states |z〉 of the single-mode field fully play their simplifyingrole [80]. We have for a trace class operator A,

Tr A =∑nv0

〈n|A|n〉 =∫

d2zπ〈z|A|z〉 . (7.42)

Thus, the partition function can be written as

Z (N , T ) =∑s1=±1

. . .∑

sN =±1

∫d2zπ〈s1 . . . sN | 〈z|e–�H |z〉 |s1 . . . sN 〉 , (7.43)

where the sums are taken over all possible atomic sites (for simplicity, we haveavoided a tensor-product notation). Therefore, it is necessary to estimate the partialmatrix element

〈z|e–�H |z〉 =∞∑r=0

(–�)r

r !〈z|Hr |z〉 , (7.44)

bearing in mind the basic coherent state property a|z〉 = z|z〉. To take into ac-count the constancy of the ratio ρ = N/L, let us rescale the mode operators asb = a√

N, b† = a†√

N, so that

bb† = b†b +1N

. (7.45)

Then the terms in the expansion of Hr , using normal ordering, are shown to be

〈z|Hr |z〉 =

⎛⎝zz +N∑j=1

h j

⎞⎠r

+ O

(1N

), (7.46)

where the h j ’s are the individual atomic Hamiltonians

h j =12

ε σzj +

λ2√

N(zσ+

j + zσ–j ) .

Hence, we get the estimate

〈z|e–�H |z〉 = e–�zz e–�

∑N

j=1h j + O

(1N

), (7.47)


and, for the partition function,

Z (N , T ) =∫

C

d2zπ

e–�|z|2∑s1=±1

. . .∑

sN =±1

⎛⎝ N∏j=1

〈s j |e–�h j |s j 〉

⎞⎠ + O

(1N

)

=∫

C

d2zπ

e–�|z|2 (Tr e–�h )N + O

(1N

).

(7.48)

The generic atomic Hamiltonian, h = 12 ε σz + λ

2√

N(zσ+ + zσ–), has the eigenvalues

±12

ε (1 + 4λ2|z|2/ε2N )1/2 .

Using this, we get, upon performing the angular integration and making use ofstandard asymptotic methods (Laplace or steepest descent [82]),

Z (N , T ) =∫

C

d2zπ

e–�|z|2(

2 cosh12

�ε(

1 + 4λ2 |z|2ε2N

)1/2)N

+ O

(1N

)W const.

√N max

0u|z|2/N u∞exp Nϕ

(|z|2N

), for large N ,

(7.49)

where ϕ(y ) = –�y + ln(

2 cosh 12 �ε

(1 + 4 λ2

ε2 y)1/2

).

7.3.3The Critical Temperature

Let us now obtain the value of y that maximizes ϕ(y ), that is, for which

ϕ′(y ) = �(

–1 +λ2

εηtanh

�εη2

)= 0 ,

where we introduce the intermediate variable

η =

√(1 +

4λ2

ε2 y

)v 1 .

It is necessary to examine the equationελ2 η = tanh

�ε2

η. This is done through the

graphical study shown in Figure 7.1.Let us comment on the appearance of three different regimes.

7.3.3.1Weak CouplingIf ε/λ2 > 1, that is, λ2 < ε (weak coupling), the linear function ε/λ2 η is alreadystrictly above 1 at η = 1. Thus, eϕ(y ) is monotone decreasing and its maximum


��

��

��

��

��

��

��

��

��

��

��

1

�� K

�η0

�

�

η

��ε2

ηελ2 η

tanh�ε2

η

Fig. 7.1 Graphical study of the equationελ2 η = tanh

�ε2

η.

holds at y 0 = 0. So

Z (N , T ) = const.√

N(2 cosh 1

2 �ε)N

, (7.50)

and the free energy f (T ) per atom is given by

– � f (T ) def= limN→∞

1N

ln Z (N , T )

= limN→∞

1N

(N ln (2 cosh 1

2 �ε) + lnconst.√

N

)= ln (2 cosh 1

2 �ε) . (7.51)

This value would also be obtained with a free Hamiltonian

H = a† a +N∑j=1

12

ε σzj .

7.3.3.2Intermediate CouplingIf λ2 > ε (intermediate or strong coupling), the solution of ϕ′(y ) = 0 now dependson the value of �. If the linear function ε/λ2 η is above the point K, this means that

ε/λ2 > tanh�ε2

,

that is, � < �c , where �c is determined by the equation ε/λ2 = tanh �ε2 . So, for

� u �c , the free energy per atom is still given by

–� f (T ) = ln (2 cosh 12 �ε) .


7.3.3.3Strong CouplingIf � > �c , there exists a solution 1 < η0 <∞ such that

2σ def=ελ2 η0 = tanh

�ε2

η0

and so for y,

y = y 0 = λ2σ2 –ε2

4λ2 .

Hence, in the case � > �c the free energy per atom is now different and is given by

–� f (T ) = ln (2 cosh 12 �ε) – �λ2σ2 + �

ε2

4λ2 , (7.52)

with 2σ = tanh �λ2σ =/ 0 .

7.3.3.4SummaryIn summary, there does or does not exist a critical temperature. We have obtainedthe value of y that maximizes ϕ(y ), according to whether the coupling λ is strong(λ >

√ε) or weak (λ <

√ε). In the former case, one obtains a critical temperature

T c given by

ε/λ2 = tanhε

2kBT c, (7.53)

for which the system jumps from a “normal” state at T > T c to a superradiativestate at T < T c , whereas there is no such phase transition for weak coupling. Aswith every phase transition, a physical quantity presents a discontinuity at T > T c ,namely, in the present case, the specific heat.

7.3.4Average Number of Photons per Atom

The physical difference between weak, intermediate, and strong regimes is betterunderstood through the following expressions, obtained by using similar coherentstate methods, for the average number of photons per atom:⟨(

a†aN

)r⟩= δr0 , (7.54)

for λ2 > ε and all �, or for λ2 > 0, � < �c , while⟨(a†aN

)r⟩= (λ2σ2 – ε2/4λ)r (7.55)

7.4 Application of Spin Coherent States to Quantum Magnetism 109

for λ2 > ε and � > �c , where σ is such that

2σ = tanh �λ2σ =/ 0 . (7.56)

These expressions clarify the terminology. In the “normal” radiant state, where⟨(a†aN

)r⟩= δr0

holds, the number of photons emitted goes to zero as N →∞.This is not so in the superradiant regime,⟨(

a†aN

)r⟩= (λ2σ2 – ε2/4λ)r , (7.57)

where an infinite number of photons are emitted, as a consequence of the coher-ence of the maser light, a truly collective effect: as the number of photons rises ina kind of chain reaction, Dicke described this phenomenon as an “optical bomb.”

7.3.5Comments

First of all, a generalization to multimode fields is straightforward, the samemethod based on the use of the standard coherent state again being useful forcarrying out explicit computations.

Second, more general results have been given by Hepp and Lieb [81]. These au-thors have made mathematically rigorous the thermodynamical limit N →∞ car-ried out previously by Wang and Hioe. They have generalized the results to multi-level atoms and to the case of an infinite number of modes in finding the appro-priate estimate. They have also extended the results to Hamiltonians (in the case offinite multimode) that have translation degrees of freedom for atoms and withoutthe rotating-wave approximation. Finally, they have made use of the spin coherentstates for multilevel atoms and standard coherent states for the multimode electro-magnetic field.

7.4Application of Spin Coherent States to Quantum Magnetism

In this third example of applications of coherent states, we consider a quantumspin in a variable magnetic field [10]. Let a particle of spin j (or simply a “spin”),with magnetic moment μ, be subjected to a variable magnetic field H(t). The timeevolution of states is ruled by the Schrödinger equation:

iddt|Ψ(t)〉 = –M · J |Ψ(t)〉 , M def=

μj

H . (7.58)


WithM‖def= i

2 (Mx – iMy ), one writes

iddt|Ψ(t)〉 = i (M‖ J+ –M‖ J– + iMz Jz) |Ψ(t)〉 . (7.59)

Furthermore, let us assume that the field H(t) tends to a definite limit in t =∞ so

that asymptotic states∣∣∣Ψ in

out

⟩exist.

We now introduce in this model the spin coherent states in complex parameter-ization, of which we recall the expression

|�〉 =j∑

m=– j

√(2 j

j + m

)� j–m

(1 + |�|2) j | j , m〉 . (7.60)

They will be precisely used to build a solution of the Schrödinger solution. Let usput as an ansatz

|Ψ(t)〉 = e–iφ(t) |�(t)〉 , (7.61)

where the functions φ(t) and |�(t)〉 remain to be determined.Spin coherent states are not eigenstates of any lowering operator. We instead

have

J–|�〉 = �–1( j – Jz) |�〉 , J+|�〉 = �( j + Jz) |�〉 . (7.62)

On the other hand,

ddt|�(t)〉 =

[–

j1 + |�|2

ddt

(1 + |�|2) +��

( j – Jz)

]|�(t)〉 , (7.63)

an identity that holds for any spin coherent state.Now, behind (7.58) and the ansatz (7.61) lies a classical dynamical system ob-

tained as follows. By identifying to zero the coefficients of the two operators in-volved, namely, the identity and Jz , one gets a differential system describing thetime evolution of this system in the set of parameters φ and �:

iφ = j (M‖ � –M‖ � – iMz)

� = –M‖ – iMz � –M‖�2 .(7.64)

Going back to the parameterization sphere S2, the above equation is to be inter-preted as the coherent state quantization of a precession motion for a magnetic mo-ment μ = (μ/ j ) JCl subjected to the field H. Such a precession is described by theequation

˙n = μH ~ n , where n =JCl

j. (7.65)

At a first view of the system, if the interaction is such thatM‖(t)→ 0 andMz(t)→const. in the limit t → ∞, then the coherent state parameter localizes on a circlein the complex plane |�| → const.. Indeed, � � –iMz � and so � � �0 e–iMz t .

7.5 Application of Spin Coherent States to Classical and Thermodynamical Limits 111

For the treatment of this “classical-like” dynamical system on the sphere, wefollow a procedure similar to that presented in Section 7.2 for the driven oscillator.One easily derives the transition probability P ( j)

jm for an extremal state as an initialstate:

|Ψin〉 = | j , j〉 –→ |Ψout〉 = | j , m〉 , (7.66)

P ( j )jm =

(2 j

j + m

) (cos2 θ

2

) j+m (sin2 θ

2

) j–m

=

(2 j

j + m

)|� j–m |2

(1 + |�|2)2 j .

(7.67)

We recognize here the binomial distribution discussed in Section 6.4 with the con-struction of coherent spin states. Actually we get more, since, for the general tran-sition

|Ψin〉 = | j , m′〉 –→ |Ψout〉 = | j , m〉 ,

the corresponding probability P ( j )m′m is precisely given by the squared modulus of a

spin spherical harmonic:

P jm′m =

4π2 j + 1

∣∣m′Y j , j–m (r)∣∣2

=∣∣∣D j

j–m,m′ (� (Rr ))∣∣∣2

. (7.68)

Hence, we find in this elementary model a nice and deep physical interpretationof the discrete probability distribution appearing in the structure of general spincoherent states.

Let us end this section by presenting an elementary example of the above model.For a field of the form

H(t) = A k + B (cos ωt ı + sin ωt j) , (7.69)

we obtain

�(t) =–ω⊥ sin Ωt ei(ω–ω‖)t

2Ω cos Ωt + i (ω – ω‖) sin Ωt(7.70)

with Ω = 12

√(ω – ω‖)2 + ω2

⊥, ω‖ = (μ/ j ) A, ω⊥ = (μ/ j ) B . If we choose j = 12 , we

recover the well-known formula for the quantum precession “spin-flip”:

P – 12

12(t) =

ω2⊥ sin2 Ωt

(ω – ω‖)2 + ω2⊥

. (7.71)

7.5Application of Spin Coherent States to Classical and Thermodynamical Limits

We now consider a system of N quantum spins, all of them sharing the same spinvalue, say, j. The model was studied by Lieb [19] and revisited in [83, 84]. Lieb


derived rigorous results concerning the intertwining between thermodynamical(N → ∞) and classical (� → 0 or equivalently j → ∞, whereas � j = const.)limits. One can reasonably hope that in the j → ∞ limit and after rescaling spinoperators as

J→ Jj

, (7.72)

one gets the classical counterpart of the system, precisely objects living in thatworld where the quantum spin observables are replaced by classical vectors andintegrals on the unit sphere S2 are substituted for traces of operators. This has beenproven for the Heisenberg model at fixed N. But what about the commutativitybetween the thermodynamical limit N →∞ and the classical limit j →∞?

7.5.1Symbols and Traces

As a preliminary to the control of the commutativity of the two limits, Lieb hasproven the following inequalities, derived also and independently by Berezin [90,91]:

Z Cl( j )︸︷︷︸classical-like partition fct.

u Z Qt( j )︸︷︷︸quantum partition fct.

u Z Cl( j + 1) . (7.73)

From them one can infer that, in the limit j →∞, the three quantities involved goto the same classical limit. Think of the analogy

‖JCl‖2 = j 2 u j ( j + 1) = ( j + 1/2)2 – 1/4︸︷︷︸eigenvalue of (JQt)2

u ( j + 1)2 . (7.74)

The proof makes use of the spin (or Bloch) coherent states:

|r〉 = |θ, φ〉 =j∑

m=– j

√(2 j

j + m

)(cos

θ2

) j+m (sin

θ2

) j–m

ei( j–m)φ| jm〉 .

(7.75)

With Berezin and Lieb, let us associate with an operator A acting in the Her-mitian space H j ~= C2 j+1 its two “symbols” with respect to the frame or “filter”provided by the overcomplete set of states |r〉:(L) Covariant or lower symbol or expected value in state |r〉:

A(r) = 〈r |A|r 〉 , (7.76)

(U) Contravariant or upper symbol◦A(r)9):

A =2 j + 1

4π

∫S2

dr◦A(r)|r〉〈r | . (7.77)

9) We adopt here an alternative notation forupper symbols to avoid the appearance of toomany hats!


Note here that◦A(r) is not necessarily positive, even though A is a positive operator.

If this integral of operators makes sense (in a weak sense), there is no reason to

have uniqueness of the upper symbol◦A(r). Nevertheless, it is always possible to

choose it so that it is infinitely differentiable with respect to the angular coordinatesθ, φ of r or with respect to the real and imaginary parts of the complex parameter� = tan θ

2 eiφ.In the computation of operator traces, one may use alternatively lower and upper

symbols. Indeed, since the kernel K (r ′, r ) def= 〈r ′|r〉 is reproducing (owing to theresolution of the identity),

K (r ′, r ) =2 j + 1

4π

∫S2

dr K (r ′, r ) K (r , r ′′) , (7.78)

the following result for the trace of an operator A (if, of course, the latter is traceclass):

(i)

tr A =2 j + 1

4π

∫S2

dr 〈r |A|r〉 =2 j + 1

4π

∫S2

dr A(r) , (7.79)

on one hand, and

(ii)

tr A =2 j + 1

4π

∫S2

dr◦A(r) tr |r〉〈r | = 2 j + 1

4π

∫S2

dr◦A(r) K (r , r)︸︷︷︸

=1

=2 j + 1

4π

∫S2

dr◦A(r) (7.80)

on the other hand.

Let us give some examples of computation of lower A(r) and upper◦A(r) symbols

for expressions involving spin operators, such as those appearing in Hamiltoniansfor spin systems. One easily proves that

J(r) = 〈r |J|r〉 = j r , (7.81)

whereas

◦J(r) = ( j + 1) r . (7.82)

For the dyad JJ def= { Ji J j , i , j = x , y , z},

ˇ︷︸︸︷JJ (r ) = j ( j – 1

2 ) r r +j2

, (7.83)

◦︷︸︸︷JJ (r ) = ( j + 1)( j + 3

2 ) r r –j + 1

2. (7.84)


7.5.2Berezin–Lieb Inequalities for the Partition Function

Let us first fix the Hilbertian framework for the quantum partition function. Inconsidering the system of N quantum spins Ji , i = 1, 2, . . . , N , we suppose that theHamiltonian of the system is polynomial in the 3N spin operators. We have

Z Qt = αN tr e–� H , αNdef=

N∏i=1

(2 j i + 1)–1 , (7.85)

where the coefficient αN is needed for normalization. The Hilbertian framework isprecisely

HN =N⊗

i=1

H ji

~=N⊗

i=1

C2 j+1 . (7.86)

We next denote by |rN 〉 the overcomplete set of normalized states inHN built fromthe individual spin coherent states:

|rN 〉 =N⊗

i=1

|r i 〉 , rN ∈ SNdef= S2 ~ S2 ~ · · · ~ S2︸︷︷︸

N

. (7.87)

7.5.2.1Classical Lower Bound for the Partition FunctionFirst let us recall the Peierls–Bogoliubov inequality:

〈ψ|eA|ψ〉 v e〈ψ|A|ψ〉 , (7.88)

which holds for any unit norm state ψ and any self-adjoint operator A in a finite-dimensional space. This inequality is derived from the spectral decomposition of A.The following inequality obeyed by the partition function results:

Z Qt =1

(4π)N

∫SN

drN 〈rN |e–�H |rN 〉 v1

(4π)N

∫SN

drN e–�H (rN ) , (7.89)

where H(rN ) def= 〈rN |H |rN 〉. Now, if the quantum Hamiltonian H is linear in thecomponents of each of the spin operators Ji (this case is said to be normal), thenthe lower symbol H(rN ) is readily obtained from the expression of H by replacingeach Ji by j i r i . So,

Z Qt v Z Cl( j 1, j 2, . . . , jN ) . (7.90)

7.5.2.2Classical Upper Bound for the Partition FunctionTo evaluate an upper bound, we consider the quantum Z Qt as the limit of familiarapproximations of the exponential

Z Qt = tr e–�H = limn→∞

Z (n)


where

Z (n) = αN tr

(1 –

�Hn

)ndef= αN tr

(F (n)

)= αN

∫dr1

N

∫dr2

N · · ·∫

drnN

n∏μ=1

◦F n

(rμ

N

)L j

(rμ

N , rμ+1N

)with n + 1 == 1 in the last factor. The factor

◦F n(rμ

N ) =

(1 –

�◦

H(rμN )

n

)n

is the upper symbol of

F (n) =

(1 –

�Hn

)n

.

We now observe that

L j (r ′N , rN ) def=1

(4π)N αN

N∏i=1

K (r ′i , r i )

is reproducing since it inherits from the set of kernel factors K (r ′i , r i ) = 〈r ′i |r i 〉their reproducing properties. We then write Z (n) = αn tr(F n L j )n in consideringL j as the kernel of a compact self-adjoint operator10) L j on L2(SN , drN ) and F n asa multiplication operator. We next make use of the following set of inequalities(derived from Cauchy–Schwarz inequality and others, see, e.g., [92]) for two self-adjoint operators A and B:∣∣tr(AB )2m

∣∣ u tr(A2 B2

)mu tr A2m B2m , m = 2l , l = 0, 1, 2, . . . .

We eventually infer from them the upper bound:

Z Qt u1

(4π)N

∫SN

drN e–�◦

H (rN ) .

In the normal case, it is enough to replace each spin operator Ji appearing in H

by ( j i + 1) r i to get the upper symbol◦

H(rN ) of the Hamiltonian. We can concludethat

Z Qt u Z Cl( j1 + 1, j 2 + 1, . . . , j N + 1) , (7.91)

and from (7.90) that the Berezin–Lieb inequalities hold true:

Z Cl( j 1, j2, . . . , j N ) u Z Qt u Z Cl( j 1 + 1, j 2 + 1, . . . , j N + 1) . (7.92)

10) An operator A in a Hilbert space is saidto be compact if it can be expanded asA =

∑n

λn | f n〉〈g n |, where {| f n〉} and{|g n〉} are (not necessarily complete)

orthonormal sets. The λn ’s form a sequenceof positive numbers, called the singularvalues of the operator. The singular valuescan accumulate only at zero.


7.5.3Application to the Heisenberg Model

Let us apply the inequalities (7.92) to the Heisenberg model in the elementary caseof one spin value J only. We rescale the spin operators through Ji = Si/ J to get forthe N-spins Hamiltonian

H =1J2

∑i , j

Si · S j == HQtN ( J) . (7.93)

The quantum partition function reads as

Z QtN ( J) =

1(2 J + 1)N e–�HQt

N ( J) == e–�N f QtN ( J ) , (7.94)

where f QtN ( J) is the corresponding free energy per spin. Now consider the specific

Berezin–Lieb inequalities:

Z Cl( J) =1

(4π)N

∫SN

drN e–�H (rN )︸︷︷︸==Z Cl

(J2

J2 �)

u Z Qt( J) u Z Cl( J + 1)

=1

(4π)N

∫SN

drN e–�◦

H (rN )︸︷︷︸==Z Cl

(( J+1)2

J2 �)

. (7.95)

The lower Z Cl(�) and upper Z Cl(

( J+1)2

J2 �)

bounds are uniform with respect to the

size N of the system. They are readily computed from

H(rN ) =1J2

∑i , j

J r i · J r j =∑

i , j

r i · r j == HCl ,

◦H(rN ) =

1J2

∑i , j

( J + 1) r i · ( J + 1) r j W J largeHCl .

Hence, the combined classical and thermodynamical limits of the quantum spinsystem are just reached through a simple bound estimate for the free energy perspin:

limJ→∞

limN→∞

f QtN ( J) = f Cl = lim

N→∞–

1�

ln Z ClN . (7.96)

117

8SU(1,1) or SL(2,R) Coherent States

8.1Introduction

This chapter is devoted to the third most known family of coherent states, namely,the SU (1, 1) coherent states as they were established by Perelomov [10] in a group-theoretical approach. We adopt instead the construction set out in Chapter 4, choos-ing as an observation set the unit disk in the complex plane. Then we describethe main properties of these coherent states, that is, we list and comment on thesequence of properties as we did in Section 3.1, probabilistic aspects, link withSU (1, 1) representations, classical aspects, and so on. Finally, we make a short in-cursion in signal analysis by exploiting the fact that the unit disk has an unboundedrepresentation that is the Poincaré half-plane of time-scale parameters. In this rep-resentation, the group SU (1, 1) is transformed into its real copy, namely, SL(2, R).We then recover the continuous wavelet or time-scale transform of signals, which isprecisely based on the subgroup of SL(2, R) describing the affine transformationsof the real line, R � t �→ b + a t , b , a ∈ R with a > 0

Note that there exists another family of coherent states associated with SU (1, 1),namely, the Barut–Girardello coherent states. They will be considered in the nextchapter through their appearance in the quantum motion problem of a particle inan infinite square well potential and also in the Pöschl–Teller potentials.

8.2The Unit Disk as an Observation Set

Besides the complex plane, the infinite cylinder, and the sphere, there is the (open)

unit disk D def= {z ∈ C , |z| < 1}. There exist many situations in physics where theunit disk is involved as a fundamental model or at least is used as a pedagogicaltoy. For instance, it is a model of phase space for the motion of a material parti-cle on a one-sheeted two-dimensional hyperboloid viewed as a (1 + 1)-dimensionalspace-time with negative constant curvature, namely, the two-dimensional anti deSitter space-time [93–95]. In signal analysis, the time-scale half-plane, which repre-


118 8 SU (1,1) or SL(2, R) Coherent States

sents a nonbounded version of the unit disk, is the set of variables for (continuous)wavelet transform [11].

As a simple illustration of two-dimensional hyperbolic (Lobatcheskian) geome-try, the unit disk has nice properties analogous to those of the sphere, except thefact that it is a noncompact bounded domain. It is commonly used as a modelfor the hyperbolic plane, by introducing a new metric on it, the Poincaré metric.The Poincaré metric is the metric tensor describing a two-dimensional surface ofconstant negative curvature. It reads in the present case (up to a constant factor) as

ds2 =dz dz

(1 – |z|2). (8.1)

The corresponding surface element is given by

μ(d2z) =d(Rz) d(Iz)(1 – |z|2)2

==i2

dz ∧ dz(1 – |z|2)2 . (8.2)

These quantities both emerge from a so-called Kählerian potential KD:

KD(z, z) def= π–1(1 – |z|2)–2 ,

ds2 =12

∂2

∂z ∂zlnKD(z, z) dz dz ,

μ(d2z) =i4

∂2

∂z ∂zlnKD(z, z) dz ∧ dz .

The unit disk equipped with such a potential has the structure of a two-dimensionalKählerian manifold [10, 96], an appellation shared by the complex plane, for whichthe potential isKC = π–1 e–z z , the sphere S2, or equivalently the projective complexline CP1, for which KS2 = π–1(1 + |z|2)–2, and the torus C/Z2 � S1 ~ S1. Note thatany Kählerian manifold is symplectic and so can be given a sense of phase spacefor some mechanical system. We will examine later the properties of these aspectsin terms of symmetries of the unit disk.

Besides the unit disk, there are two other equivalent representations common-ly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane,already mentioned, and defining a model of hyperbolic space on the upper half-plane. The disk D and the upper half-plane P+ = {Z ∈ C , IZ > 0} are related bya conformal map, called Möbius transformation,

P+ � Z �→ z = eiφ Z – Z 0

Z – Z 0∈ D , (8.3)

φ and Z 0 being arbitrary. The canonical mapping is given by Z 0 = i and φ = π/2.It takes i to the center of the disk and the origin O to the bottom of the disk. Theother representation is the punctured disk model, defined by z = eiπZ , IZ > 0.

8.3 Coherent States 119

8.3Coherent States

Let η be a real parameter such that η > 1/2 and let us equip the unit disk witha measure proportional to (8.2):

μη(d2z) def=2η – 1

πμ(d2z) =

2η – 1π

d2z(1 – |z|2)2 . (8.4)

Consider now the Hilbert space L2η = L2(D, μη) of all functions f (z, z) on D that

are square-integrable with respect to μη. Within this “large” Hilbert space we selectall functions of the form

φ(z, z) = (1 – |z|2)η g (z) , (8.5)

where g (z) is holomorphic on D. The closure of the linear span of such functionsis a Hilbert subspace of L2

η denoted here byK+. An orthonormal basis ofK+ is givenby the countable set of functions

φn(z, z) ==

√(2η)n

n!(1 – |z|2)ηzn with n ∈ N, (8.6)

where (2η)n = Γ(2η+n)Γ(2η) is the Pochhammer symbol [18]. The proof is readily derived

from the integral representation of the beta function,

B (x , y ) =Γ(x)Γ(y )Γ(x + y )

=∫ 1

0tx–1(1 – t)y–1 dt .

Note that∞∑

n=0

|φn(z, z)|2 = 1 . (8.7)

We are now in a position to define the coherent states resulting from this choice.They read as the following superpositions of vectors |en〉 forming an orthonormalbasis in some separable Hilbert space H:

|z; η〉 def=∞∑

n=0

φn(z, z)|en〉 = (1 – |z|2)η∞∑

n=0

√(2η)n

n!zn |en〉 . (8.8)

Again, by construction, these states are normalized and solve the identity IH inH:

〈z; η|z; η〉 = 1 ,∫D

μη(d2z) |z; η〉〈z; η| = IH . (8.9)

Their mutual overlap offers an explicit representation of these states as elementsof the Hilbert space K+. It is readily obtained from the binomial expansion:

〈z′; η|z; η〉 = (1 – |z|2)η (1 – z′ z)–2η (1 – |z′|2)η . (8.10)

It is also a reproducing kernel, for which the Hilbert space K+ is a Fock–Bargmannspace, analogous to those encountered for the standard and spin coherent states.


8.4Probabilistic Interpretation

As we could have guessed from the choice of the orthonormal set, we find beneaththe structure of the states (8.8) a duality between two types of probability distribu-tions. The first one is discrete and reads as

n �→ |φn(z, z)|2 =(2η)n

n!(1 – |z|2)2η|z|2n == P (2η, n; 1 – |z|2) . (8.11)

It is, when 2η is an integer v 1, a negative binomial distribution (see Appendix A).Recall that for a fixed integer m v 1, the negative binomial distribution is given by

P (m, n; λ) =Γ(m + n)

Γ(n + 1)Γ(m)λm (1 – λ)n , n = 0, 1, 2, . . . , (8.12)

where the parameter λ lies in the interval (0, 1). The quantity P (m, n, λ) can bethought of as being the probability that m + n is the number of independent trialsthat are necessary to obtain the result of m successes (the (m + n)th trial beinga success) when λ is the probability of success in a single trial. The term negativebinomial stems from the fact that

(1 – λ)–k =∞∑

n=0

Γ(k + n)Γ(n + 1)Γ(k)

λn ,

from which it also follows that

∞∑n=0

P (m, n; λ) = 1 . (8.13)

The second distribution is continuous in the variable |z|2,

|z|2 �→ 2η – 1(1 – |z|2)2 |φn (z, z)|2

=Γ(2η + n)

Γ(2η – 1)n!(1 – |z|2)2η–2|z|2n == �(1 – |z|2; 2η – 1, n + 1) . (8.14)

It is a beta distribution in the variable λ = 1 – |z|2 ∈ [0, 1], with parameters 2η –1 and n + 1 = 1, 2, 3, . . . ,∞. We recall that the beta distribution in the variableλ ∈ [0, 1], with discrete parameters m, n = 1, 2, 3, . . ., is derived from the integralrepresentation of the beta function and is given by

�(λ; m, n) =1

B (m, n)λm–1(1 – λ)n–1 ,

∫ 1

0�(λ; m, n) dλ = 1 . (8.15)

Thus, we observe that the prior measure on the parameter space [0, 1] is just dλ,whereas the associated Bayesian posteriors are the quantities 2η–1

(1–|z|2)2 |φn(z, z)|2

with λ = 1 – |z|2.

8.5 Poincaré Half-Plane for Time-Scale Analysis 121

8.5Poincaré Half-Plane for Time-Scale Analysis

As indicated in the introduction, we can use the Poincaré half-plane P+ as a set ofparameters for the coherent states described in this chapter. We choose the canon-ical Möbius transformation mapping the unit disk D onto P+:

D � z �→ Z =z + i

iz + 1∈ P+ , (8.16)

and conversely z =Z – i1 – iZ

. (8.17)

Note that when extended to the boundaries, the bijection (8.16) is a Cayley trans-formation that maps in a stereographic way the unit circle S1 onto the real line

S1 � eiθ �→ t =eiθ + i

ieiθ + 1∈ R , θ ∈ [0, 2π) , (8.18)

where θ = 0 �→ t = 1, θ = π/2 �→ t =∞, θ = π �→ t = –1 and θ = 3π2 �→ 0.

Let us introduce the (x , y ), y > 0, variables as the real and imaginary partsof Z: Z = x + i y . In continuous wavelet analysis the real part x would have themeaning of a time variable, whereas y would stand for a scale. These coordinatesare expressed in terms of the preimage z of Z as

x =2R(z)

1 + |z|2 – 2I(z), y =

1 – |z|21 + |z|2 – 2I(z)

. (8.19)

The relation between respective Poincaré metrics is given by

ds2 = (1 – |z|2)–1dz dz =dx2 + d y 2

4y 2 . (8.20)

The Lebesgue measures in the half-plane and in the open disk are related by

d2Z =4

(|z|2 + 1 – 2I(z))2 d2z , (8.21)

which, in terms of x and y, gives the relation between the respective Poincaré sur-face elements:

(1 – |z|2)–2 d2z =dx d y4y 2 . (8.22)

Inversely, going from the half-plane to the disk, we have

z = z(x , y ) =2x

x2 + (1 + y )2 + ix2 + y 2 – 1

x2 + (1 + y )2 . (8.23)

We also note also the useful formulas

|z|2 =x2 + (y – 1)2

x2 + (y + 1)2 , 1 – |z|2 =4y

x2 + (y + 1)2 . (8.24)


In terms of parameters x and y, the coherent states (8.8) read as

|x , y ; η〉 def= (4y )η∞∑

n=0

√(2η)n

n!(2x + i (y 2 + x2 – 1))n

(x2 + (1 + y )2)η+n |en〉 . (8.25)

The resolution of the unity now reads as

2η – 12π

∫P+

|x , y ; η〉〈x , y ; η| dxd y4y 2 = IH . (8.26)

The expression (8.25) is a lot more involved than the one on the disk. However, itmakes more transparent the time-scale content of such objects in view of utilizationin analyzing a temporal signal s(t) belonging to the Hilbert space L2(R, dt). Choos-ing the abstractH as the latter and some explicit orthonormal basis {|en〉 , n ∈ N},we derive from the decomposition |s〉 =

∑∞n=0 sn |en〉 a time-scale transform of the

signal as the following function of time and scale parameters (borrowing fromwavelet analysis the usual notation b == x , a == y ):

Sη(b , a) def= 〈b , a; η|s〉

= (4a)η∞∑

n=0

sn

√(2η)n

n!(2b – i (a2 + b2 – 1))n

(b2 + (1 + a)2)η+n . (8.27)

However, whereas the sensu stricto continuous wavelet transform is based on thesubgroup of SL(2, R) describing the affine transformations of the real line, R �t �→ b + a t , b , a ∈ R with a > 0, as will be described in the last section of thischapter, the time-scale representation of signals based on the SU (1, 1) � SL(2, R)coherent states, as exemplified by (8.27), is of a different nature. As a matter offact, it does not possess the affine covariant properties of the continuous wavelettransform. Furthermore, it depends on a prior Hilbertian decomposition of thesignal versus a certain basis, and, at small scale a << 1 (habitually considered asthe most interesting part of the analysis), it discriminates signals by their timeevolution only, yielding a portrait of the signal in terms of the Fourier series:

Sη(b , a) W( 4a

b2 + 1

)η ∞∑n=0

sn

√(2η)n

n!einθb , with θb = arctan

(2b

b2 – 1

).

(8.28)

However, it offers the opportunity to play with the extra parameter η > 1/2.

8.6Symmetries of the Disk and the Half-Plane

Like the sphere S2 is invariant under space rotations forming the group SO (3) �SU (2)/Z2, Z2 = {1, –1}, the unit disk D is invariant under transformations of thehomographic or Möbius type:

D � z �→ z′ = (α z + �) (� z + α)–1 ∈ D , (8.29)

8.7 Group-Theoretical Content of the Coherent States 123

with α, � ∈ C and |α|2 – |�|2 =/ 0. Since a common factor of α and � is unimportantin the transformation (8.29), one can associate with the latter the 2 ~ 2 complexmatrix (

α �� α

)def= g , with det g = |α|2 – |�|2 = 1 , (8.30)

and we will write z′ = g · z. These matrices form the group SU (1, 1), the simplestexample of a simple, noncompact Lie group (see Appendix B). It should be notedthat SU (1, 1) leaves invariant the boundary S1 � U (1) of D under the transforma-tions (8.29).

The invariance of D under (8.29) is not only geometrical. It also holds for thePoincaré metric (8.1) and surface element (8.2), since both emerge from the invari-ant Kählerian potential KD:

KD(z, z) = π–1(1 – |z|2)–2 = π–1(1 – |z′|2)–2 ,

ds2 =dz dz

(1 – |z|2)=

dz′ dz′

(1 – |z′|2),

μ(d2z) =d(Rz) d(Iz)(1 – |z|2)2 =

d(Rz′) d(Iz′)(1 – |z′|2)2 .

This invariance is the essence of Lobatchevskian geometry [96].Let us now turn our attention to the corresponding symmetries in the Poincaré

half-plane. Let us write the canonical Möbius transformation (8.17), as

z =Z – i1 – iZ

==1√2

(1 –i–i 1

)· Z == m · Z , Z = m–1 · z . (8.31)

Therefore, the transformation z′ = g · z, where g ∈ SU (1, 1), becomes in thehalf-plane the transformation Z ′ = s · Z , with

s = m–1 g m =

(Rα + I� Iα + R�–Iα + R� Rα – I�

)==

(a bc d

), a , b , c , d ∈ R . (8.32)

Since det s = 1, the set of such 2 ~ 2 real matrices form the group SL(2, R), whichleaves invariant the upper half-plane, and its Poincaré metric (8.20) and surfaceelement (8.22) as well.

8.7Group-Theoretical Content of the Coherent States

8.7.1Cartan Factorization

In semisimple group theory there exists a well-known group factorization calledthe Cartan decomposition (see Appendix B) or “phase-space” decomposition in the


present context when the unit disk, as a symplectic manifold, is given a phase-space meaning. The Cartan decomposition of SU (1, 1), denoted by SU (1, 1) = P H ,means that any g ∈ SU (1, 1) can be written as the product g = p h , with p ∈ Pand h ∈ H . It is defined by the Cartan involution iph : g �→ (g †)–1 in the sense thatP is made of all p ∈ SU (1, 1) such that iph( p) = p–1, that is, p = p† is Hermitian,while H has all its elements unchanged under iph, that is, h† = h–1, which meansthat h is unitary. In consequence, H ~= U (1) is the unitary subgroup of SU (1, 1).The decomposition reads explicitly

SU (1, 1) � g =

(α �� α

)= p(z) h(θ) , (8.33)

with

p(z) =

(δ δz

δz δ

), z = �α–1 , δ = (1 – |z|2)–1/2 (8.34)

and

h(θ) =

(eiθ/2 0

0 e–iθ/2

), θ = 2 arg α , 0 u θ < 4π . (8.35)

The bundle section11)D ∈ z �→ p(z) ∈ P gives the unit disk D a symmetricspace realization identified with the coset space SU (1, 1)/H . We remark that p2 =g g † and that ( p(z))–1 = p(–z). We can exploit the Cartan factorization by makingSU (1, 1) act on D through a left action on the set of matrices p(z). Explicitly,

g : p(z) �→ p(z′) defined by g p(z) = p(z′) h ′ . (8.36)

It is then easily verified that z′ is given by the Möbius action (8.29): z′ = g · z.

8.7.2Discrete Series of SU(1, 1)

We now consider a class of unitary irreducible representations of SU (1, 1), pre-cisely indexed by the parameter η appearing in the measure on the unit disk, andinvolved in the construction of the coherent states, to which this chapter is devot-ed. For a given η > 1, we introduce the Fock–Bargmann Hilbert space FBη of allanalytical functions f (z) on D that are square-integrable with respect to the scalarproduct:

〈 f 1| f 2〉 =2η – 1

2π

∫D

f 1(z) f 2(z) (1 – |z|2)2η–2 d2z . (8.37)

11) A nontrivial fiber bundle consists of fourobjects, (E , B , π, F ), where E, B, and Fare topological spaces and π : E �→ Bis a continuous surjection such that E islocally (but not globally!) homeomorphicto the Cartesian product B ~ F . B is called

the base space of the bundle, E the totalspace, and F the fiber. The map π is calledthe projection map (or bundle projection).A section (or cross section) is a continuousmap, s : B �→ E , such that π(s(x)) = x for allx in B.


Note that the elements of this space are just the conjugate of the elements of K+

cleared of their nonanalytical factor (1–|z|2)2η. The orthonormal basis given by (8.6)is now made of powers of z suitably normalized:

pn(z) ==

√(2η)n

n!zn with n ∈ N . (8.38)

We define, for η = 1, 3/2, 2, 5/2, . . . the unitary irreducible representation

g =

(α �� α

)�→ U η(g )

of SU (1, 1) on FBη by

FBη � f (z) �→(U η(g ) f

)(z) = (–� z + α)–2η f

(αz – �

–�z + α

). (8.39)

This countable set of representations constitutes the “almost complete” holomor-phic discrete series of representations of SU (1, 1) [97–99]. It is “almost complete”because the lowest one, which corresponds to the value η = 1/2, requires a specialtreatment owing to the nonexistence of the inner product (8.37) in this case: thereis no Fock–Bargmann realization in that case. We will come back to this importantquestion in the last section. Had we considered the continuous set η ∈ [1/2, +∞),we would have been led to involving the universal covering of SU (1, 1) [100, 101].

The matrix elements of the operator U η(g ) with respect to the orthonormal ba-sis (8.38) are given (see, e.g., [102] or Appendix A in [103]) in terms of hypergeo-metrical polynomials by

U ηnn′ (g ) = 〈pn |U η(g )|pn′ 〉 =

(n>! Γ(2η + n>)n<! Γ(2η + n<)

)1/2

α–2η–n> αn<

~(γ(�, �))n>–n<

(n> – n<)! 2F 1

(–n< , n> + 2η ; n> – n< + 1 ;

|�|2|α|2

), (8.40)

where

γ(�, �) =

{–� n> = n′

� n> = n, n>

<=

maxmin

(n, n′) v 0 .

Owing to the relation |�|2|α|2 = 1 – 1

|α|2 , this expression is alternatively given in termsof Jacobi polynomials as follows:

U ηnn′ (g ) =

(n<! Γ(2η + n>)n>! Γ(2η + n<)

)1/2

α–2η–n> αn<

~(γ(�, �))n>–n<√

(n> – n<)!P (n>–n< , 2η–1)

n<

(1 – |�|21 + |�|2

). (8.41)


8.7.3Lie Algebra Aspects

Any element g ∈ SU (1, 1) can also be factorized, in a nonunique way, in terms ofthree one-parameter subgroup elements: g = ± h(θ) s(u) l(v ). Besides the sign ±,the first factor was already encountered in the Cartan decomposition (8.35), where-as the others are of noncompact hyperbolic type and are given by

s(u) =

(cosh u sinh usinh u cosh u

), l(v ) =

(cosh v i sinh v

–i sinh v cosh v

), u , v ∈ R .

(8.42)

The first subgroup is isomorphic to U (1), whereas the two others are isomorphicto R. Their respective generators, N μ, μ = 0, 1, 2, are defined by

h(θ) = eθ N 0 , s(u) = eu N 1 , h(θ) = ev N 2 , (8.43)

and are given in terms of the Pauli matrices by

N 0 =i2

σ3 , N 1 =12

σ1 , N 2 = –12

σ2 . (8.44)

They form a basis of the Lie algebra su(1, 1) and obey the commutation relations

[N 0, N 1] = N 2 , [N 0, N 2] = –N 1 , [N 1, N 2] = –N 0 . (8.45)

Their respective self-adjoint representatives under the unitary irreducible represen-tation (8.39), defined generically as –i ∂/∂t U η(g (t)), are the following differentialoperators on the Fock–Bargmann space FBη

i N 0 �→ K 0 = zd

dz+ η , (8.46a)

i N 1 �→ K 1 = –i2

(1 – z2)d

dz+ iηz , (8.46b)

i N 2 �→ K 2 =12

(1 + z2)d

dz+ ηz , (8.46c)

and obey the commutation rules

[K 0, K 1] = i K 2 , [K 0, K 2] = –i K 1 , [K 1, K 2] = –i K 0 . (8.47)

We may check that the elements of the orthonormal basis (8.38) are eigenvectorsof the compact generator K 0 with equally spaced eigenvalues:

K 0 |pn〉 = (η + n) |pn〉 . (8.48)

The particular element |p0〉 of the basis is a lowest weight or “vacuum” state for therepresentations U η. Indeed, let us introduce the two operators with their commu-tation relation:

K ± = ∓i (K 1 ± iK 2) = K 2 ∓ iK 1 , [K +, K –] = –2K 0 . (8.49)


As differential operators, they read as K + = z2 d/dz + 2ηz, K – = d/dz. Adjoint ofeach other, they are raising and lowering operators, respectively,

K + |pn〉 =√

(n + 1)(2η + n) |pn+1〉 ,

K – |pn〉 =√

n (2η + n – 1) |pn–1〉 ,(8.50)

and, as announced, K – |p0〉 = 0. The states |pn〉 are themselves obtained by suc-cessive ladder actions on the lowest state as follows:

|pn〉 =

√Γ(2η)

Γ(2η + n) n!(K +)n |p0〉 . (8.51)

One can also check directly from (8.46a) that the Casimir operator

C def= K 21 + K 2

2 – K 20 =

K +K – + K –K +

2– K 2

0 (8.52)

is fixed at the value C = –η(η – 1) Id on the space FBη that carries the unitaryirreducible representation U η.

8.7.4Coherent States as a Transported Vacuum

We are now in the position to explain the group-theoretical content of the coherentstates (8.8), that is, the rationale behind the appellation SU (1, 1) coherent states, asthey were introduced by Perelomov. To each element z in the unit disk correspondsthe element p(z) (note the conjugate variable) of SU (1, 1), defined in (8.34) fromthe Cartan decomposition. Let us now apply to the lowest state |p0〉 the operatorsof the representation U η restricted to the set P of such matrices, and expand the“transported” state in terms of the Fock–Bargmann basis:

U η( p(z)) |p0〉 =∞∑

n=0

U ηn0( p(z)) |pn〉 = (1 – |z|2)η

∞∑n=0

√(2η)n

n!zn |pn〉 . (8.53)

Thus, the coherent state defined in (8.8) is exactly this transported state:

U η( p(z)) |p0〉 = |z; η〉 . (8.54)

In the same spirit, we could think of transporting any other element of the Fock–Bargmann basis under the action of the unit disk through the bundle sectionz �→ p(z). We thus obtain a discretely indexed set of families of coherent states,defined by

|z; η; m〉 def= U η( p(z)) |pm〉 =∞∑

n=0

U ηnm ( p(z)) |pn〉 , (8.55)

with

U ηnm ( p(z)) =

(n>! Γ(2η + n>)n<! Γ(2η + n<)

)1/2

(1 – |z|2)η |z|n>–n<

(n> – n<)!ei(n–m)φ

~ (sgn(n – m))n–m2F 1 (–n< , n> + 2η ; n> – n< + 1 ; |z|2

), (8.56)


with z = |z|eiφ. To fully justify the adjective “coherent,” it is necessary to prove thatthese normalized states resolve the unity in FBη:∫

Dμη(d2z) |z; η; m〉〈z; η; m| = IFBη . (8.57)

Indeed, from the representation property combined with (8.36), we have

U η(g ) U η( p(z)) = U η(g p(z)) = U η( p(g · z) U η(h ′) ,

which holds for any g ∈ SU (1, 1) and where h ′ ∈ H . Now, with

h ′ =

(eiθ′/2 0

0 e–iθ′/2

), U η(h ′) |pm〉 = e–i(η+m)θ′ |pm〉 .

So this phase factor disappears from

U η(g )∫D

μη(dz dz) |z; η; m〉〈z; η; m|U η(g–1)

=∫D

μη(dz dz) |g · z; η; m〉〈g · z; η; m| .

From the invariance of the measure and Schur’s lemma, we deduce that the left-hand side of (8.57) is a multiple of the identity. It is straightforward to check thatthis factor is 1. Mutatis mutandis, we deduce from the resolution of the unity thatthe set of functions {U η

nm ( p(z)) , n ∈ N} is orthonormal with respect to the mea-sure μη on the unit disk. The coherent states (8.55) are the counterparts of SU (2)sigma-spin coherent states |r ; σ〉 defined in (6.9) and Weyl–Heisenberg coherentstates |z; m〉 defined in (7.31). It is possible to give the discrete probability distribu-tion n �→ |U η

nm ( p(z))|2 with parameter |z|2 or its counterpart on the Poincaré half-plane a physical meaning in terms of transition probability and S matrix, like wedid for the two other types of coherent states in Chapter 7. Examples are providedby the Morse Hamiltonian (see [104] and references therein) and its supersymmet-ric aspects [105, 106], and also by the Schrödinger operator with a magnetic fieldon the Poincaré upper half-plane [107, 108].

There exists an alternative expression for the “SU (1, 1) displacement” operatorU η( p(z)) used to build coherent states. It is, in the present context, the counterpartof the displacement operator D (z) = exp(za† – za) involved in the construction ofthe standard coherent states. With an element z of the unit disk D we associate thecomplex variable � with the same argument φ and modulus |�| = tanh–1 |z|, that is,z = tanh |�| eiφ. Then we have the relation

U η( p(z)) = e� K +–� K – == D η(�) . (8.58)

The demonstration exploits the 2 ~ 2 matrix representation of the elements ofSU (1, 1) and its Lie algebra and the various factorizations on the group. Indeed,the result of the action of D η(�) on the lowest state |p0〉 will be made almost trivialif we are able to factorize it as

D η(�) = eA(z)K + eB (z)K 0 eC (z)K – ,

8.8 A Few Words on Continuous Wavelet Analysis 129

since then, from K – |p0〉 = 0, K 0 |p0〉 = η |p0〉, and (8.51), we get

Dη(�) |p0〉 = eA(z)K + eB (z)K 0 eC (z)K – |p0〉

= eB (z)η∞∑

n=0

√(2η)n

n!(A(z))n |pn〉 . (8.59)

Now, from the inverse of the correspondences (8.46a),

� K + – � K – �→ i (� N + – � N –) =

(0 �� 0

),

we obtain by exponentiation

D η(�) �→(

cosh |�| e–iφ sinh |�|eiφ sinh |�| cosh |�|

)== Δ(�) .

This matrix is easily factorized as

Δ(�) = eA(z)iN + eB (z)iN 0 eC (z)iN –

=

(1 0

A(z) 1

)(e–B (z)/2 0

0 eB (z)/2

)(1 –C (z)0 1

)=

(e–B/2 –C e–B/2

Ae–B/2 eB/2 – Ae–B/2

).

A simple identification gives A(z) = z, e–B (z)/2 = cosh |�| = (1 – |z|2)–1/2, and C (z) =–z. We thus recover in the action (8.59) of D η(z) the SU (1, 1) coherent states (8.8).

This kind of disentangling technique for factorizing elaborate representation oper-ators is well known [9], and we will use it again in the chapters devoted to squeezedstates and to fermionic coherent states.

8.8A Few Words on Continuous Wavelet Analysis

The inner product (8.37) vanishes identically at the limit value η = 1/2. Howev-er, there exists a well-defined unitary irreducible representation of SU (1, 1) cor-responding to this value. Its carrier space can be realized as the Hilbert spaceL2(S1, dθ/2π) of the exponential Fourier series, that is, the space of square-inte-grable complex-valued functions on the boundary S1 of the unit disk D. Indeed,the action (8.29) of

g =

(α �� α

)∈ SU (1, 1)

on D extends to the boundary as

g · eiθ = (α eiθ + �) (� eiθ + α)–1 == eiθ′ ∈ S1 , (8.60)


and so leaves the latter invariant. It is then easy to check that the transformationUη=1/2(g ) == U (g ) in L2(S1, dθ/2π), defined by

(U (g ) f )(eiθ) = (–� eiθ + α)–1 f (g–1 · eiθ) == f ′(eiθ) , (8.61)

is unitary with respect to the inner product for Fourier series,

〈 f 1| f 2〉 =1

2π

∫ 2π

0f 1(eiθ) f 2(eiθ) dθ = 〈 f ′1| f ′2〉 . (8.62)

This is due to the transformation of the measure:

dθ′

dθ= |αeiθ – �|–2 .

Let us now transport this material onto the real line, the boundary of the upper half-plane P+, by restricting the transformations (8.31) and (8.32) to S1 � eiθ �→ x ∈ R:

x =eiθ + i1 + ieiθ =

cos θ1 – sin θ

, dθ =2 dx

1 + x2 . (8.63)

This gives on the level of the inner product (8.62) the relation

12π

∫ 2π

0f 1(eiθ) f 2(eiθ) dθ =

1π

∫ +∞

–∞f 1(m · x) f 2(m · x)

dx1 + x2 , (8.64)

where

m =1√2

(1 –i–i 1

)is the Möbius–Cayley matrix introduced in (8.31). Hence, any square-integrablefunction on the circle yields a square-integrable function on the real line with re-spect to the Lebesgue measure along the map:

L2(S1, dθ/2π) � f (eiθ) �→ F (x) def= (1 – ix) f (m.x) ∈ L2(R, dx) . (8.65)

On the level of group representations, the unitary representation (8.61) of SU (1, 1)is transported into the following unitary representation of SL(2, R) on L2(R, dx):

(U (s) F )(x) = (–cx + a)–1 f(s–1 · x

)== F ′(x) , s =

(a bc d

). (8.66)

We now restrict the above representation to the subgroup Aff(R) of affine transfor-mations of the real line, which is defined as

Aff(R) def=

{(b , a) ==

(sgn(|a|)

√a b

0 1√|a|

), b ∈ R , a =/ 0

}, (8.67)


with the action resulting from the Möbius transformation of the real line of thetype (8.29) (b , a) · x = b + a x and the group law

(b , a)(b ′, a′) = (b + ab ′, aa′) . (8.68)

Thus, Aff(R) is a semidirect product of the translation group R by the dilationgroup R∗ : Aff(R) = R � R∗. The unit element is (0, 1) and the inverse of (b , a) is(–a–1b , a–1). Continuous wavelet analysis rests on the essential result that [11, 109,110]:

Continuous Wavelet Analysis rests on the essential result that [11, 109], up tounitary equivalence, Aff(R) has a unique UIR, acting in L2(R, dx), namely

(U (b , a) f

)(x) = |a|–1/2 f

(x – b

a

)== f ba(x) (a =/ 0, b ∈ R) , (8.69)

or, on Fourier transforms, f (�) = 1√2π

∫ +∞–∞ e–i�x f (x) dx :

(U (b , a) f

)(�) = |a|1/2 f (a�)e–ib� (a =/ 0, b ∈ R) . (8.70)

The representation U is square integrable, i.e., for all admissible ψ ∈ L2(R, dx), thefunction (b , a) �→ 〈ψ|U (b , a)|ψ〉 is square integrable on Aff(R) w.r.t. its left Haarmeasure db da/a2. Now, a vector ψ ∈ L2(R, dx) is said admissible if it satisfies thecondition

cψ == 2π∫ ∞

–∞|ψ(�)|2 d�

|�| <∞ . (8.71)

In practice, the admissibility condition (8.71) (plus some regularity: ψ ∈ L1 ∩ L2

suffices) is equivalent to a zero mean condition:

ψ admissible(⇐)⇒ ψ(0) = 0 ⇔

∫ +∞

–∞ψ(x) dx = 0 . (8.72)

An admissible function will be called a wavelet. Thus a wavelet ψ is by necessityan oscillating function, real or complex-valued (see the examples below), and this isin fact the origin of the term “wavelet”. Let us explain in what sense this admis-sibility condition is crucial in signal analysis. Let ψ be a wavelet and s ∈ L2(R) afunction viewed as a signal within the present context. Then the continuous wavelettransform (CWT) of s with respect to ψ is the function S == T ψs on the time-scalehalf-plane, which is given by the scalar product of s with the transformed waveletψba:

S(b , a) = 〈ψba|s〉 =∫ +∞

–∞

ψ(a–1(x – b))√|a|

s(x) dx

=√|a|

∫ +∞

–∞ψ(a�) s(�) eib� d� . (8.73)


Then (8.71) ensures that the wavelet transform preserves the “energy” of the signal:

‖s‖2 =1cψ

∞∫–∞

db

∞∫0

daa2 |S(b , a)|2 . (8.74)

To a certain extent, this equation is equivalent to the resolution of the unity by thecontinuous family {ψba}:

Id =1cψ

∞∫–∞

db

∞∫0

daa2 |ψba〉〈ψba| . (8.75)

In this regard, the states |ψba〉 can be viewed as coherent states for the affine group.In practice one often imposes on the analyzing wavelet ψ a number of additional

properties, for instance, restrictions on the support of ψ and of ψ. Or ψ may berequired to have a certain number N v 1 of vanishing moments (by the admissibilitycondition (8.72), the moment of order 0 must always vanish):∫ ∞

–∞xnψ(x) dx = 0, n = 0, 1, . . . , N . (8.76)

This property improves its efficiency at detecting singularities in the signal. In-deed, the transform (8.73) is then blind to the smoothest part of the signal that ispolynomial of degree up to N — and less interesting, in general. Only the sharp-er part remains, including all singularities (like jumps in the signal or one of itsderivatives). For instance, if the first moment (n = 1) vanishes, the transform willerase any linear trend in the signal.

Let us give two well-known examples of wavelets, which are depicted in Fig-ure 8.1.

The Mexican hat or Marr wavelet: This is simply the second derivative of a Gaus-sian:

ψH (x) = (1 – x2) e–x2/2, ψH (�) = �2 e–�2/2 . (8.77)

−10 0 10−10 0 10

Fig. 8.1 Two usual wavelets: The Mexican hat or Marr wavelet(left); the real part of the Morlet wavelet (right), (�0 = 5.6).


0 0.5 10

0.5

1

x

(a)

log 2(a

)

x

(b)

0 0.5 1

6

8

10

log 2(a

)

x

(c)

0 0.5 1

6

8

10

Fig. 8.2 Continous wavelet transform of a fractal function:(a) the devil’s staircase; (b) its wavelet transform (with thefirst derivative of a Gaussian); (c) the corresponding skeleton,that is, the set of lines of local maxima (by courtesy of PierreVandergheynst, EPFL).

It is a real wavelet, with two vanishing moments (n = 0, 1). Similar wavelets, withmore vanishing moments, are obtained by taking higher derivatives of the Gaus-sian:

ψ(m)H (x) =

(1i

ddx

)m

e–x2/2, ψ(m)H (�) = �m e–�2/2. (8.78)

The Morlet wavelet: This is just a modulated Gaussian:

ψM (x) = π–1/4(

ei�0x – e–�20/2

)e–x2/2

ψM (�) = π–1/4[

e–(�–�0)2/2 – e–�2/2 e–�20/2

].

(8.79)

In fact the first term alone does not satisfy the admissibility condition; hence thenecessity for a correction. However, for �0 large enough (typically �0 v 5.5), thiscorrection term is numerically negligible (u 10–4). The Morlet wavelet is complex;hence, the corresponding transform S(b , a) is also complex. This enables one todeal separately with the phase and the modulus of the transform, and the phaseturns out to be a crucial ingredient in most algorithms used in applications suchas feature detection.

We give in Figure 8.2 an example of such a “time-scale” representation of a sig-nal, namely, the quite transient “devil’s staircase”, an increasing fractal functionwhose the derivative is zero everywhere with the exception of a Cantor set ofLebesgue measure zero.

135

9Another Family of SU(1,1) Coherent Statesfor Quantum Systems

9.1Introduction

This chapter is devoted to another family of SU (1, 1) coherent states that natural-ly emerges from the quantum motion in infinite-well and trigonometric Pöschl–Teller potentials. The latter are defined by [111]:

V (x) == V λ,κ(x) =12

V 0

(λ(λ – 1)cos2 x

2a+

κ(κ – 1)sin2 x

2a

), 0 � x � πa , (9.1)

and are represented in Figure 9.1 for different values of the parameters (λ, κ).The material is mainly borrowed from [112], and from more recent research on

the subject. It is also a direct illustration of a construction of coherent states thatwas proposed by Klauder and the author in [58]. In the construction of these states,we take advantage of the simplicity of the solutions, which ultimately stems fromthe fact they share a common SU (1, 1) symmetry à la Barut–Girardello [113]. In-deed, the Pöschl–Teller potentials share with their infinite-well limit the nice prop-erty of being analytically integrable. The reason behind this can be understoodwithin a group-theoretical context: these potentials possess an underlying dynam-ical algebra, namely, su(1, 1) and the discrete series representations of the latter.We know from the previous chapter that the discrete series unitary irreducible rep-resentations of the Lie algebra su(1, 1) are labeled by a parameter η, which takesits values in { 1

2 , 1, 32 , 2, . . .} for the discrete series sensu stricto, and in [ 1

2 , +∞) forthe extension to the universal covering of the group SU (1, 1). The relation betweenthe Pöschl–Teller parameters and η is given by 2η – 1 = λ + κ, and the limit caseλ, κ→ 1+ corresponds to η = 3

2 .

9.2Classical Motion in the Infinite-Well and Pöschl–Teller Potentials

Let us consider a particle trapped in an infinite square well, that is, confined in theinterval 0 < x < πa, and also in the Pöschl–Teller family (9.1) viewed as regulariza-tions of such an infinite potential.


136 9 Another Family of SU(1,1) Coherent States for Quantum Systems

0

200

400

600

800

1000

1200

1400

1600

1800

2000

y

0.2 0.4 0.6 0.8 1x

Fig. 9.1 The Pöschl–Teller potential V (x) =12 V 0

[λ(λ – 1)cos–2 x

2a + κ(κ – 1)sin–2 x2a

], with a = π–1 and

for (λ, κ) = (4, 4), (4, 8), (4, 16) (from bottom to top). SourceAntoine et al. [112] (reprinted with permission from [Antoine,J.-P., Gazeau, J.-P., Monceau, P., Klauder J.R., Penson K.A., Tem-porally stable coherent states for infinite well, J. Math. Phys.,42, p. 2349, 2001], American Institute of Physics).

9.2.1Motion in the Infinite Well

Let us first review the classical behavior of a particle of mass m trapped in an infi-nite well of width πa, an elementary but not so trivial model. For a nonzero energy

E = 12 mv2, there corresponds a speed v =

√2Em for a position 0 < x < πa. There

are perfect reflections at the boundaries of the well. So the motion is periodic withperiod (the “round-trip time”) T equal to

T =2πa

v= 2πa

√m2E

. (9.2)

9.2 Classical Motion in the Infinite-Well and Pöschl–Teller Potentials 137

�

�

��

��

��

��

��

��

��

��

��

��

��

��

��

0

πa –

x

t�T2

T

�3T2

2T

�

Fig. 9.2 The position x(t) of the particle trapped in an infinitesquare well of width πa, as a function of time.

With the initial condition x(0) = 0, the time behavior of the position is then givenby (see Figure 9.2)

0 � t � 12 T : x = vt ,

12 T � t � T : x = 2πa – vt ,

(9.3)

and of course x(t + nT ) = x(t).Consequently the velocity is a periodized Haar function:

v = v+∞∑n=0

[�[nT ,(n+ 1

2 )T ] – �[(n+ 12 )T ,(n+1)T ]

](9.4)

(here �B denotes the characteristic function of a set B ∈ R), whereas the accelera-tion is the superposition of two Dirac combs on the half-line:

γ =+∞∑n=0

[δnT – δ(n+ 1

2 )T

].

The average position and average velocity of the particle are then

x =1T

∫ T

0x(t) dt =

πa2

, v = 0 , (9.5)

whereas the mean square dispersions are√x2 – x2 =

πa2√

3,

√v2 – v2 =

√2Em

. (9.6)

Figure 9.3 shows the phase trajectory of the system. This trajectory encirclesa surface of area equal to the action variable A = 1

2π

∮pdq = mva, where q = x and

p = mv are canonically conjugate. Note the other expressions for A:

A =2πa2m

T=

mv2T2π

= a√

2mE . (9.7)


�

�

�

�

0

mv

–mv

p = mv

πa x

Fig. 9.3 Phase trajectory of the particle in an infinite square well.

9.2.2Pöschl–Teller Potentials

The solution to the equations of motion with the potentials (9.1) is straightforward,in spite of the rather heavy expression of the latter. The turning points x± of theperiodic motion at a given energy E are given by

x± = a arccos

[α – �

2±√

Δ

], (9.8)

where

Δ = (1 – 12 (√

α +√

�)2)(1 – 12 (√

α –√

�)2) ,

α =V 0

Eλ(λ – 1), � =

V 0

Eκ(κ – 1). So the motion is possible only if

E >V 0

2

(√λ(λ – 1) +

√κ(κ – 1)

)2. (9.9)

The time evolution of the position is given by

x(t) = a arccos

[α – �

2+√

Δ cos

(√2Em

ta

)], x(0) = x– . (9.10)

Hence, the period is

T = 2πa

√m2E

. (9.11)

It is remarkable that the period T does not depend on the strength V 0, nor on λand κ.

9.2 Classical Motion in the Infinite-Well and Pöschl–Teller Potentials 139

The action variable A satisfies the relationdAdE

=T2π

, and thus A = a√

2mE +

const. The constant is determined by the condition that A = 0 for E = V min, thatis, const. = –a

√2mV min. The Pöschl–Teller potential V (x) reaches its minimum at

the location xo defined by

tan2 xo

2a=

√κ(κ – 1)λ(λ – 1)

. (9.12)

So we have, in agreement with (9.9),

V min = V (xo) =V 0

2

[√λ(λ – 1) +

√κ(κ – 1)

]2, (9.13)

and consequently

A = a√

2mE – a√

mV 0[√

λ(λ – 1) +√

κ(κ – 1)] . (9.14)

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 1 2 3 4 5 6t

Fig. 9.4 The position x(t) of the particle in the symmetricPöschl–Teller potential λ = κ = 2 for energy E = 8V 0 and peri-od T = π

2 (cf. Figure 9.2). Source Antoine et al. [112] (reprintedwith permission from [Antoine, J.-P., Gazeau, J.-P., Monceau,P., Klauder J.R., Penson K.A., Temporally stable coherent statesfor infinite well, J. Math. Phys., 42, p. 2349, 2001], AmericanInstitute of Physics).


1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

p

1 1.2 1.4 1.6 1.8 2 2.2q

Fig. 9.5 Upper part of the phase trajectory of the particle inthe symmetric (2,2) Pöschl–Teller system, for the same val-ues of E and T as in Figure 9.4 (cf. Figure 9.3). Source Antoineet al. [112] (reprinted with permission from [Antoine, J.-P.,Gazeau, J.-P., Monceau, P., Klauder J.R., Penson K.A., Tem-porally stable coherent states for infinite well, J. Math. Phys.,42, p. 2349, 2001], American Institute of Physics).

It is worthwhile comparing (9.11) and (9.14) with their respective infinite-wellcounterparts (9.2) and (9.7). We should also check that the time behavior (9.10) ofx(t) goes into (9.3) at the limits α, �→ 0. In Figures 9.4 and 9.5 we show the timeevolution of the position of the particle trapped in a Pöschl–Teller potential and thecorresponding phase trajectory in the plane (q = x , p = mv ).

Note that, in the general case, the equation for the latter reads (at energy E)

p = ±√

2mEsin q

a

[1 – (α + �) + (α – �) cos

qa

– cos2 qa

]1/2. (9.15)

Finally, let us give the canonical transformation leading to the action–angle vari-ables:

ϕ = arccos1√Δ

[cos

qa

–α – �

2

]A = a [ p2 + 2mV (q)]1/2 . (9.16)

9.3 Quantum Motion in the Infinite-Well and Pöschl–Teller Potentials 141

9.3Quantum Motion in the Infinite-Well and Pöschl–Teller Potentials

9.3.1In the Infinite Well

Any quantum system trapped inside the infinite well 0 � x � πa must have itswave function equal to zero outside the well. It is thus natural to impose on thewave functions the boundary conditions

ψiw(x) = 0, x � πa and x � 0 . (9.17)

Since the movement takes place only inside the interval [0, πa], we may also ignorethe rest of the line and replace the conditions (9.17) by the following ones:

ψiw ∈ L2([0, πa], dx) , ψiw(0) = ψiw(πa) = 0 . (9.18)

Alternatively, one may consider the periodized well and impose the same periodicboundary conditions, namely, ψiw(nπa) = 0, ∀n ∈ Z.

In either case, stationary states of the trapped particle of mass m are easily foundfrom the eigenvalue problem for the Schrödinger operator. For reasons to be justi-fied in the sequel, we choose the shifted Hamiltonian:

H == Hw = –�2

2md2

dx2 –�2

2ma2 . (9.19)

Then

Ψ iw(x , t) = e– i� Ht Ψ iw(x , 0) , (9.20)

where Ψ iw(x , 0) == ψiw(x) obeys the eigenvalue equation

Hψiw(x) = E ψiw(x) , (9.21)

together with the boundary conditions (9.17). Normalized eigenstates and corre-spxonding eigenvalues are then given by

ψiwn (x) =

√2

πasin(n + 1)

xa

== 〈x |ψiwn 〉 , 0 � x � πa , (9.22)

H |ψiwn 〉 = E n |ψiw

n 〉 , n = 0, 1, . . . , (9.23)

E n =�2

2ma2 n(n + 2) == �ωxn , (9.24)

with

ω =�

2ma2==

2πT r

and xn = n(n + 2), n = 0, 1, . . . ,

where T r is the “revival” time to be compared with the purely classical round-trip


time given in (9.2). Now the Bohr–Sommerfeld quantization rule applied to theclassical action gives

a√

2mE = A = (n + 1)� , (9.25)

so

E = (n + 1)2 �2

2ma2 = E n +�2

2ma2 , n = 0, 1, . . . (9.26)

Thus, here the Bohr–Sommerfeld quantization is exact [17], despite the presenceof the extra term �2/2ma2 that follows from our particular choice of zero in theenergy scale (see (9.19)).

9.3.2In Pöschl–Teller Potentials

Pöschl–Teller potentials were originally introduced in a quantum molecular physicscontext. The energy eigenvalues and corresponding eigenstates are solutions to theSchrödinger equation[

–�2

2md2

dx2 +V 0

2

(λ(λ – 1)cos2 x

2a+

κ(κ – 1)sin2 x

2a

)–

�2

8ma2 (λ + κ)2

]ψpt

n (x) = E ψptn (x) ,

(9.27)

with 0 � x � πa and where we have also shifted the Hamiltonian of the trappedparticle of mass m by an amount equal to – �2

8ma2 (λ +κ)2. Here too, as for the infinitewell, we have the choice of putting the potential equal to infinity outside the interval[0, πa], or periodizing the problem, with period 2πa.

Since the potential strength is overdetermined by specifying V 0, λ, and κ simul-taneously, we can freely put for convenience, as in [111, 114],

V 0 =�2

4ma2 . (9.28)

With this choice, and the boundary conditions ψpt(0) = ψpt(πa) = 0, the normalizedeigenstates, and the corresponding eigenvalues, all of them simple, are given by

ψptn (x) = [cn(κ, λ)]–

12

(cos

x2a

)λ (sin

x2a

)κ

~ 2F 1

(–n, n + λ + κ ; κ +

12

; sin2 x2a

), (9.29)

where cn(κ, λ) is a normalization factor that can be given analytically when κ and λare positive integers, and

E n =�2

2ma2 n(n + λ + κ) == � ωxn , n = 0, 1, . . . , (9.30)

9.4 The Dynamical Algebra su(1, 1) 143

with

ω =�

2ma2 , xn = n(n + λ + κ), λ, κ > 1 . (9.31)

Note that the Bohr–Sommerfeld rule applied to the canonical action (9.14) yields(here we do not impose the normalization (9.28))

a√

2mE – a√

mV o

[√λ(λ – 1) +

√κ(κ – 1)

]= �(n + 1

2 ) ,

that is,

E n =�2

2ma2 (n + 12 )2 +

�

ma

√mV 0(n + 1

2 )[√

λ(λ – 1) +√

κ(κ – 1)]

(9.32)

+V 0

2

[√λ(λ – 1) +

√κ(κ – 1)

]2. (9.33)

This formula is interesting on two counts at least.(a) The first term in (9.32) gives, apart from the term 1

2 in (n + 12 ), the exact

spectrum of the infinite well. More precisely, these values of the energy maybe obtained simply by letting V 0 → 0 in V (x) and keeping in mind thatV =∞ outside [0, πa].

(b) In the limit V 0 → ∞, the first term in (9.32) can be neglected and one isleft, up to a global, V 0-dependent, shift, with the spectrum of a harmonicoscillator with elementary quantum

�ω = �

√V 0

ma2

[√λ(λ – 1) +

√κ(κ – 1)

].

Hence, the Pöschl–Teller potential interpolates between the square well andthe harmonic oscillator.

9.4The Dynamical Algebra su(1, 1)

Behind the spectral structure of the infinite well or Pöschl–Teller Hamiltonians,there exists a dynamical algebra generated by lowering and raising operators actingon |en〉 == |ψiw

n 〉 or |ψptn 〉. The latter are defined by

a|en〉 =√

xn|en–1〉 , (9.34)

a†|en〉 =√

xn+1|en+1〉 , (9.35)

with

xn = n(n + 2) , for the infinite well ,

xn = n(n + λ + κ) , for the Pöschl–Teller potential , n = 0, 1, 2, . . .


Then we observe that the operator X N = a†a is diagonal with eigenvalues xn :X N |en〉 = xn |en〉. Note that the number operator N,

N |en〉 = n|en〉 , (9.36)

is given in terms of X N by

N = –12

(λ + κ) +

(X N +

14

(λ + κ)2

)1/2

. (9.37)

For any diagonal operator Δ with eigenvalues δn

Δ|en〉 = δn |en〉 , (9.38)

we denote its finite difference by Δ′. The latter is defined as the diagonal operatorwith eigenvalues δ′n == δn+1 – δn ,

Δ′|en〉 = δ′n |en〉 . (9.39)

More generally, the mth finite difference Δ(m) will be recursively defined by

Δ(m) = (Δ(m–1))′ . (9.40)

Now, from the infinite matrix representation (in the basis {|en〉}) of the operators aand a†,

a =

⎛⎜⎜⎝0√

x1 0 0 . . .

0 0√

x2 0 . . .

0 0 0√

x3 . . .

. . . . . . . . . . . . . . .

⎞⎟⎟⎠ , (9.41)

a† =

⎛⎜⎜⎜⎜⎝0 0 0 0 . . .√x1 0 0 0 . . .

0√

x2 0 0 . . .

0 0√

x3 0 . . .

. . . . . . . . . . . . . . .

⎞⎟⎟⎟⎟⎠ , (9.42)

it is easy to check that

[a, a†] =

⎛⎝x1 – x0 0 . . .

0 x2 – x1 . . . 00 0 x3 – x2

⎞⎠ = X ′N , (9.43)

X ′N |en〉 = x ′n |en〉, x ′n = xn+1 – xn = 2n + 3 , or 2n + 1 + λ + κ . (9.44)

We also check that, for any diagonal operator Δ, we have

[a, Δ] = Δ′a , [a†, Δ] = –a†Δ′ . (9.45)

9.4 The Dynamical Algebra su(1, 1) 145

Therefore,

[a, X ′N ] = X ′′N a ,

with

X ′′N |en〉 = x ′′n |en〉 = (x ′n+1 – x ′n)|en〉 = 2|en〉 . (9.46)

So X ′′N = 2Id , X ′′′N = 0, and [a, X ′N ] = 2a. Similarly, [a†, X ′N ] = –2a†. In summary,there exists a “dynamical” Lie algebra, which is generated by {a, a†, X ′N}. Then thecommutation rules

[a, a†] = X ′N , [a, X ′N ] = 2a, [a†, X ′N ] = –2a† (9.47)

clearly indicate that it is isomorphic to

su(1, 1) ~ sl(2, R) ~ so(2, 1) . (9.48)

A more familiar basis for (9.48) is given (in so(2, 1) notation) by

L– =1√2

a , L+ =1√2

a†, L12 =12

X ′N , (9.49)

where L12 is the generator of the compact subgroup SO (2), namely,

[L±, L12] = ∓L±, [L–, L+] = L12 . (9.50)

If we add the operator X N (i.e., the Hamiltonian H) to the set {a, a†, X ′N}, weobtain an infinite-dimensional Lie algebra contained in the enveloping algebra.Indeed

[a, X N ] = X ′N a , [a†, X N ] = –a†X ′N[a, X ′N a] = 2a2 , [a†, X ′N a] = –X ′N

2 – 2X N ,etc . . .

(9.51)

Note also the relation between X N and X ′N :

X N = 14

(X ′N

2 – 2X ′N – 3)

, or 14

(X ′N

2 – 2X ′N – (λ + κ + 1)(λ + κ))

. (9.52)

In the same vein, we note that the condition X ′′′N = 0 is necessary to obtain a gen-uine Lie algebra (instead of a subset of the enveloping algebra). Therefore, su(1, 1)is the only dynamical Lie algebra that can arise in such a problem.

It follows from the considerations above that the space H of states |en〉 carriessome representation of su(1, 1). The latter is found by examining the formulas forthe su(1, 1) discrete series representation [99].

Given η = 12 , 1, 3

2 , . . ., the discrete series unitary irreducible representation U η isrealized on a generic separable Hilbert spaceHη with basis {|η, n〉, n ∈ N} through


the following actions of the Lie algebra elements in the realization (9.49) (slight-ly different from the Fock–Bargman realization (8.50) introduced in the previouschapter),

L12|η, n〉 = (η + n)|η, n〉 , (9.53)

L–|η, n〉 =1√2

√(2η + n – 1)n |η, n – 1〉 , (9.54)

L+|η, n〉 =1√2

√(2η + n)(n + 1) |η, n + 1〉 . (9.55)

The representation U η fixes the Casimir operator Q = –L12(L12 – 1) + 2L+L– tothe following value: QHη = η(η – 1)Hη. Using (9.54) and (9.55), and comparingwith (9.34), (9.35), and (9.44), we obtain the specific value of η for the infinite-wellproblem, namely, η = 3

2 , so we can make the identifications H3/2 == H, | 32 , n〉 ==|ψiw

n 〉. On the other hand, we obtain a continuous range of values for the Pöschl–Teller potentials,

η =λ + κ + 1

2>

32

, (9.56)

and we shall denote the corresponding Hilbert spaces and states (9.22) by Hη and|η, ψpt

n 〉, respectively. The relation (9.56) simply means that here we are in the pres-ence of the (abusively called) discrete series representations of the universal cover-ing of SU (1, 1), except for the interval η ∈

(12 , 3

2

).

9.5Sequences of Numbers and Coherent States on the Complex Plane

In a general setting, consider a strictly increasing sequence of positive numbers

0 = x0 < x1 < x2 . . . < xn < . . . , (9.57)

that are eigenvalues of a self-adjoint positive operator X N in some separable Hilbertspace H,

X N |en〉 = xn |en〉 , (9.58)

where the set {|en〉, n ∈ N} is an orthonormal basis of H. There correspondsto (9.57) a (generically infinite) dynamical Lie algebra with basis {a, a†, X ′N , . . .},with the notation of the previous section. There also corresponds the sequence of

“factorials” xn ! = x1x2 . . . xn with x0! def= 1 and the “exponential”

N (t) =+∞∑n=0

tn

xn !, (9.59)

with a radius of convergence

R = lim supn→+∞

n√

xn ! (9.60)

9.5 Sequences of Numbers and Coherent States on the Complex Plane 147

that is assumed to be nonzero, of course. We next suppose that there exists a prob-ability distribution t �→ π(t) on [0, R) such that

xn ! =∫ R

0un π(u) du . (9.61)

Otherwise said, the Stieltjes moment problem [115, 116] has a solution for the se-quence of factorials (xn !)n∈N. We know that a necessary and sufficient condition forthis is that the two matrices⎛⎜⎜⎜⎜⎜⎝

1 x1! x2! . . . xn !x1! x2! x3! . . . xn+1!x2! x3! x4! . . . xn+2!

......

.... . .

...xn ! xn+1! xn+2! . . . x2n!

⎞⎟⎟⎟⎟⎟⎠ , (9.62)

⎛⎜⎜⎜⎜⎜⎝x1! x2! x3! . . . xn+1!x2! x3! x4! . . . xn+2!x3! x4! x5! . . . xn+3!

......

.... . .

...xn+1! xn+2! xn+3! . . . x2n+1!

⎞⎟⎟⎟⎟⎟⎠ (9.63)

have strictly positive determinants for all n.Let us consider in the complex plane the open disk D√R with radius

√R and

centered at the origin, and the Hilbert space L2(D√R , μπ(d2z)

)of complex-valued

functions that are square-integrable onD√R with respect to the measure μπ(d2z) def=π(|z|2) d2z/π. We then choose in this space the set of monomials in z

φn(z) =zn√

xn !. (9.64)

By construction, this set is orthonormal with respect to the measure μπ, and itsatisfies

∞∑n=0

|φn(z)|2 = N (|z|2) .

Hence, we are in the Bayesian statistical context described in Section 5.3, that is,the interplay between two probability distributions:

• For each z, the discrete distribution,

n �→ |φn (z)|2N (|z|2)

, (9.65)

which could be considered as concerning experiments performed on thesystem within some experimental protocol, say, E , to measure the spectralvalues of X N .


• For each n, the continuous distribution on (D√R , μπ),

D√R � z �→ |φn (z)|2 . (9.66)

Following our general scheme of coherent state construction, we get in the presentsituation the family {|z〉, z ∈ D√R} of coherent states in the Hilbert spaceH:

|z〉 =1√N (|z|2)

∑n�0

φn(z, z) |en〉 =1√N (|z|2)

∑n�0

zn√

xn!|en〉 . (9.67)

They are normalized, 〈z|z〉 = 1, and resolve the unity in H with respect to themeasure N (|z|2) μπ(d2z):∫

D√R

d2zπ

π(|z|2)N (|z|2) |z〉〈z| = IH . (9.68)

Moreover, these coherent states are eigenvectors of the operator a defined in (9.34):

a|z〉 = z|z〉 . (9.69)

Now suppose that X N is (up to a factor) the Hamiltonian for a quantum system,

H = �ωX N . (9.70)

Then the coherent states (9.67) evolve in time as

e– i� Ht |z〉 =

1√N (|z|2)

∑n�0

zn√

xn !e–iωxn t |en〉 . (9.71)

We know that if xn ∝ n, that is, in the case of the harmonic oscillator, the tempo-ral evolution of the coherent state |z〉 reduces to a rotation in the complex plane,namely, e–iHt/� |z〉 = |z e–iωt〉. In general, however, we lose the temporal stabilityof our family of coherent states (9.67). Hence, to restore it, we must extend ouroriginal definitions to the entire family of temporal evolution orbits:

{e–i H� t |z〉, z ∈ D√R , t ∈ I} . (9.72)

The interval I is the whole real line when xn is generic, whereas it can be restrictedto a period, that is, a finite interval [a, b ],

b – a =2πωα

(9.73)

if xn ∈ αN. A straightforward calculation now shows that

〈z|H |z〉 = 〈z|�ωX N |z〉 = �ω|z|2 . (9.74)

9.5 Sequences of Numbers and Coherent States on the Complex Plane 149

Therefore, the quantity |z|2 is the average energy evaluated in the elementaryquantum unit �ω. Note that

�|z|2 == J (9.75)

is simply the action variable in the case where H is the Hamiltonian of the har-monic oscillator and the variable z is given the meaning of a classical state in thephase space C.

On the other hand, introducing the dimensionless number

γ = ωt , γ ∈ ωI , (9.76)

we are naturally led to study the continuous family of states (9.72)

|z, γ〉 def= e–iγ H�ω |z〉 =

1√N ( J/�)

∑n�0

zn e–iγxn

√xn !

|en〉 . (9.77)

These states, parameterized by (z, γ) ∈ D√R ~ I , may also be called “coherent” forseveral reasons. First they are, by construction, eigenvectors of the operator

a(γ) == e–iγH/�ω a eiγH/�ω , (9.78)

namely,

a(γ)|z, γ〉 = z|z, γ〉 . (9.79)

They obey the temporal stability condition

e–iHt/� |z, γ〉 = |z, γ + ωt〉 . (9.80)

Again, if we consider the harmonic oscillator case, we do not make any distinctionbetween the argument of the complex parameter z and the angle variable γ, sincethen xn = n and zne–iγn = (ze–iγ)n , so the only parameters we need are J = �|z|2and γ. The latter are easily identified with the classical action–angle variables. Fromnow on, we will stick to the minimal parameterization set in the present general-ization and shall denote our coherent states by

| J, γ〉 =1√N ( J)

∑n�0

Jn/2 e–iγxn

√xn !

|en〉 , (9.81)

where we have put � = 1 for convenience. However, we now have to identify themeasure on the observation set X == {( J , γ), J ∈ [0, R), γ ∈ I} for which theyresolve the unity, or, up to the factor N ( J), for which the set of functions

�n( J , γ) =Jn/2 e–iγxn

√xn !

(9.82)

is orthonormal. For a generic real sequence of xn ’s and for I = R, the following


measure is defined, in the Bohr’s sense [117], on functions on X by

f �→ μB ( f ) =∫

XμB (d J dγ) f ( J , γ) def= lim

Γ→∞

12Γ

∫ Γ

–Γdγ

∫ R

0f ( J , γ)π( J ) d J .

(9.83)

The orthogonality of the functions �n derives from the orthogonality of the Fourierexponentials:

limΓ→∞

12Γ

∫ Γ

–Γeiγ(xn –xn′ )dγ = lim

Γ→∞

sin Γ (xn – xn′ )Γ (xn – xn′ )

= δnn′ . (9.84)

Of course, for a sequence {xn , n ∈ N} of integers, the measure on R in the sense ofBohr reduces to the ordinary normalized measure on a period interval, for example,[0, 2π).

We thus obtain the resolution of the unity by the coherent states | J , γ〉:∫X

μB (d J dγ)N ( J)| J, γ〉〈 J , γ| = IH . (9.85)

In a suitable way [58] (see also the discussion in Section 9.7.4), it is also accept-able to regard the parameterization ( J , γ) as “action–angle” variables, and it is con-venient to refer to them as such, even when keeping in mind the possibility ofextending

√J to the complex plane, that is, replacing

√J by z.

9.6Coherent States for Infinite-Well and Pöschl–Teller Potentials

9.6.1For the Infinite Well

Let us now adapt the material of the previous section to our problem of the infinitewell. In that case,

xn ! = x1 x2 . . . xn =n!(n + 2)!

2, (9.86)

| J , γ〉 =1√N ( J)

∑n�0

Jn/2 e–iγn(n+2)√n!(n+2)!

2

|ψiwn 〉 . (9.87)

The normalization factor is easily calculated in terms of the modified Bessel func-tion Iν [18].

N ( J) = 2+∞∑n=0

Jn

n!(n + 2)!=

2J

I2(2√

J) . (9.88)

9.6 Coherent States for Infinite-Well and Pöschl–Teller Potentials 151

The radius of convergence R = lim supn→+∞n

√n!(n+2)!

2 is of course infinite. More-over, since the xn ’s are here natural numbers, the interval of variation of the evolu-tion parameter γ can be chosen as I = [0, 2π].

The generalized factorials xn ! arise as moments of a probability distributionπiw(u),

xn ! =∫ ∞

0un πiw(u) du , (9.89)

and πiw(u) is explicitly given in terms of the other modified Bessel function K ν [18],

πiw(u) = uK 2(2√

u) . (9.90)

It results from the previous section that the family {| J, γ〉, J ∈ R+, γ ∈ [0, 2π]}resolves the unity operator,

Id =∫| J, γ〉〈 J , γ| N ( J) μiw(d J dγ) , (9.91)

with ∫(·) μiw(d J dγ) =

12π

∫ 2π

0dγ

∫ +∞

0πiw( J)(·) d J . (9.92)

As is well known, the overlap of two coherent states does not vanish in general.Explicitly, we have

〈 J ′, γ′| J, γ〉 =2√

N ( J)N ( J ′)

∑n�0

( J J ′)n2

n!(n + 2)!e–in(n+2)(γ–γ′) . (9.93)

If γ = γ′, we obtain a Bessel function

〈 J ′, γ | J , γ〉 =2

( J J ′)12

√N ( J)N ( J ′)

I2 (2( J J ′)14 ) . (9.94)

If γ =/ γ′, we can give an integral representation of (9.93) in terms of a thetafunction and Bessel functions [18]:

〈 J ′, γ′| J , γ 〉 =ei(γ–γ′)/4

iπ√N ( J)N ( J ′)

∫ π

0dϕ θ1

(ϕ

π, –

γ – γ′

π

)~

[–e–i(ϕ–γ+γ′)

( J J ′)12

I2

(2( J J ′)

14 ei(ϕ– γ–γ′

2 + π2 ))

+ei(ϕ+γ–γ′)

( J J ′)12

I2

(2( J J ′)

14 e–i(ϕ+ γ–γ′

2 – π2 ))]

. (9.95)


9.6.2For the Pöschl–Teller Potentials

The relations (9.86) and (9.87) of the previous section are easily generalized to thepresent case. We shall list them without unnecessary comments.

From the energies E n = � ω xn given by (9.30), we get the moments

xn ! = x1 x2. . .xn = n!Γ(n + ν + 1)

Γ(ν + 1), (9.96)

with ν = λ + κ > 2.Thus, the coherent states read as

| J , γ〉 =[Γ(ν + 1)]1/2√N ( J)

∑nv0

Jn/2e–iγn(n+ν)

[n! Γ(n + ν + 1)]12

|ψptn 〉 . (9.97)

The normalization is then given by

N ( J) = Γ(ν + 1)∑n� 0

Jn

n! Γ(n + ν + 1)=

Γ(ν + 1)Jν/2

Iν(2√

J) . (9.98)

The radius of convergence R is infinite. The interval of variation of the evolutionparameter γ is generically the whole real line, unless the parameter ν is an integer.

The numbers xn ! are moments of a probability distribution πpt(u) involving themodified Bessel function K ν:

xn ! =∫ ∞

0un πpt(u)du , (9.99)

with (cf. (9.90))

πpt(u) =2

Γ(ν + 1)uν/2K ν(2

√u) . (9.100)

It might be useful to recall here the well-known relation between modified Besselfunctions [18],

K ν(z) =π

2 sin πν[I–ν(z) – Iν(z)] , ν �∈ Z . (9.101)

The resolution of the unity is then explicitly given by

Id =∫| J, γ〉〈 J , γ| N ( J) μpt(d J dγ) , (9.102)

with ∫(·)μpt(d J dγ) = lim

Γ→∞

12Γ

∫ Γ

–Γdγ

[∫ +∞

0πpt( J) (·) d J

]. (9.103)

9.7 Physical Aspects of the Coherent States 153

Finally, the overlap between two coherent states is given by the series

〈 J ′, γ′| J, γ〉 =Γ(ν + 1)√N ( J)N ( J ′)

∑n� 0

( J J ′)n/2

n!Γ(n + ν + 1)e–in(n+ν)(γ–γ′) , (9.104)

which reduces to a Bessel function for γ = γ′:

〈 J ′, γ | J , γ〉 =Γ(ν + 1)√N ( J)N ( J ′)

Iν(2( J J ′)1/4)( J J ′)ν/4

. (9.105)

At this point, we should emphasize the fact that, when γ = 0 and J is takenas a complex parameter, our temporally stable families of coherent states (9.87)and (9.97) are nothing other but the temporal evolution orbits of the well-knownBarut–Girardello coherent states for SU (1, 1) [113].

In addition, we should also quote Nieto and Simmons [118], who considered theinfinite square well and the Pöschl–Teller potentials as examples of their construc-tion of coherent states. The latter are required to minimize an uncertainty rela-tion or, equivalently, to be eigenvectors of some “lowering operator” A (à la Barut–Girardello [113]). However, those states have a totally different meaning and shouldbe considered only in the semiclassical limit.

9.7Physical Aspects of the Coherent States

In this section, we shall review some of the spatial and temporal features of thecoherent states described in [112], treating together the infinite-well coherent states(9.81) and the Pöschl–Teller coherent states (9.97), the former being obtained fromthe latter simply by putting ν = λ + κ = 2.

9.7.1Quantum Revivals

As the (infinite) superposition of stationary states which are spatially and temporal-ly periodic for integer values of ν, they should display nonambiguous revivals andfractional revivals. Let us first recall the main definitions concerning the notion ofrevival, as given in [119]. For other related works, see [120–125, 127] and the recentreview by Robinett [128].

A revival of a wave function occurs when a wave function evolves in time to a stateclosely reproducing its initial form. A fractional revival occurs when the wave func-tion evolves in time to a state that can be described as a collection of spatially dis-tributed sub-wave functions, each of which closely reproduces the shape of the ini-tial wave function. If a revival corresponds to phase alignments of nearest-neighborenergy eigenstates that constitute the wave function, it can be asserted that a frac-tional revival corresponds to phase alignments of nonadjacent energy eigenstatesthat constitute this wave function.


For a general wave packet of the form

|ψ(t)〉 =∑n�0

cn e–iE n t/�|en〉 , (9.106)

with∑

n�0 |cn |2 = 1, the concept of revival arises from the weighting probabilities|cn |2. Suppose that the expansion (9.106) is strongly weighted around a mean value〈n〉 for the number operator N , N |en〉 = n|en〉:

〈ψ|N |ψ〉 =∑n�0

n |cn |2 == 〈n〉 . (9.107)

Let n ∈ N be the integer closest to 〈n〉. Assuming that the spread σ W Δn ==[〈n2〉 – 〈n〉2

]1/2is small compared with 〈n〉 W n, we expand the energy E n in a Tay-

lor series in n around the centrally excited value n:

E n � E n + E ′n(n – n) +12

E ′′n(n – n)2 +16

E ′′′n (n – n)3 + . . . , (9.108)

where each prime on E n denotes a derivative. These derivatives define distincttime scales [120], namely, the classical period T cl = 2π�/|E ′n |; the revival time t rev =2π�/ 1

2 |E ′′n |; the superrevival time tsr = 2π�/ 16 |E ′′′n |; and so on. Inserting this expan-

sion into the evolution factor e–iE nt/� of (9.106) allows us to understand the possibleoccurrence of a quasiperiodic revival structure of the wave packet (9.106) accordingto the weighting probability n �→ |cn |2. In the present case, we have

E n =�

2ma2 n(n + ν) =�

2ma2

[n(n + ν) + (2n + ν)(n – n) + (n – n)2

]. (9.109)

So the first characteristic time is the “classical” period

T cl =2π�

2n + ν2ma2

�2 =2πma2

�(n + ν2 )

, (9.110)

which should be compared with the actual classical (Bohr–Sommerfeld) counter-part deduced from (9.11) and (9.14),

T =2πma2

A + a√

mV 0[√

λ(λ – 1) +√

κ(κ – 1)]. (9.111)

The second characteristic time is the revival time

t rev =4πma2

�= (2n + ν)T cl . (9.112)

There is no superrevival time here, because the energy is a quadratic function of n.With these definitions, the wave packet (9.106) reads in the present situation (up

to a global phase factor)

|ψ(t)〉 =∑n�0

cn e–2πi[

(n–n) tT cl

+(n–n)2 ttrev

]|en〉 . (9.113)


Hence, it will undergo motion with the classical period, modulated by the revivalphase [126]. Since T cl << t rev for large n, the classical period dominates for smallvalues of t (mod t rev), and the motion is then periodic with period T cl. As t increasesfrom zero and becomes nonnegligible with respect to t rev, the revival term (n –n)2 t

trevin the phase of (9.113) causes the wave packet to spread and collapse. The

latter gathers into a series of subsidiary waves, the fractional revivals, which moveperiodically with a period equal to a rational fraction of T cl. Then, a full revivalobviously occurs at each multiple of t rev.

To put into evidence these revival structures for a given wave packet ψ(x , t) =〈x |ψ(t)〉, an efficient method is to calculate its autocorrelation function [126]:

A(t) = 〈ψ(x , 0) |ψ(x , t)〉 =∑n�0

|cn |2 e–iE n t/� .

Numerically, |A(t)|2 varies between 0 and 1. The maximum |A(t)|2 = 1 is reachedwhen ψ(x , t) exactly matches the initial wave packet ψ(x , 0), and the minimum 0corresponds to nonoverlapping: ψ(x , t) is far from the initial state. On the otherhand, fractional revivals and fractional “superrevivals” appear (in the general case)as periodic peaks in |A(t)|2 with periods that are rational fractions of the classicalround-trip time T cl and the revival time t rev.

9.7.2Mandel Statistical Characterization

Since the weighting distribution |cn |2 is crucial for understanding the temporalbehavior of the wave packet (9.106), it is worthwhile also giving some general pre-cisions of a statistical nature [129–133]. before examining the special case of ourcoherent states. It is clear that the revival features will be more or less apparent, de-pending on the value of the deviation (n – n) (relative to n) that is effectively takeninto account in the construction of the wave packet. In this respect, it is interestingto compare |cn |2 with the Poissonian case 〈n〉n e–〈n〉/n! and with the Gaussian case,(2π(Δn)2)–1/2 exp[–(n – 〈n〉)2/2(Δn)2].

A quantitative estimate is given by the so-called Mandel parameter Q [129, 131,132] defined as follows:

QMdef=

(Δn)2

〈n〉 – 1 , (9.114)

where the average is calculated with respect to the discrete probability distributionn �→ |cn |2. In the Poissonian case, we have QM = 0, that is, Δn = 〈n〉1/2. We saythat the weighting distribution is sub-Poissonian (or super-Poissonian) if QM < 0(or QM > 0). In the super-Poissonian case, that is, Δn > 〈n〉1/2, the set of states|en〉 that contribute significantly to the wave packet can be rather widely spreadaround n � 〈n〉, and this may have important consequences for the properties oflocalization and temporal stability of the wave packet.


When the wave packets are precisely our coherent states (where |en〉 stands forboth |ψiw〉 and |ψpt〉),

| J , γ〉 =1√N ( J)

∑n� 0

Jn/2 e–ixn γ

√ρn

|en〉 , (9.115)

the weighting distribution depends on J,

|cn |2 =Jn

N ( J) ρn, (9.116)

and we can see the interesting statistical interplay with the probability distributionπ( J) of which the xn ! are the moments; see (9.89).

The following mean values are easily computed, together with their asymptoticvalues for large J [18]:

〈n〉 =JN ( J)

dd JN ( J) = J

dd J

lnN ( J)

=√

JIν+1(2

√J)

Iν(2√

J)=

√J –

ν2

–14

+ O

(1√

J

). (9.117)

〈n2〉 =JN ( J)

dd J

Jd

d JN ( J)

=√

JIν+1(2

√J)

Iν(2√

J)+ J

Iν+2(2√

J)Iν(2√

J)

= 〈n〉 + JIν+2(2

√J)

Iν(2√

J)W

√J(

√J + 1) ( J >> 1) . (9.118)

So, the dispersion is

(Δn)2 = JIν+2(2

√J)

Iν(2√

J)+ 〈n〉 – 〈n〉2

=J

[Iν(2√

J)]2

(Iν+2(2

√J)Iν(2

√J) – [Iν+1(2

√J)]2

)+

√J

Iν+1(2√

J)Iν(2√

J)

W√

J2

, for J large.

(9.119)

Finally, the Mandel parameter is given explicitly by

QM = Jd

d Jln

dd J

lnN ( J) =√

J

[Iν+2(2

√J)

Iν+1(2√

J)–

Iν+1(2√

J)Iν(2√

J)

]. (9.120)

It is easily checked that (Iν+1(x))2 � Iν(x)Iν+2(x), for any x � 0, and, thus, QM � 0for any J � 0. Note that QM � 0 for large J, while QM � – J for small J. Therefore,|cn |2 is sub-Poissonian in the case of our coherent states, whereas a quasi-Poisso-nian behavior is restored at high J. This fact is important for understanding thecurves presented in Figure 9.6a, which show the distributions

D (n, J, ν) == |cn |2 =1

n! Γ(n + ν + 1)Jn+ν/2

Iν(2√

J)(9.121)


(a)

J = 300

J = 50

J = 6

0

0.1

0.2

0.3

0.4

5 10 15 20 25

n

(b)

α = 17.3

α = 7.03

α = 2.34

0

0.05

0.1

0.15

0.2

50 100 150 200 250 300 350

n

Fig. 9.6 (a) The weighting distribution|cn |2 == D (n, J , ν) given in (9.121) for the infi-nite square well ν = 2 and different values of J.Note the almost Gaussian shape at J = 300,centered at n = 〈n〉 =

√J – ν

2 – 14 � 16,

a width equal to 2Δn =√

2 J1/4 � 5.9.(b) The same for the harmonic oscillator:|cn |2 = 1

n! |α|2ne–|α|2 . The values of α are cho-

sen so as to get essentially the same meanenergy values as in (a): α =

√J . Source An-

toine et al. [112] (reprinted with permissionfrom [Antoine, J.-P., Gazeau, J.-P., Monceau,P., Klauder J.R., Penson K.A., Temporally sta-ble coherent states for infinite well, J. Math.Phys., 42, p. 2349, 2001], American Institute ofPhysics).


for ν = 2 and different values of J. For the sake of comparison, we show in Fig-ure 9.6b the corresponding distribution |cn|2 = 1

n! |α|2ne–|α|2 for the harmonic os-cillator. Exactly as in the latter case, it can be shown easily that the distributionD (n, J , ν) tends for J → ∞ to a Gaussian distribution. This Gaussian is centeredat√

J – ν2 – 1

4 and has a half-width equal to 1√2

J1/4:

D (n, J, ν) W1√

π√

Je–

[n–

(√J– ν

2 – 14

)]2/√

J (n >> 1) . (9.122)

9.7.3Temporal Evolution of Symbols

We consider now the probability density |〈x | J, γ〉|2 as a function of the evolutionparameter γ = ωt for increasing values of J. This evolution is shown in Figure 9.7 inthe case of the infinite square well, for J = 2, 10, and 50. We can see at γ = π = 1

2 t rev

a perfect revival of the initial shape at γ = 0. This revival takes place near theopposite wall, as expected from the symmetry with respect to the center of the well.On the other hand, the ruling of the wave-packet evolution by the classical period

T cl =t rev

2n + ν=

πn + 1

becomes more and more apparent as J increases. We also note that, at multiplesof the half reversal time 1

2 t rev = π, the probability of localization near the wallsincreases with the energy J.

In Figure 9.8, we show the squared modulus

|〈 J , 0|e– i� Ht | J, 0〉|2 = |〈 J, 0 | J , ωt〉|2 (9.123)

=Γ(ν + 1)N ( J)2

∣∣∣∣ ∑n� 0

Jn

n!Γ(n + ν + 1)e–iωn(n+ν)t

∣∣∣∣2

(9.124)

of the autocorrelation versus γ = ωt for the infinite well, for J = 2, 10, and 50.Like in Figure 9.7, we draw attention to the large-J regime. Here fractional revivalsoccur as intermediate peaks at rational multiples of the classical period

T cl =π

n + 1W

π√J

, J >> 1 ,

and they tend to diminish as J increases, which is clearly the mark of a quasi-classical behavior. The same quantity is shown in Figure 9.9 for the Pöschl–Tellerpotential, for J = 20 and 40. Note that, in actual calculations like this, one has tochoose a finite number of orthonormal eigenstates of the Pöschl–Teller potential,denoted here by nmax. Correspondingly, the normalization of the coherent state| J, γ〉 has then to be modified as

nmax∑p=0

J p

p! Γ( p + κ + λ)=

Iκ+λ(2√

J)

J12 (κ+λ)

–Jnmax+1

(nmax + 1)! Γ(nmax + κ + λ + 2)

~ 1F 2(1; nmax + 2, nmax + κ + λ + 2; J) , (9.125)

where 1F 2 is the hypergeometrical function.


γ = πγ = π/2

γ = π/3 γ = π/4

γ = π/6

γ = 0

0

1.10.1

0.90.1

1.1

0.5 1 1.5 2 2.5 3x

(a)

γ = π

γ = π/2

γ = π/3 γ = π/4

γ = π/6γ = 0

0

2.20.2

0.70.1

2.2

0.5 1 1.5 2 2.5 3x

(b)

γ = π

γ = π/2

γ = π/3γ = π/4

γ = π/6

γ = 0

0

4.30.3

1.40.2

4.3

0.5 1 1.5 2 2.5 3x

(c)

Fig. 9.7 The evolution (versus γ) of the prob-ability density |〈x | J, γ〉|2, in the case of theinfinite square well for (a) J = 2, (b) J = 10,and (c) J = 50. We note the perfect revival atγ = π = 1

2 trev (in suitable units), symmetricallywith respect to the center of the well. Source

Antoine et al. [112] (reprinted with permissionfrom [Antoine, J.-P., Gazeau, J.-P., Monceau,P., Klauder J.R., Penson K.A., Temporally sta-ble coherent states for infinite well, J. Math.Phys., 42, p. 2349, 2001], American Institute ofPhysics).


J=10

J=50

J=2

1 2 3 4 5 6γ

Fig. 9.8 Squared modulus |〈 J , 0 | J , ωt〉|2 ofthe autocorrelation versus γ = ωt for the in-finite square well, for J = 2, 10, and 50. As inFigure 9.7, the large-J regime is characterizedby the occurrence of fractional revivals. SourceAntoine et al. [112] (reprinted with permission

from [Antoine, J.-P., Gazeau, J.-P., Monceau,P., Klauder J.R., Penson K.A., Temporally sta-ble coherent states for infinite well, J. Math.Phys., 42, p. 2349, 2001], American Institute ofPhysics).

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6t

(a)

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6t

(b)

Fig. 9.9 Squared modulus |〈 J, 0 | J, ωt〉|2 of the autocorre-lation for the Pöschl–Teller potential with nmax = 10, for (a)J = 20 and (b) J = 40. Source Antoine et al. [112] (reprintedwith permission from [Antoine, J.-P., Gazeau, J.-P., Monceau,P., Klauder J.R., Penson K.A., Temporally stable coherent statesfor infinite well, J. Math. Phys., 42, p. 2349, 2001], AmericanInstitute of Physics).


J=10

J=2

J=50

1 2 3 4 5 6γ

Fig. 9.10 Temporal behavior of the aver-age position of the particle in the infinitesquare well (in the Heisenberg picture),〈 J , 0 |Q(t) | J , 0〉 = 〈 J , ωt = γ |Q | J , ωt = γ〉,as a function of γ = ωt , for J = 2, 10, and50. Source Antoine et al. [112] (reprinted with

permission from [Antoine, J.-P., Gazeau, J.-P.,Monceau, P., Klauder J.R., Penson K.A., Tem-porally stable coherent states for infinite well,J. Math. Phys., 42, p. 2349, 2001], AmericanInstitute of Physics).

Most interesting is the temporal behavior of the average position 〈Q〉 and moregenerally of the average of a well-defined operator A(t) in such coherent states:

〈 J , 0|A(t)| J, 0〉 = 〈 J , 0|e i� Ht Ae– i

� Ht | J, 0〉= 〈 J , ωt = γ|A| J, ωt = γ〉 . (9.126)

This temporal behavior is shown in Figure 9.10 for the average position in the in-finite square well, for J = 2, 10, and 50. We note the tendency to stability aroundthe classical mean value 1

2 πa, except for strong oscillations of ultrashort durationbetween the walls near γ = nπ. The latter increase with J as expected when one ap-proaches the classical regime. For the sake of comparison, we show in Figure 9.11the temporal behavior of the average position in the asymmetric Pöschl–Teller po-tential (κ, λ) = (4, 8), for J = 20 and 50.

As some final information (but not the least important!), we show in Figure 9.12the temporal behavior of the average position 〈 J, 0 |Q (t) | J , 0〉 for the infinitesquare well, for a very high value of J = 106, near γ = ωt = 0. Here the quasi-classical behavior is striking in the range of values considered for γ. These tem-poral oscillations are clearly governed by T cl � π√

J= 3 ~ 10–3 and should be

compared with their purely classical counterpart in Figure 9.2.We do not include in this study the temporal behavior of the momentum opera-

tor. Indeed, in the case of the infinite well, its tentative expression as an essentiallyself-adjoint operator raises well-known problems (see, e.g., [112, 134–137]). We willcome back to this fundamental question in Chapter 15.


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

(a)

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

(b)

Fig. 9.11 Temporal behavior of the average position for theasymmetric Pöschl–Teller potential (λ, κ) = (4, 8) with nmax =10, for (a) J = 20 and (b) J = 50. Source Antoine et al. [112](reprinted with permission from [Antoine, J.-P., Gazeau, J.-P., Monceau, P., Klauder J.R., Pernon K.A., Temporally stablecoherent states for infinite well, J. Math. Phys., 42, p. 2349,2001], American Institute of Physics).

9.7.4Discussion

The specific choices we have made here for the set of coherent states are based ontwo additional guiding principles besides continuity and resolution of unity, as wasproposed in [58]. The first of these is “temporal stability”, which in words assertsthat the temporal evolution of any coherent state always remains a coherent state.The second of these, referred to as the “action identity” in [58], chooses variablesfor the coherent state labels that have as close a connection as possible to classical“action–angle” variables. In particular, for a single degree of freedom, the label pair( J , γ) is used to identify the coherent state | J, γ〉. Temporal stability means that,under the dynamics chosen, temporal evolution proceeds according to | J , γ + ωt〉,for some fixed parameter ω. To ensure that ( J , γ) describes action–angle variables,it is sufficient to require that the symplectic potential induced by the coherent statesthemselves is of Darboux form, or specifically that

i�〈 J , γ| d | J, γ〉 = J dγ ,

where d | J, γ〉 == | J + d J, γ + dγ〉 – | J , γ〉. Temporal stability is what fixes the phasebehavior of the coherent states, that is, the factor e–iγxn (cf. (9.77)), while ensuring


0

0.5

1

1.5

2

2.5

3

3.5

0 0.01 0.02 0.03 0.0 .05γ

Fig. 9.12 Temporal behavior of the average position in the caseof the infinite square well, for a very high value of J = 106 .Source Antoine et al. [112] (reprinted with permission from[Antoine, J.-P., Gazeau, J.-P., Monceau, P., Klauder J.R., PensonK.A., Temporally stable coherent states for infinite well, J. Math.Phys., 42, p. 2349, 2001], American Institute of Physics).

that ( J , γ) are canonical action–angle variables is what fixes the amplitude behaviorof the coherent states, that is, Jn e–iγ xn /(N ( J)

√[xn ]!) (cf. (9.81)).

In order for coherent states to interpolate well between quantum and classicalmechanics, it is necessary for values of the action J >> � that the quantum motionbe well approximated by the classical motion. In particular, for a classical systemwith closed, localized trajectories, a suitable wave packet should, if possible, re-main “coherent” for a number of classical periods. For the systems under study inthis chapter, we have demonstrated the tendency for improved packet coherencewith increasing J values within the range studied. For significantly larger values ofJ, we notice that the packet coherence substantially improves. Interesting resultswere obtained independently in a related study by Fox and Choi [138], who founda similar packet coherence for 10 or more classical periods for an infinite squarewell, even though they used a different amplitude prescription for their coherentstates. In both approaches, however, the probability distribution shows a Gaussianbehavior for large values of J, and this explains the similarity of the results.

It would appear that allowing for generalized phase and amplitude behavior inthe definition of coherent states has led us closer to the idealized goal of a set ofcoherent states adapted to a chosen system and having a large number of propertiesin common with the associated classical system, despite being fully quantum intheir characteristics.

165

10Squeezed States and Their SU(1, 1) Content

10.1Introduction

This chapter is devoted to another occurrence of SU (1, 1) in the construction ofvarious popular quantum states. The so-called squeezed states, which are to be de-scribed here, pertain again to quantum optics and have been raising interest inquantum optics and other fields for the last three decades. Squeezed states, a namegiven by Hollenhorst [139], were initially introduced in quantum optics [140] fordealing with processes in which emission or absorption of two photons is involved.The formalism underlying the process involves the square of raising and loweringoperators, a2 and a†2, for each mode of the quantized electromagnetic field. In thisregard, squeezed states might be viewed as a sort of “two-photon” coherent states.Their mathematical properties had actually been investigated before [141–143]. Theexperimental evidence for such states was provided by Slusher et al. [144]. Inter-esting applications to quantum nondemolition in view of detecting gravitationalwaves were envisaged as early as 1981 [145, 146]. In addition, it is interesting tolearn from Nieto [147] that these states were already known in 1927, 1 year afterthe Schrödinger states. For more details on theoretical and experimental aspects ofsqueezed states, see [25] and the recent report by Dell’Annoa et al. [148].

The introduction that is given here to the squeezed coherent states insists moreparticularly on their relations with the unitary irreducible representations of thesymplectic group S p(2, R), another version of SU (1, 1). We will also stress, ofcourse, their importance in quantum optics: they render possible reduction of theuncertainty on one of the two noncommuting observables present in the measure-ments of the electromagnetic field.


166 10 Squeezed States and Their SU (1, 1) Content

10.2Squeezed States in Quantum Optics

10.2.1The Construction within a Physical Context

In Chapter 7, we showed the efficiency of the coherent state formalism in the quan-tum description of the driven oscillator. An ingoing free oscillator state, |Ψin〉, thatexperiences during a certain time the action of a driving force F (t) is transformedinto an outgoing free oscillator state, |Ψout〉, through the unitary action of an Smatrix given by the displacement operator:

|Ψout〉 = S†|Ψin〉 , S† = D

(–i

F (ω)√2m�ω

), (10.1)

with F (ω) =∫ +∞

–∞ eiωt F (t) dt . Therefore, starting from the ground state, |Ψin〉 = |0〉,ones ends up with a coherent state |α〉 with parameter α = –i F (ω)√

2m�ω. For instance,

for a constant force with unit duration, F (t) = F 0 �[0,1](t), where �S is the character-istic function of the set S, we obtain

|Ψout〉 =

∣∣∣∣– F 0e–i ω2

√2m�ω

sin ω2

ω2

⟩. (10.2)

As an example, we can imagine a charged particle, with mass m and charge e, sub-jected to a harmonic restoring force and acted on during a certain time by a con-stant electric field E 0 so that F 0 = eE 0. In the classical description, the action ofthis constant force amounts to displacing the potential energy parabola as

H in =p2

2m+

12

mω2x2 → Hpert =p2

2m+

12

mω2x2 – F 0x . (10.3)

The quantum counterpart of this reads as

H in = �ω(

a†a +12

)→ Hpert = �ω

(a†a +

12

)– F 0

√�

2mω(a + a†) . (10.4)

The coherent states corresponding to the shifted parabola retain a Gaussian shape(e.g., in position representation and up to a phase factor), the same as for the origi-nal ground state. Now, one can explore the consequences of adding, at the classicallevel, a quadratic term to the potential:

H in =p2

2m+

12

mω2x2 → Hpert =p2

2m+

12

mω2x2 – F 0(g1x – g2x2) . (10.5)

This actually amounts, besides the shifting of the parabola, to modifying the springconstant as

k → k′ = k + 2g2F 0 . (10.6)

10.2 Squeezed States in Quantum Optics 167

On the quantum level, the Hamiltonian changes as

H in = �ω(a†a + 12 )→ Hpert = �ω(a†a + 1

2 ) + P 2(aa†) , (10.7)

where P 2 is quadratic polynomial. Then, the original ground state |0〉 changes intoa kind of “squeezed state,” for which we now give a mathematical definition.

Definition 10.1 A squeezed (coherent) state |α, �〉, α, � ∈ C, results from the com-bined action on the Fock vacuum state of the two following unitary operators:

|α, �〉 = D (α) S(�)|0〉 , D (α) = eαa†–αa , S(�) def= e12 (�a†2–�a2 ) , (10.8)

with |0, 0〉 == |0〉.

The squeezing operator S(�) is unitary since it is the exponential of the anti-Hermi-tian combination �a†2 – �a2. Note that its inverse is given, like for the displacementoperator D (α), by S–1(�) = S†(�) = S(–�).

Let us examine more closely the interest in such states before pursuing the de-velopment of the formalism. The elliptic deformation of the Gaussian results fromthe construction of the squeezed states. There follows a decreasing value for oneof the two variance factors in the Heisenberg uncertainty inequalities. This factcan be exploited in the detection of weak signals; see, for instance, [139]. Examplesdrawn from quantum optics are standard in regard to this notion of squeezing;as a matter of fact, squeezed light can be generated from light in a coherent stateor a vacuum state by using certain optical nonlinear interactions. For the sake ofsimplicity, let us consider a one-mode electromagnetic field propagating in one-dimensional volume L, with vector potential

A(x , t) = –i

(4π�c2

2V ω

)1/2 (aei(kx–ωt) – a†e–i(kx–ωt)

).

Neglecting spatial dependence, which is not important for our purpose, the electricfield operator reads as

E (x , t) = –1c

∂

∂tA(x , t) = E

(ae–iωt + a†eiωt

),

where E =(

4π�ω2L

)1/2.

Let us now introduce the following two self-adjoint observables

X = 12 (a + a†) , Y =

12i

(a – a†) , (10.9)

which would stand, in the context of one-particle quantum mechanics and up to thefactor 1√

2and appropriate units, for position and momentum operator, respectively.

Here, they describe the two components in phase quadrature of the field. Theycan, in principle, be measured separately through homodyne detection [43], but


not simultaneously, of course. Indeed, the commutation rule and the Heisenberginequality read, respectively,

[X , Y ] =i2

Id , ΔX ΔY v14

. (10.10)

In optical interferometry, the term homodyne signifies that the reference radiation(the “local oscillator”) is derived from the same source as the signal before the mod-ulating process. For example, in a laser scattering measurement, the laser beam issplit into two parts. One is the local oscillator and the other is sent to the system tobe probed. The scattered light is then mixed with the local oscillator at the detector.

In terms of these “quadratures,” the electric field reads

E (t) = 2E (X cos ωt + Y sin ωt) . (10.11)

Its mean value and variance in a certain state are given by

〈E (t)〉 = 2E (〈X 〉 cos ωt + 〈Y 〉 sin ωt) , (10.12)

(ΔE (t))2 = (2E )2(cos ωt sin ωt

)((ΔX )2 Δ{X Y }

Δ{X Y } (ΔY )2

)(cos ωtsin ωt

),

(10.13)

where Δ{X Y } = 〈 12 (X Y + Y X )〉 – 〈X 〉〈Y 〉. The Hermitian square matrix in (10.13)

is called a covariance matrix. It is an important object for the statistical descrip-tion of the quantum field. Let us now compare these expressions when we choosea coherent state, a pure squeezed state, or a mixed (coherent + squeezed) state.

10.2.1.1In a Coherent StateThe mean value of the electric field in a coherent state |α〉 = D (α)|0〉 is given by

〈E (t)〉/2E = R(αe–iωt

)=

(R(α) I(α)

)(cos ωtsin ωt

). (10.14)

The corresponding covariance matrix is equal to((ΔX )2 Δ{X Y }

Δ{X Y } (ΔY )2

)=

14

(1 00 1

), (10.15)

with (ΔE )2 = E2.

10.2.1.2In a “Pure” Squeezed StateA pure squeezed state is defined as produced by the sole action of the unitary oper-ator S(�) on the vacuum |0〉:

|0, �〉 def= S(�)|0〉 = e12 (�a†2–�a2 )|0〉 . (10.16)


In view of the computation of the mean value and the covariance matrix for theelectric field, we need to determine the unitary transport of the lowering and raisingoperators, S†(�)aS(�) and S†(�)a†S(�). By applying the formula

eABe–A = B + [A, B ] +12!

[A, [A, B ]] + · · · ,

and, with � = reiθ, one finds

S†(�)aS(�) = a cosh r + a†eiθ sinh r , S†(�)a†S(�) = a† cosh r + ae–iθ sinh r .

Hence, with U + iV == (X + iY )e–iθ/2, we have

S†(�)(U + iV )S(�) = Uer + iVe–r .

There follows for the field mean value

〈0, �|E (t)|0, �〉 = 0 , (10.17)

and for the covariance matrix((ΔX )2 Δ{X Y }

Δ{X Y } (ΔY )2

)=

14

(cosh 2rI2 + sinh 2r

(cos θ sin θsin θ cos θ

)).

(10.18)

In particular, we have for the product of variances

(ΔX )2 (ΔY )2 =116

(1 + sinh2 2r sin2 θ) ,

or equivalently

ΔU ΔV =14

. (10.19)

In consequence, with pure squeezed states, there is saturation of the Heisenberginequalities, but, contrarily to the coherent states, with ΔU =/ ΔV .

10.2.1.3In a General Squeezed StateIn a general squeezed state, |α, �〉 = D (α)S(�)|0〉, the average value of the field isthe same as in a coherent state:

〈α, �|E (t)|α, �〉 = 〈α, 0|E (t)|α, 0〉 = 2E R(αe–iωt

)=

(R(α) I(α)

) (cos ωtsin ωt

).

On the other hand, the variance is the same as for a pure squeezed state:

(ΔE )2 = 〈α, �|E 2 – 〈E 〉2|α, �〉 = 〈0, �|E 2 – 〈E 〉2|0, �〉 .

The same holds for the covariance matrix.Note that these results are independent of the manner in which one would define

the squeezed states, |α, �〉 def= D (α) S(�)|0〉 or |�, α〉 def= S(�) D (α)|0〉.


10.2.1.4A General Definition of SqueezingOne can now give a general definition of squeezing with respect to a pair of quan-tum observables A and B with commutator [A, B ] = iC =/ 0. When variancesare calculated in a generic state, one obtains from Cauchy–Schwarz inequalityΔA ΔB v 1

2 |〈C 〉|. Now a state will be called squeezed (with respect to the pair (A, B ))if (ΔA)2 (or (ΔB )2) < 1

2 |〈C 〉|. A state will be called ideally squeezed (with respect tothe pair (A, B )) if the equality ΔA ΔB = 1

2 |〈C 〉| is reached together with (ΔA)2 (or(ΔB )2) < 1

2 |〈C 〉|. Hence, the class of ideally squeezed states contains the set of puresqueezed states.

10.2.1.5Squeezing the UncertaintiesTo conclude this section, let us briefly explain the interest in squeezed states invarious applications, particularly in the coding and transmission of informationthrough optical devices. From the above study of the electric field, Glauber (stan-dard) optical coherent states have, in the phase quadrature plane, circularly sym-metric uncertainty regions, so the uncertainty relation dictates some minimumnoise amplitudes, for instance, for the amplitude and phase (Figure 10.1). A fur-ther reduction in amplitude noise is possible only by “squeezing” the uncertaintyregion, reducing its width in the amplitude, that is, radial, direction while increas-ing it in the orthogonal, that is, angular or phase, direction, so that the phase un-certainty is increased. Such light is called amplitude-squeezed (Figure 10.2). Con-versely, phase-squeezed light (Figure 10.3) has decreased phase fluctuations at theexpense of increased amplitude fluctuations. There is also the so-called squeezedvacuum, where the center of the uncertainty region (corresponding to the averageamplitude) is at the origin of the coordinate system, and the fluctuations are re-duced in some direction. The mean photon number is larger than zero in this case;a squeezed vacuum is a “vacuum” only in the sense that the average amplitude (but

��

� 〈X〉

�

〈Y〉

��

��

��

��

Fig. 10.1 Coherent light in the plane of phase quadratures. Theuncertainty region around one point of the plane is circularlysymmetric.


�

�

�

�

� 〈X〉

�

〈Y〉

��

��

��

��

Fig. 10.2 Amplitude-squeezed light in the plane of phasequadratures. The uncertainty ellipse is stretched in the angu-lar direction.

�

��

�

� 〈X〉

�

〈Y〉

��

��

��

��

Fig. 10.3 Phase-squeezed light in the plane of phase quadra-tures. The uncertainty ellipse is stretched in the radial direc-tion.

not the average photon number) is zero. For example, an optical parametric ampli-fier with a vacuum input can generate a squeezed vacuum with a reduction in thenoise of one quadrature components on the order of 10 dB.

In summary, the main feature of the squeezed states lies in the fact that theyoffer the opportunity of deforming (squeezing!) in a certain direction the circle of“quantum uncertainty” that so becomes an uncertainty ellipse. Hence, they providea way of reducing the quantum noise for one of the two quadratures.

10.2.2Algebraic (su(1, 1)) Content of Squeezed States

We have seen that squeezed states are obtained by forcing two-photon processeswith time-dependent classical sources (from this the appellation two-photon coherent


states). The most general form of the corresponding Hamiltonian reads

H = �ω(

a†a +12

)+ f 2(t) a†

2+ f 2(t) a2 + f 1(t) a† + f 1(t) a . (10.20)

The production of coherent states or of squeezed states will depend on the re-spective importance granted to the factors f 1(t) and f 2(t). Now, the Hamiltonianin (10.20) has clearly the form of an element in the “two-photon” Lie algebra, denot-ed by h6, generated by the set of operators {a, a†, Id , N = a†a, a2, a†2}, and alreadymentioned in Section 2.2.4. From the (nontrivial) commutation rules,

[a, a†] = Id , [a, N ] = a , [a†, N ] = –a† ,

[a2, a†] = 2a , [a†2, a] = –2a† , [a2, N ] = 2a2 , [a†

2, N ] = –2a†

2,

(10.21)

we see that h6 is a representation of the semidirect sum of su(1, 1) with the Weyl–Heisenberg algebra. The corresponding group, denoted by H6, is the semidirectproduct H6 = W � SU (1, 1).

The existence of such an algebraic tool in the construction of squeezed states isvery useful in solving problems involving Hamiltonians such as (10.20). Supposewe have to deal with an evolution equation of the type

i∂

∂tU (t , t0) = H(t) U (t , t0) , U (t0, t0) = Id , (10.22)

where the mathematical objects have a “h6” nature: the time-dependent Hamilto-nian H is an element of the Lie algebra h6, whereas the evolution operator U (t , t0),as a solution to (10.22), should be an element of a unitary representation of theLie group H6. The trick is of disentangling nature [9], like in (8.7.4). It amountsto solving the equation by choosing among linear faithful representations of H6 orh6 the simplest one, namely, the four-dimensional one in which group and algebraelements are realized as 4 ~ 4 matrices. Let us associate with a generic element Xin h6, written as

X = η(N + 1

2

)+ δId + Ra†

2+ La2 + ra† + la , η, δ, R, L, r , l ∈ C , (10.23)

the following matrix in M(4, C):

M(X ) =

⎛⎜⎜⎝0 0 0 0r η 2R 0–l –2L –η 0

–2δ –l –r 0

⎞⎟⎟⎠ . (10.24)

This representation of X is made possible because of the Lie algebra isomorphismbetween basic operators defining h6 and elementary projectors E i j with matrixelements

(E i j

)kl

= δik δ j l generating the Lie algebra M(4, C):

N +12�→ E 22 – E 33 , Id �→ –2E 41 ,

a†2 �→ 2E 23 , a† �→ E 21 – E 43 ,

a2 �→ –2E 32 , a† �→ –E 31 – E 42 .

(10.25)


Owing to the general form (10.20) of the Hamiltonian, (10.22) is equivalent toa dynamical system in the space of time-dependent parameters (η, δ, R, L, r , l). Thisdynamical system is easily made explicit thanks to the map M : X �→ M(X ), whichpermits one to write (10.22) in its 4 ~ 4-matrix representation:

i∂

∂tM(U ) = M(H) M(U ) , M(U 0) = I4 , (10.26)

where U = eX . Now, the exponential of a matrix such as (10.24) has the generalform ⎛⎜⎜⎝

1 0 0 00

P 01

⎞⎟⎟⎠ ,

where P is a 3 ~ 3 matrix. Therefore, the solution M(U ) to (10.26) should be ofthis type, and its parameters (η, δ, R, L, r , l) should be, in general, determined nu-merically. Inverting the map M allows one to find the unitary operator U (t , t0) andeventually the resulting S matrix for a specific physical process. Note that there ex-ist various forms for U = eX to be used as an ansatz in dealing with (10.26), forinstance,

U = exp[

η(N + 1

2

)+ δId + Ra†

2+ La2 + ra† + la

]= exp[R′a†

2+ r ′a†] exp

[η′

(N + 1

2

)+ δ′Id

]exp[L′a2 + l ′a]

= exp[r ′′a† + l ′′a] exp[R′′a†2

+ L′′a2] exp[η′′

(N + 1

2

)+ δ′′Id

].

We give below the matrix P appearing through the map M for each of these expres-sions:

P =

⎛⎜⎜⎝eG – I2

G

(r–l

)eG

–2δ +(–l –r

) eG – I2 – GG

(r–l

) (–l –r

) eG – I2

G

⎞⎟⎟⎠=

⎛⎝ –2R′l ′e–η′ eη′ – 4L′R′e–η′ 2R′e–η′

–l ′e–η′ –2L′e–η′ e–η′

–2δ′ + r ′l ′e–η′ 2r ′L′e–η′ –r ′e–η′

⎞⎠

=

⎛⎜⎜⎜⎝r ′′ cosh θeη′′ 2R′′

sinh θθ

e–η′′

–l ′′ –2L′′sinh θ

θeη′′ cosh θe–η′′

–2δ′′ –l ′′ cosh θeη′′ – 2L′′r ′′sinh θ

θ–r ′′ cosh θe–η′′ – 2R′′l ′′

sinh θθ

⎞⎟⎟⎟⎠,

with

G =

(η 2R′

–2L –η

),


and eG = cosh θI2 +G sinh θθ , θ2 = η2 –4LR. Such results, obtained from tractable dis-

entangling factorizations of 4 ~ 4 matrices, permit us to calculate easily mean valuesand variances of arbitrary powers of elements of h6 or H6. These disentangling ma-nipulations are an easy alternative to the Baker–Campbell–Hausdorff–Zassenhausformulas mentioned in Section 6.5. Let us start with the mean value 〈X 〉 of X ∈ h6

in the state |Ψ(t)〉 = U (t , t0)|0〉. We can write

〈X 〉 = 〈0|U †X U |0〉 =∂

∂γ〈0|U †eγX U |0〉

∣∣∣∣γ=0

.

Now, the expression U †eγX U , as a product of operator-valued exponentials, is pre-cisely computed by using the finite-dimensional representation M:

U †eγX U M→ (M(U ))–1M(eγX

)M(U ) .

The appropriate disentangling relation is that which leads to X = η(N + 1/2) + δId ,U = eLa2 +la , and

M(

eLa† 2+ra†)

M(eη(N +1/2)+δId

)M

(eLa2 +ra

),

where η and δ are functions of γ such that η(0) = 0 = δ(0). Then, by inverting themap M, we get the trivial calculation of the mean value:

〈0|eLa† 2+ra† eη(N +1/2)+δId eLa2 +ra|0〉 = eη(γ)/2+δ(γ) .

The successive moments of the observable X are then given by

〈X n〉 =

(∂

∂γ

)n

eη(γ)/2+δ(γ)

∣∣∣∣γ=0

. (10.27)

For the first two, one gets

〈X 〉 = 12 η′(0) + δ′(0) , 〈X 2〉 =

(12 η + δ

)′′∣∣∣γ=0

+ (〈X 〉)2 . (10.28)

Therefore, the variance is given by

(ΔX )2 = 〈X 2〉 – 〈X 〉2 =(

12 η + δ

)′′∣∣∣γ=0

. (10.29)

As a complementary expression to these formulas, let us consider the case inwhich we calculate the mean value of the operator eγa†+γa in a squeezed state |Ψ〉:

F (γ, γ) def=⟨

eγa†+γa⟩

.

(i) First expression of squeezed states. If

|Ψ〉 = D (α)S(�)|0〉 ,

then

F (γ, γ) =(γα + 1

2 γ2eiθ sinh ρ)

+ (c.c.) + 12 |γ|

2 cosh 2ρ ,

with � == ρeiθ.


(ii) Second expression of squeezed states. If

|Ψ〉 = S(�)D (α)|0〉 ,

then

F (γ, γ) =(γ(α cosh ρ + αeiθ sinh ρ) + 1

2 γ2eiθ sinh ρ)

+ (c.c) .

(iii) Most general expression of squeezed states. If

|Ψ〉 = eRa† 2+ra†–Ra2 –ra|0〉 ,

then

F (γ, γ) =

(γr + 2rR1 – 4RR

+ 12 γ2 2R

1 – 4|R|2

)+ c.c. .

10.2.3Using Squeezed States in Molecular Dynamics

Let us give a nice illustration (see [9] and references therein for more details) ofthe use of the previous formulas in the domain of molecular dynamics. Let usconsider a diatomic molecule (e.g., O2, NO, N2) hit by an inert atom (e.g., He). TheHamiltonian of the system reads as

H = H target + Hprojectile + H int ,

where

H int(R, q) = V (R, q) – V (∞, q) ,

R being the relative position of the projectile with respect to the center of massof the molecule, whereas q = q(t) is the position operator corresponding to thedisplacement of the intermolecular relative position with respect to the equilibri-um. Within the usual framework of the semiclassical approach, namely, the Born–Oppenheimer approximation, the trajectory of the projectile R = R(t) is consid-ered as classical, and we can deal with the series expansion of H int in terms of thesmall displacement q:

H int = h0(t) + h1(t)q +12

h2(t)q2 + · · · .

By putting q =√

�mω (a+a†)

/√2 and neglecting terms beyond the quadratic ones,

we get the following expression for the Hamiltonian:

H =12

MR2 +

(�ω +

�

mωB (t)

)(a†a +

12

)+

�

2mωA(t)(a + a†) +

�

2mωB (t)(a2 + a†

2) . (10.30)


From this (approximate) Hamiltonian it becomes possible to determine, for in-stance, the probability of the target of changing its vibrational ground state to anexcited state |n〉 after the collision. The calculation goes through the S-matrix for-malism:

|Ψout〉 == |Ψ(t = –∞)〉 = S†|Ψin〉 == S†|Ψ(t = –∞)〉 . (10.31)

Thus the S matrix is the limit (in a certain functional sense)

S† = limt0→–∞t→+∞

U (t , t0)

of the evolution operator associated with the Hamiltonian through the equation

i�∂

∂tU (t , t0) = H(t)U (t , t0) , U (t0, t0) = Id . (10.32)

In dealing with this equation, one should consider the constraint resulting fromthe conservation of the classical energy:

12 M

(R2(∞) – R2(–∞)

)+ ΔV = 0 . (10.33)

Fig. 10.4 Transition probabilities for ex-citations of H2 by He scattering. (a) Tran-sition probabilities P 0→n f (n f = 0 to6) as a function of the total energy E tot ofthe He + H2 system. All curves are drawnsmoothly through the data points of Gilmoreand Yuan [149, 150]. • and � denote resultsobtained by Gazdy and Micha [151, 152].

(b) Transition probabilities P ni→n f as a func-tion of n f for E = 12 and 16. The three plotson the left are for E = 12 and those on theright are for E = 16 [149, 150] (reprinted withpermission from [Zang, W.-M., Feng, D.-H.,and Gilmore, R., Rev. Mod. Phys., 26, 867,1990], American Institute of Physics).


The functionR(t) is determined classically by taking into account a loss of kineticenergy suitably smoothed during the collision. Once R(t) is known, one proceedsto the numerical integration of the 4 ~ 4-matrix version of (10.32):

i�∂

∂tM [U (t , t0)] = M [H(t)] M [U (t , t0)] , M [U (t0, t0)] = I4 . (10.34)

Taking the limits t → +∞, t0 → –∞, one obtains the parameters (R, r , η, δ, L, l) ofthe S matrix, as they were introduced in (10.23):

S = eRa† 2+ra† eη(n+1/2)+δId eLa2 +la . (10.35)

Its transition 0→ n matrix element is then given by

〈n|S|0〉 =� n

2 �∑s=0

rn–2sRs√

n!(n – 2s)!s!

eη/2+δ . (10.36)

Some examples of curves giving the probability P 0 n = |〈n|S|0〉|2 are shown inFigure 10.4 for the collision He + H2 →He + H2

�.

179

11Fermionic Coherent States

11.1Introduction

Quantum elementary systems can be divided into two classes, bosons and fermi-ons, according to the integral or half-integral value of their spins. This divisionrules many fundamental physical evidences, such as the possibility of particle colli-sions or nuclear reactions or the stability of matter. Fermions are particles that obeyFermi–Dirac statistics (identical particles are forbidden from being in the same state),while bosons are particles that obey Bose–Einstein statistics (identical particles canbe in the same state). One can state that the standard coherent states are bosonic,in the sense that they are superpositions of number states for which there is nolimitation on the number of excitations involved.

We present here the so-called fermionic coherent states, which are, on their side,superpositions of number states |n〉 where the number n assumes two values only,n = 0 or n = 1. After their algebraic description, we will give some insight intotheir utilization in the study of many-fermion systems (e.g., the Hartree–Fock–Bogoliubov approach).

Fermionic models are essentially encountered in atomic and nuclear physics.Typically, they account for assemblies of about 100 particles. Therefore, it is nec-essary to proceed to approximations such as variational methods. The criteria thatare retained for acceptable test functions are simplicity and optimal considerationof quantum correlations, criteria that are met by coherent states.

11.2Coherent States for One Fermionic Mode

The Pauli exclusion principle requires the creation and annihilation operators fora fermionic mode to obey the following anticommutation rules:

[a, a†]+def= aa† + a†a = Id , [a, a]+ = 0 = [a†, a†]+ . (11.1)

Any system having such commutation rules has the group SU (2) as a “dynami-cal” group, that is, a group whose elements connect states with different energies.


180 11 Fermionic Coherent States

The generators of this group are in the present context [2]{a†, a, a†a – 1/2}, withthe commutation rules

[a†, a] = 2(a†a – 1

2

) [a†a – 1

2 , a]

= –a[a†a – 1

2 , a†]

= a† . (11.2)

It is important to note that these commutation rules are derived from the anticom-mutation rules (11.1) and not from the usual canonical commutation rules.

Let us now proceed to the construction of the corresponding coherent states.Since a†2 = 0 = a2, the only pertinent Hermitian space is the two-dimensionalHermitian one, H j=1/2 ~= C2. Thus, it is enough to repeat the construction of thestandard coherent states, but keeping the constraint a†2 = 0 = a2, or the construc-tion of the SU (2) coherent states with spin generators J± and j = 1/2. Hence, weassociate with � ∈ C the states

|�〉± = e�a†–�a |±〉 , (11.3)

where |±〉 (== | 12 ±12 〉 in SU (2) notation) are such that a†|–〉 = |+〉 , a|+〉 = |–〉. Since

(�a† – �a)2 = –|�|2(a†a + aa†) = –|�|2, we have

e�a†–�a = cos |�|Id +�|�| sin |�| a† –

�|�| sin |�| a .

Putting � = – θ2 e–iφ, we get the two expressions

|�〉+ = cosθ2|+〉 + sin

θ2

eiφ|–〉 , |�〉– = sinθ2

e–iφ|+〉 + cosθ2|–〉 , (11.4)

in which we recognize the spin or Bloch coherent states for j = 1/2. Therefore,we inherit the properties of spin coherent states in the present fermionic context,like the overlap relation (6.31), the way they behave under the action of an SU (2)transformation (6.28), and the resolution of the identity (6.10). Note that we couldalso use the other complex parameterization � = tan θ

2 eiφ, a notation that we willrecover on a more general level in the next sections.

11.3Coherent States for Systems of Identical Fermions

11.3.1Fermionic Symmetry SU(r)

We now consider a system of r fermionic states or modes occupied by at mostone fermion. To each mode corresponds one pair (a†i , ai ) of operators obeying theanticommutation rules (11.1):

[ai , a†j ]+ = δi j , [ai , a j ]+ = 0 = [a†i , a†j ]+ , 1 u i , j u r . (11.5)

11.3 Coherent States for Systems of Identical Fermions 181

Table 11.1 A short glossary of matrix groups and algebras.

GroupsSymbol Definition Real dimension

M (r , C) r ~ r matrices with complex entries 2r2

GL(r , C) ⊂ M (r , C) Invertible matrices 2r2

SL(r , C) ⊂ GL(r , C) Unit determinant 2r2 – 2

U (r ) ⊂ GL(r , C) Unitary matrices mm† = Id r2

SU (r ) = U (r )⋂

SL(r , C) Unitary and unit determinant r2 – 1

Lie algebras

Symbol Definition Real dimension

gl(r , C) ~= M (r , C) r ~ r matrices with complex entries 2r2

sl(r , C) Null trace 2r2 – 2

u(r ) Anti-Hermitian matrices m† = –m r2

su(r ) Anti-Hermitian matrices with null trace r2 – 1

Several Lie algebras (see Table 11.1) can be built from the set of operators ai , a†i .The most immediate one is the algebra u(r), linear span of the r2 quadratic opera-tors

a†i a j , 1 u i , j u r . (11.6)

They obey the commutation rules

[a†i a j , a†kal ] = δ jk a†i al – δi l a†ka j . (11.7)

The r elements Hi = a†i ai span a maximal Abelian subalgebra of u(r). The waythis subalgebra acts on the other elements of u(r) is described by the commutationrelations

[Hi , a†j ak ] =(δi j – δik

)a†j ak . (11.8)

This algebra acts on the Hermitian spaces describing r fermionic states, moreprecisely that which carries a completely antisymmetric fundamental representa-tion of u(r), denoted Λ = {λ1, λ2, . . . , λr}, with λi ∈ {0, 1}. The latter are the valuesassumed by the r fundamental weights of the Lie algebra u(r) (see Appendix B).Thus, when p u r identical fermions are involved,

Λ = {λ1, λ2, . . . , λr} = {1, 1, . . . , 1, 0, 0 . . . , 0} == {1p , 0r– p} , p u r . (11.9)

This space of states has its dimension equal to( r

p

), and a suitable basis is formed

by the kets |n1, n2, . . . , nr 〉, where ni = 0 or 1 denotes the occupation number of


the ith level, and the constraint∑r

i=1 ni = p expresses the conservation of the totalnumber of fermions. These basis states are common eigenstates of the completeset {Hi , 1 u i u r} of commuting observables. With respect to this basis, thematrix representative E i j of the generator a†i a j of u(r) has all its entries equal to 0with the exception of entry (i , j ), which is equal to 1:

a†i a j ←→ E i j with(E i j

)kl

= δik δ j l , (11.10)

a notation previously used in (10.25). Let us consider the following “extremal” statefor the representation Λ = {1p , 0r– p}:

|extr〉 = |p︷︸︸︷

1, 1, . . . , 1,

r– p︷︸︸︷0, . . . , 0〉 . (11.11)

It is actually the nonperturbed ground state of a p-fermion Hamiltonian expressedin a mean-field approximation:

H =p∑

i , j=1

ki j a†i a j +p∑

i , j,l,m=1

V i jlm a†i a†j amal . (11.12)

The mean-field approach is necessary when we deal with a many-body system withinteractions, since in general there is no way to solve exactly the model, exceptfor extremely simple cases. A major drawback (e.g., when computing the partitionfunction of the system) is the treatment of combinatorics generated by the interac-tion terms in the Hamiltonian when summing over all states. The goal of mean-field theory (also known as self-consistent-field theory) is precisely to resolve thesecombinatorial problems. The main idea of mean-field theory is to replace all inter-actions with any one body with an average or effective interaction. This reduces anymultibody problem into an effective one-body problem.

State (11.11) is extremal in the sense that it is annihilated by all elements asa†i a j |extr〉 = 0 for all i, j, such that 1 u i =/ j u p or p + 1 u i , j u r . Togetherwith the Hi ’s, 1 u i u p , these generators span the subalgebra u( p)⊕ u(r – p). Thecorresponding group, U ( p)⊗U (r – p), is the subgroup of elements in U (r) leavinginvariant, up to a phase, the extremal state |extr〉, that is, it is the stability group ofthe latter. One then defines a family of coherent states for the group U (r) by

∣∣� ==(�i j

)⟩= exp

⎡⎢⎢⎣ ∑1u ju p

p+1uiur

(�i j a†i a j – �i j a

†j ai )

⎤⎥⎥⎦ |extr〉 . (11.13)

These states are in one-to-one correspondence with the points of the coset U (r)/(U ( p)

⊗U (r – p)) [153]. They represent in a certain sense a “quantum” version of

this compact manifold. Note that for a system of bosonic states, the manifold to beconsidered would be the coset U (r)/(U (1) ⊗U (r – 1)).

Given the set of complex parameters {�i j , 1 u j u p , p + 1 u i u r} viewed asthe entries of a (r – p) ~ p matrix �, let us introduce two other (r – p) ~ p matrices:

Z = �sin

√�†�√

�†�, sin

√�†� =

√�†� –

(√

�†�)3

3!+ . . . , (11.14)


� = Z (I p – Z †Z )– 12 . (11.15)

The matrix � is well defined because I p –Z †Z is by construction a positive matrix. Itcan be coordinatized by p(r – p) complex parameters, denoted �α, p(r – p) being thecomplex dimension of the coset U (r)/(U ( p)⊗U (r – p)). The �α’ represent a kindof projective coordinates for this manifold. With this notation, if one considers thematrix representation a†i a j ←→ E i j of the u(r) basis elements, we find that thedisplacement operator

exp

⎡⎢⎢⎣ ∑1u ju p

p+1uiur

(�i j a†i a j – �i j a

†j ai )

⎤⎥⎥⎦or equivalently the coherent state |�〉 == |�〉, is represented by the following unitaryr ~ r matrix:(√

Ir– p – Z Z † Z–Z †

√I p – Z †Z

)== γ(Z ) ∈ U (r) . (11.16)

The notation � is particularly suitable for describing the action of the group U (r)on its coset manifold U (r)/(U ( p) ⊗ U (r – p)). Let us write an element of U (r) inthe following block matrix form:

U (r) � g =

⎛⎝ A(r– p)~(r– p)

B

C D( p)~( p)

⎞⎠ . (11.17)

Its action on the elements � of the coset reads as a projective one:

�g�→ �′ == g · � = (A� + B )(C � + D )–1 . (11.18)

11.3.1.1RemarkThis action of the group on one of its cosets is not mysterious. We already describedan example of it in (8.36). On a general level, let G be a group and H be a subgroupof G. The right coset G/H is the set of equivalence classes “modulo H” defined bythe relation g ==

Hg ′ if and only if there exists h ∈ H such that g ′ = gh . Thus, G/H =

{gH def= {gh , h ∈ H}, g ∈ G}. Now, let g0 ∈ G and g0H be its equivalence class.An element g ∈ G “transports” this class by left multiplication:

g0Hg�→ g (g0H) = (g g0)H .

In the present case, it is enough to note that any element

g =

(A BC D

)∈ U (r)


with nonsingular D can be factorized as g = γ(Z g )h g with �g = BD –1, γ(Z g ) likein (11.16) and h g ∈ U ( p)⊗U (r – p). So, by simple matrix multiplication, we havegγ(Z ) = γ(Z ′)h ′ ==

Hγ(Z ′), where �′ is like in (11.18).

The compact manifold U (r)/(U ( p) ⊗ U (r – p)) has a rich structure. It is, likeC or the sphere S2 or the Poincaré half-plane, a Kähler manifold: its Riemanniansymplectic structure is simply encoded by the Kähler potential F = F (�, �) given by

F (�, �) = ln det(I p + �†�) . (11.19)

The metric element and the 2-form derive from it as follows:

ds2 =∑α,�

gα� d�α d�� , gα� =∂2

∂�α∂��F (�, �) , (11.20a)

ω =i2

∑α,�

gα� d�α ∧ d�� . (11.20b)

The volume element is given by

dμ(�, �) =dim V Λ

Vol(U (r)/U (r – p)

) (det

(I p + �†�

))–r ∏α

d�α d�� . (11.21)

In the normalization factor, V Λ denotes the finite-dimensional space carryingthe unitary irreducible representation Λ of U (r). The explicit form of the coher-ent states is established thanks to the existence of a Baker–Campbell–Hausdorff–Zassenhaus factorization formula for the group U (r). This formula expresses thedisplacement operator appearing in (11.13) in terms of the variables �i j :

exp

⎡⎢⎢⎣ ∑1u ju pp+1uiur

(�i j a

†i a j – �i j a

†j ai

)⎤⎥⎥⎦ =

exp


�i j a†i a j

⎤⎥⎥⎦ exp

⎡⎢⎢⎣ ∑1ui , ju p

p+1ui , jur

λi j a†i a j

⎤⎥⎥⎦ exp

⎡⎢⎢⎣ ∑1uiu p

p+1u jur

–�i j a†i a j

⎤⎥⎥⎦ .

(11.22)

Equation (11.22) is proven through the use of the following factorization in thefinite matrix representation of the operators involved:(√

Ir– p – Z Z † Z–Z †

√I p – Z †Z

)=

(Ir– p �

0 I p

) (eλ1 00 eλ2

) (Ir– p 0–�† I p

),

(11.23)

with eλ1 = (Ir– p – Z Z †)–1/2 and eλ2 = (I p – Z †Z )1/2.


By using the factorization (11.22) and the fact that the two right factors stabilizethe extremal state

a†i a j |extr〉 = 0 for 1 u i =/ j u p or p + 1 u i , j u r ,

one can write the coherent states as

|�〉 =1

(N (�, �))1/2exp


�i j a†i a j

⎤⎥⎥⎦ |extr〉 , (11.24)

withN (�, �) = det(I p~ p – �†�

). The scalar product (overlap) between two coherent

states is then given in terms of the Kähler potential by

〈�′|�〉 =eF (�,�′)

(N (�, �))1/2 (N (�′, �′))1/2. (11.25)

These coherent states, as an overcomplete family of vectors in V Λ, resolve theidentity:∫

U (r )/(U ( p)⊗U (r– p))dμ(�, �) |�〉〈�| = Id . (11.26)

This implies that any state |Ψ〉 in the Hermitian space of fermionic levels can beexpanded in terms of those states,

|Ψ〉 =∫

dμ(�, �) Ψ(�) |�〉

where the “contravariant” symbol of |Ψ〉 is given by

Ψ(�) def= 〈�|Ψ〉 == (N (�, �))–1/2 f (�) .

Here, f (�) is an analytical function of �, thus providing the space V Λ with a real-ization of the Fock–Bargmann type.

11.3.2Fermionic Symmetry SO(2r)

A second relevant algebra, namely, so(2r), is constructed from the operators a†i anda j . Its generators are the r(2r – 1) operators

a†i a j –12

δi j , 1 u i , j u r (11.27)

ai a j and a†i a†j , 1 u i =/ j u r . (11.28)

The maximal Abelian subalgebra, that is, Cartan subalgebra, is generated by the roperators Hi = a†i ai – 1/2. The relevant Hermitian spaces, denoted V Λ± , are those


which carry the two fundamental spinor representations of so(2r), denoted by Λ± ={12 , 1

2 , . . . ,± 12

}. The sign ± corresponds to the parity of the number n =

∑ri=1 ni ,

where the components ni are the respective eigenvalues of the generators Hi . Theyassume their values in {0, 1} and are used to label the basis states |n1, n2, . . . , nr 〉.Thus, the dimension of V Λ± is equal to 2r–1 = 1

2

∑p

( rp

): here, the number of

existing fermions varies from 0 to r.The matrix representation of the generators of so(2r) is as follows:

a†i a j – 12 δi j ↔ E i j – E r+ j r+i ,

a†i a†j ↔ E ir+ j – E jr+i

ai a j ↔ E r+i j – E r+ j i ,

where the matrices E i j were defined in (11.10), and with 1 u i , j u r . Extremalstates can be chosen as

|extr〉 =

{|0, 0, . . . , 0〉 == |0〉 for Λ+

|1, 0, . . . , 0〉 == |1〉 for Λ– .(11.29)

These extremal states are nonperturbed ground states of an r fermionic stateHamiltonian of the type (11.12), already considered for su(r), with the restrictionthat the parity of the total number of fermions now be fixed to be even (for V Λ+ )or odd (for V Λ– ). These two extremal states are annihilated or left invariant in thefollowing way:

a†i a j |0〉 = 0 for 1 u i , j u r , for the Λ+ ground state ,⎧⎪⎪⎪⎨⎪⎪⎪⎩a†1a1|1〉 = |1〉 ,

a†i a j |1〉 = 0 for 2 u i , j u r ,

a†1a†i |1〉 = 0 for 2 u i u r ,

ai a1|1〉 = 0 for 2 u i u r ,

for the Λ– ground state .

Thus, the stability subgroup of the state |0〉 is generated by the operators a†i a j –12 δi j , 1 u i , j u r and so is isomorphic to U (r). Similarly, the subgroup that stabi-lizes the state |1〉 is generated by the operators a†1a1 – 1/2, a†i a j – 1

2 δi j , 2 u i , j u r ,a1ai , and a†1a†i , 2 u i u r , and so is also isomorphic to U (r). Both are subgroupsof Spin(2r), the double covering of SO (2r). In consequence, the family of coher-ent states involved is in one-to-one correspondence with the points of the cosetSpin(2r)/U (r):

Spin(2r)/U (r) � � �→ |�〉 = exp

⎛⎝∑1ui=/ j

�i j a†i a†j – h.c.

⎞⎠ |extr〉 . (11.30)

In what follows, we choose as the extremal state the even one, |0〉. By using thematrix representation, one easily checks that a coherent state corresponds to the


following point in the coset (11.30):

|�〉 ↔

(√Ir – Z Z † Z

–Z †√

Ir – Z †Z

), Z def= �

sin√

�† �√�† �

, (11.31)

where � and Z are r ~ r antisymmetric matrices, �i j = –� ji . As for U (r), weintroduce the projective matrix variable,

� = Z (I p – Z †Z )– 12 == (�α) , (11.32)

and we will denote coherent states as well by |�〉 == |�〉.Here too, the coset Spin(2r)/U (r) has the geometrical structure of a Kähler man-

ifold with potential equal to F (�, �) = 12 ln det(Ir + �†�). From this potential the

Riemannian metric,

ds2 =∑α,�

gα,� d�α d�� , gα,� =∂2

∂�α∂��F (�, �) ,

and the 2-form,

ω = 12

∑α,�

gα,�d�α ∧ d�α ,

are derived. The volume element is given by

dμ(�, �) =dim V Λ

Vol(Spin(2r)/U (r)

) (det

(Ir~r + �†�

))– 12

∏α

d�α d�α .

As expected, we have the resolution of the identity,∫dμ(�, �) |�〉〈�| = Id ,

and the resulting analytical Fock–Bargmann realization of the space of states V Λ:

V Λ � Ψ �→ Ψ(�) def= 〈�|Ψ〉 .

11.3.3Fermionic Symmetry SO(2r + 1)

With the adding of the operators ai , a†j , 1 u i , j u r , to the generators of so(2r), oneobtains the algebra so(2r+1). The commutations rules for so(2r) are now completedby

[a†i , a j ] = 2(a†i a j – 12 δi j ) , (11.33)

[a†i a j – 12 δi j , ak ] = –δika j , (11.34)


[a†i a j – 12 δi j , a†k] = δ j ka†i . (11.35)

The Cartan subalgebra is generated by the r operators

Hi = a†i ai –12

i = 1, 2, . . . , r . (11.36)

The apposite Hermitian space V Λ of states is now the one that carries the spinorialrepresentation Λ ==

[12 , 1

2 , . . . 12

], for which the basis states are |n1, n2, . . . , nr 〉, with

ni = 0 or 1. There are 2r such states. The matrix representation is expressed interms of matrices E i j of order 2r + 1, for example,

a†i ↔ E i0 – E 0r+i , etc .

By choosing as the extremal state |0〉 == |0, 0, . . . , 0〉, the stability group of the latteris a subgroup isomorphic to U (r) and is generated by the operators

a†i a j –12

δi j , 1 u i , j u r .

Hence, the coherent states built from the state |0〉 are in one-to-one correspondencewith the points � in the coset Spin(2r + 1)/U (r).

11.3.4Graphic Summary

In Table 11.2 we summarize the chain of dynamical groups for the Hamiltonian

H =∑

i , j

εi j a†i a j +

∑i , j ,l,m

V i jlm a†i a†j al am

for a system of r identical fermions in the mean field approximation.

Table 11.2 Dynamical groups for systems of identical fermions.

Chain of groups and representation carrier spaces

Group Carrier space Dimension Conserved quantity

U (r ) V Λk(

rk

)Number k of fermions

↓SO (2r ) V Λ+ ⊕ V Λ– = ⊕r

k=0V Λk 2r–1 + 2r–1 = 2r Parity of the total number

of occupied levels↓

SO (2r + 1) ⊕rk=0V Λk 2r

11.4 Application to the Hartree–Fock–Bogoliubov Theory 189

11.4Application to the Hartree–Fock–Bogoliubov Theory

For most of the N fermion systems (e.g., nuclei, atoms, molecules), the shellstructure emerges in a systematic way from approximations of the mean-fieldtype. Hence, by restricting ourselves to one- or two-body interactions, we considerHamiltonians of a form that is a particular case of (11.12):

H =r∑

j=1

ε j a†j a j + 14

∑i , j ,k,l

V i jkl a†i a†j ak al == H0 + H int . (11.37)

The maximal dynamical symmetry group is SO (2r + 1). If one excludes from thescope of our analysis the transfer of one sole particle, then the dynamical symme-try group reduces to SO (2r). If the pairing is also neglected, then the dynamicalsymmetry group becomes U (r).

Let us consider the case SO (2r) with its fundamental spinorial representationΛ+ =

[12 , 1

2 , . . . , 12

]. We have defined the coherent states by

|�〉 == |�〉 = T |0〉 , where T = exp

⎛⎝∑j<k

(� jka†j a†k – � jka j ak

)⎞⎠ .

(11.38)

The “displacement” operator T acting on the vacuum |0〉 = |0, 0, . . . , 0〉 is the mostgeneral possible unitary transformation since U (r) is the stability subgroup of thisextremal state and any element of SO (2r) can be written as the product of theelement corresponding to T with an element of U (r).

To the transformation T there corresponds the following transformation of theone-particle annihilation and creation operators a j and a†j :

(αα†

)= T

(aa†

)T –1 , a ==

⎛⎜⎜⎜⎝a1

a2...

ar

⎞⎟⎟⎟⎠ == (a j ) , and so on . (11.39)

This is precisely the most general one among the so-called Hartree–Fock–Bogoliu-bov transformations particle �→ “quasiparticle”. Within the present framework, thistransformation is rewritten in matrix-block operatorial form as(

αα†

)= T†

(aa†

), with T† =

(U –V †

V U t

), (11.40)

and

U = cos√

�†� = U † , V = –�sin

√�†�√

�†�= –V † .


It follows that the new vacuum is precisely the coherent state |�〉. The quasiparticlesare then obtained from the resolution of the variational equation:

δ 〈�|H – λN |�〉 = 0 , (11.41)

where N is the operator number of particles and λ is a constraint parametercalled the chemical potential. Condition (11.41) leads, within the framework of theHartree–Fock–Bogoliubov approximation, to the system(

v Δ–Δ –v

) (U i

V i

)= E i

(U i

V i

), (11.42)

where v is the so-called potential, Δ is the pairing potential,

v i j = εi j – λδi j + Γi j Δi j =12

∑i ′ , j ′

V i ji ′ j ′ κi ′ j ′ ,

and Γ is the “deformation potential,”

Γi j =∑i ′ j ′

V ii ′ j j ′ ρi ′ j ′ .

In these definitions two matrices have been introduced. Their elements are lowersymbols with respect to the coherent states |�〉:

ρi j = 〈�|a†i a j |�〉 “matrix density,”

κi j = 〈�|ai a j |�〉 “pairing tensor.”

If the pairing is neglected, then the relevant dynamical group is U (r), � is a pointin the coset U (r)/(U ( p) ⊗ U (r – p)), and the above procedure is equivalent to theso-called mean-field Hartree–Fock theory.

Part Two Coherent State Quantization

193

12Standard Coherent State Quantization:the Klauder–Berezin Approach

12.1Introduction

The second part of this book is devoted to the quantization of sets, more preciselymeasure spaces, through coherent states. To introduce that other important aspectof these states, we explain in this chapter how standard coherent states allow a nat-ural quantization of the complex plane viewed as the phase space of the particlemotion on the line. We show how they offer a classical-like representation of theevolution of quantum observables. They also help to set Heisenberg inequalitiesconcerning “phase operator” and number operator for the oscillator Fock states.By restricting the formalism to the finite dimension, we present new quantuminequalities concerning the respective spectra of “position” and “momentum” ma-trices that result from such a coherent state quantization scheme for the motionon the line.

12.2The Berezin–Klauder Quantization of the Motion of a Particleon the Line

Let us go back to the quantum harmonic oscillator, and more generally the quan-tum version of the particle motion on the real line. Let us first recall some ofthe features of this basic model. On the classical level, the corresponding phasespace is X = R2 � C = {z = 1√

2(q + i p)} (in complex notation and with suitable

physical units). This phase space is equipped with the ordinary Lebesgue mea-sure on a plane that coincides with the symplectic 2-form: μ(dz dz) == 1

π d2z whered2z = dRz dIz. Strictly included in the Hilbert space L2(C, μ(dz dz)) of all com-plex-valued functions on the complex plane that are square-integrable with respectto this measure, there is the Fock–Bargmann Hilbert subspace FB of all square-

integrable functions that are of the form φ(z) = e– |z|2

2 g (z), where g (z) is analyticallyentire. As an orthonormal basis of this subspace we have chosen the normalizedpowers of the conjugate of the complex variable z weighted by the Gaussian, that


194 12 Standard Coherent State Quantization: the Klauder–Berezin Approach

is, φn(z) == e– |z|2

2 zn√

n!with n ∈ N. Standard coherent states stem from the general

construction described at length in Chapter 5:

|z〉 =1√N (z)

∑n

φn(z)|n〉 = e– |z|2

2

∑n∈N

zn√

n!|n〉 . (12.1)

For the normalized states (12.1), the resolution of the unity results:

1π

∫C

|z〉〈z| d2z = IH . (12.2)

The property (12.2) is crucial for our purpose in setting the bridge between theclassical and the quantum worlds. It encodes the quality of standard coherent statesof being canonical quantizers along a guideline established by Klauder [15, 154] andBerezin [20]. This Berezin–Klauder coherent state quantization, also named anti-Wickquantization or Toeplitz quantization [156]12) by many authors, consists in associat-ing with any classical observable f that is a function of phase space variables (q, p)or equivalently of (z, z), the operator-valued integral

1π

∫C

f (z, z) |z〉〈z| d2z = A f . (12.3)

The function f is usually supposed be be smooth, but we will not retain in thesequel this too restrictive attribute. The resulting operator A f , if it exists, at least ina weak sense, acts on the Hilbert space H of quantum states for which the set ofFock (or number) states |n〉 is an orthonormal basis. It is worthwhile being moreexplicit about what we mean by “weak sense”: the integral

〈ψ|A f |ψ〉 =∫

C

f (z, z)|〈ψ|z〉|2 d2zπ

(12.4)

should be finite for any |ψ〉 in some dense subset in H. One should note that if ψis normalized, then (12.4) represents the mean value of the function f with respectto the ψ-dependent probability distribution z �→ |〈ψ|z〉|2 on the phase space.

More mathematical rigor is necessary here, and we will adopt the following ac-ceptance criteria for a function (or distribution) to belong to the class of quantizableclassical observables.

Definition 12.1 A function C � z �→ f (z, z) ∈ C and more generally a distributionT ∈ D′(R2) is a CS quantizable classical observable along the map f �→ A f definedby (12.3), and more generally by T �→ AT :

• if the map C � z = 1√2(q + i p) == (q, p) �→ 〈z|A f |z〉 (or C � z �→ 〈z|AT |z〉) is

a smooth (i.e., ∈ C∞) function with respect to the (q, p) coordinates of thephase plane.

12) More generally, the Berezin–Toeplitzquantization is for symplectic manifolds thatadmit a Kähler structure.

12.2 The Berezin–Klauder Quantization of the Motion of a Particle on the Line 195

• and if we restore the dependence on � through z → z√�

, we must get theright semiclassical limit, which means that 〈 z√

�|A f | z√

�〉 W f ( z√

�, z√

�) as �→

0. The same asymptotic behavior must hold in a distributional sense if weare quantizing distributions.

The function f (or the distribution T) is an upper or contravariant symbol of the oper-ator A f (or AT ), and the mean value 〈z|A f |z〉 (or 〈z|AT |z〉) is the lower or covariantsymbol of the operator A f (or AT ). The map f �→ A f is linear and associates withthe function f (z, z) = 1 the identity operator in H. Note that the lower symbol ofthe operator A f is the Gaussian convolution of the function f (z, z):⟨

z√�

∣∣∣∣ A f

∣∣∣∣ z√�

⟩=

∫d2z′

π�e– |z–z′ |2

� f

(z′√�

,z′√�

). (12.5)

This expression is of great importance and is actually the reason behind the robust-ness of coherent state quantization, since it is well defined for a large class of non-smooth functions and even for a class of distributions comprising the temperedones. Equation (12.5) illustrates nicely the regularizing role of quantum mechan-ics compared with classical singularities. Note also that the Gaussian convolutionhelps us carry out the semiclassical limit, since the latter can be extracted by us-ing a saddle point approximation. For regular functions for which A f exists, theapplication of the saddle point approximation is trivial and we have⟨

z√�

∣∣∣∣ A f

∣∣∣∣ z√�

⟩W f

(z√�

,z√�

)as �→ 0 , (12.6)

as expected.At this point, we should mention the “reverse” of the coherent state quantization,that is, the coherent state dequantization. This problem consists in finding, or, atleast, proving the existence of, an upper symbol f A for an operator A in H alongthe equation

A =1π

∫C2

d2z f A(z, z) |z〉〈z| . (12.7)

This question was also examined by Sudarshan in his seminal paper [8] and bymany others, mainly within the context of quantum optics [6, 7, 157–159]. Our ap-proach should be viewed as conceptually different. It raises different mathematicalproblems because we are not subjected to the same constraints. More details aregiven in [160].

Expanding bras and kets in (12.3) in terms of the Fock states yields the expression

of the operator A f in terms of its infinite matrix elements (A f )nn′def= 〈n|A f |n′〉:

A f =∑

n,n′v0

(A f )nn′ |n〉〈n′| , (A f )nn′ =1√

n!n′!

∫C

d2zπ

e–|z|2 zn zn′ f (z, z) .

(12.8)


Alternatively, if the classical observable is “isotropic”, that is, f (z, z) == h(|z|2), thenA f is diagonal, with matrix elements given by a kind of gamma transform:

(A f )nn′ = δnn′1n!

∫ ∞

0du e–u un h(u) . (12.9)

In the case where the classical observable is purely angular-dependent, that is,f (z, z) = g (θ) for z = |z| eiθ, the matrix elements (A f )nn′ are given by

(A f )nn′ =Γ( n+n′

2 + 1)√

n!n′!cn′–n(g ) , (12.10)

where cm(g ) def= 12π

∫ 2π0 g (θ) e–imθ dθ is the Fourier coefficient of the 2π-periodic

function g.Let us explore what the coherent state quantization map (12.3) produces, starting

with some elementary functions f. We have for the most basic one

f (z) = z �→∫

C

z |z〉〈z| d2zπ

=∑

n

√n + 1|n〉〈n + 1| == a , (12.11)

which is the lowering operator, a|n〉 =√

n|n – 1〉. The adjoint a† is obtained byreplacing z by z in (12.11).

From q = 1√2(z + z) and p = 1√

2i(z – z), one easily infers by linearity that the

canonical position q and momentum p map to the quantum observables 1√2(a+a†) ==

Q and 1√2i

(a – a†) == P , respectively. In consequence, the self-adjoint operators Qand P obtained in this way obey the canonical commutation rule [Q , P ] = iIH, andfor this reason fully deserve the name of position and momentum operators ofthe usual (Galilean) quantum mechanics, together with all localization propertiesspecific to the latter, as they were listed at length in Chapter 1.

12.3Canonical Quantization Rules

At this point, it is worth recalling what quantization of classical mechanics means ina commonly accepted sense (for a recent and complete review, see [155]). In thiscontext, a classical observable f is supposed to be a smooth function with respectto the canonical variables. Here we reintroduce again the Planck constant since itparameterizes the link between classical and quantum mechanics.

12.3.1Van Hove Canonical Quantization Rules [161]

Given a phase space with canonical coordinates (q, p)

(i) to the classical observable f (q, p) = 1 corresponds the identity operator inthe (projective) Hilbert spaceH of quantum states,

12.4 More Upper and Lower Symbols: the Angle Operator 197

(ii) the correspondence that assigns to a classical observable f (q, p), supposedbe to infinitely differentiable, a (essentially) self-adjoint operator A f inH isa linear map,

(iii) to the classical Poisson bracket corresponds, at least at the order �, the quan-tum commutator, multiplied by i�:

with f j (q, p) �→ A f j for j = 1, 2, 3

we have { f 1, f 2} = f 3 �→ [A f 1 , A f 2 ] = i�A f 3 + o(�) ,

(iv) some conditions of minimality on the resulting observable algebra.

The last point can give rise to technical and interpretational difficulties.It is clear that points (i) and (ii) are fulfilled with the coherent state quantization,

the second one at least for observables obeying fairly mild conditions. To see betterthe “asymptotic” meaning of condition (iii), let us quantize higher-degree mono-mials, starting with |z|2 = p2+q2

2 , the classical harmonic oscillator Hamiltonian. Forthe latter, we get immediately from (12.9)

|z|2 �→ A|z|2 =∑nv0

(n + 1)|n〉〈n| = N + Id , (12.12)

where N = a†a is the number operator. Since N = P 2+Q2

2 – 12 Id , we see from this el-

ementary example that the coherent state quantization does not fit exactly with thecanonical one, which consists in just replacing q by Q and p by P in the expressionsof the observables f (q, p) and next proceeding to a symmetrization to comply withself-adjointness. In the present case, there is a shift by 1/2, more precisely �/2 af-ter restoring physical dimensions, of the quantum harmonic oscillator spectrum,that is, both quantizations differ by an amount on the order of �. Actually, it seemsthat no physical experiment can discriminate between those two spectra that differfrom each other by a simple shift; see [162] for a careful discussion of this point.

12.4More Upper and Lower Symbols: the Angle Operator

Since we do not retain in our quantization scheme the smoothness condition onthe classical observables, we feel free to quantize with standard coherent states an-other elementary classical object, namely, the argument θ ∈ [0, 2π) of the complexvariable z = r eiθ. Computing its quantum counterpart, say, Aarg from (12.3) isstraightforward and yields the infinite matrix:

Aarg = π Id + i∑n=/n′

Γ(

n+n′2 + 1

)√

n!n′!

1n′ – n

|n〉〈n′| . (12.13)


1 2 3 4 5 6

1

2

3

4

5

6

θ 0.0

0.5

1.0

r

0

2

4

6

θ

2

3

4

Fig. 12.1 Lower symbol of the angle operator for r = {0.5, 1, 5}and θ ∈ [0, 2π) and for (r , θ) ∈ [0, 1] ~ [0, 2π).

The corresponding lower symbol reads as the sine Fourier series:

〈z|Aarg|z〉 == 〈(r , θ)|Aarg|(r , θ)〉 = π + i e–|z|2∑n=/n′

Γ(

n+n′2 + 1

)n!n′!

zn′ zn

n′ – n

= π – 2∞∑

q=1

cq(r)q

sin qθ , (12.14)

where

cq(r) =2q

e–r2rq Γ( q

2 + 1)Γ(q + 1) 1F 1

( q2

+ 1; q + 1; r2)

.

One can easily prove that this symbol is infinitely differentiable in the variables rand θ [160].

The behavior of the lower symbol (12.14) is shown in Figure 12.1.It is interesting to evaluate the asymptotic behaviors of the function (12.14) at

small and large r, respectively. At small r, it oscillates around its average value πwith amplitude equal to

√πr :

〈(r , θ)|Aarg|(r , θ)〉 W π –√

πr sin θ .

At large r, we recover the Fourier series of the 2π-periodic angle function:

〈(r , θ)|Aarg|(r , θ)〉 W π – 2∞∑

q=1

1q

sin qθ = θ for θ ∈ [0, 2π) .

The latter result can be equally understood in terms of the classical limit of thesequantum objects. Indeed, by reinjecting into our formulas physical dimensions,we know that the quantity |z|2 = r2 acquires the dimension of an action and shouldappear in the formulas divided by the Planck constant �. Hence, the limit r →∞ in our previous expressions can also be considered as the classical limit � →

12.5 Quantization of Distributions: Dirac and Others 199

0. Since we have at our disposal the number operator N = a† a, which is, up toa constant shift, the quantization of the classical action, and an angle operator,we can examine their commutator and its lower symbol to measure the extent towhich we get something close to the expected canonical value, namely, i IH. Thecommutator reads as

[Aarg, N ] = i∑n=/n′

Γ(

n+n′2 + 1

)√

n!n′!|n〉〈n′| . (12.15)

Its lower symbol is then given by

〈(r , θ)|[Aarg, N ]|(r , θ)〉 = i∞∑

q=1

qcq(r) cos qθ == i C(r , θ) , (12.16)

with the same cq(r) as in (12.14).At small r, the function C(r , θ) oscillates around 0 with amplitude equal to

√πr :

C(r , θ) W√

πr cos θ .

At large r, the function C(r , θ) tends to the Fourier series 2∑∞

q=1 cos qθ, whose con-vergence has to be understood in the sense of distributions. Applying the Poissonsummation formula, we get at r → ∞ (or � → 0) the expected “canonical” be-havior for θ ∈ (0, 2π). More precisely, we obtain in this quasi-classical regime thefollowing asymptotic behavior:

〈(r , θ)|[Aarg, N ]|(r , θ)〉 W –i + 2πi∑n∈Z

δ(θ – 2πn) . (12.17)

The fact that this commutator is not exactly canonical was expected since we knowfrom Dirac [163] the impossibility of getting canonical commutation rules for thequantum versions of the classical action–angle pair. One can observe that the com-mutator symbol becomes canonical for θ =/ 2πn, n ∈ Z. Dirac singularities arelocated at the discontinuity points of the 2π-periodic extension of the linear func-tion f (θ) = θ for θ ∈ [0, 2π).

12.5Quantization of Distributions: Dirac and Others

It is commonly accepted that a “coherent state diagonal” representation of thetype (12.3) is possible only for a restricted class of operators in H. The reason isthat we usually impose too strong conditions on the upper symbol f (z, z) viewedas a classical observable on the phase space, and so subjected to belonging to thespace of infinitely differentiable functions on R2. We already noticed that a “rea-sonable” phase or angle operator is easily built starting from the classical discon-tinuous periodic angle function. The application of the same quantization method


to the free particle “time” q/p , which is obviously singular at p = 0, was carriedout in [160] and yields a fairly reasonable time operator for the free motion on theline.

Owing to the general expression (12.8) for matrix elements of the quantizedversion of an observable f, one can immediately think of extending the methodto tempered distributions. In particular, we will show that any simple operator

Πnn′def= |n〉〈n′| also has a coherent state diagonal representation by including dis-

tributions on R2 in the class of classical observables.In (12.8), the functions

(z, z) �→ e–|z|2 zn zn′ (12.18)

are rapidly decreasing C∞ functions on the plane with respect to the canonical co-ordinates (q, p), or equivalently with respect to the coordinates (z, z): they belongto the Schwartz space S(R2). The use of complex coordinates is clearly more adapt-ed to the present context, and we adopt the following definitions and notations fortempered distributions. Firstly, any function f (z, z) that is “slowly increasing” andlocally integrable with respect to the Lebesgue measure d2z on the plane definesa regular tempered distribution T f , that is, a continuous linear form on the vec-tor space S(R2) equipped with the usual topology of uniform convergence at eachorder of partial derivatives multiplied by polynomial of arbitrary degree [164]. Thisdefinition rests on the map

S(R2) � ψ �→ 〈T f , ψ〉 def=∫

C

d2z f (z, z) ψ(z, z) , (12.19)

and the notation is the same for all tempered distributions T. In view of (12.18),we can extend (12.19) to locally integrable functions f (z, z) that increase likeeη|z|2 p(z, z) for some η < 1 and some polynomial p, and to all distributions thatare derivatives (in the distributional sense) of such functions. We recall here thatpartial derivatives of distributions are given by⟨

∂r

∂zr

∂s

∂zs T , ψ⟩

= (–1)r+s

⟨T ,

∂r

∂zr

∂s

∂zs ψ⟩

. (12.20)

We also recall that the multiplication of distributions T by smooth functionsα(z, z) ∈ C∞(R2) is understood through

S(R2) � ψ �→ 〈αT , ψ〉 def= 〈T , α ψ〉 . (12.21)

Now, equipped with the above distributional material, we consider as “accept-able” observables those distributions in D′(R2) that obey the following condition.

Definition 12.2 (Coherent state quantizable observable) A distribution T ∈ D′(R2)is a coherent state quantizable classical observable if there exists η < 1 such that theproduct e–η|z|2 T ∈ S′(R2), that is, is a tempered distribution.

12.5 Quantization of Distributions: Dirac and Others 201

Of course, all compactly supported distributions such as the Dirac distribution andits derivatives are tempered and so are coherent state quantizable classical observable.The Dirac distribution supported by the origin of the complex plane is denoted asusual by δ (and abusively in the present context by δ(z, z)):

C∞(R2) � ψ �→ 〈δ, ψ〉 ==

∫C

d2z δ(z, z) ψ(z, z) def= ψ(0, 0) . (12.22)

Let us now coherent state quantize the Dirac distribution using the recipe pro-vided by (12.3) and (12.8):

1π

∫C

δ(z, z) |z〉〈z| d2z =∑

n,n′v0

1√n!n′!

∫C

d2zπ

e–|z|2 zn zn′ δ(z, z) |n〉〈n′|

=1π|z = 0〉〈z = 0| =

1π

Π00 .

(12.23)

We thus find that the ground state (as a projector) is the quantized version of theDirac distribution supported by the origin of the phase space. The obtaining ofall possible diagonal projectors Πnn = |n〉〈n| or even all possible operators Πnn′ =|n〉〈n′| is based on the quantization of partial derivatives of the δ distribution. First,let us quantize the various derivatives of the Dirac distribution:

U a,b =∫

C

[∂b

∂zb

∂a

∂za δ(z, z)

]|z〉〈z| d2z

=∑

n,n′v0

(–1)n+a b ! a!(b – n)!

1√n!n′!

δn–b ,n′–a Πnn′ . (12.24)

Once this quantity U a,b is at hand, one can invert the formula to get the operatorΠr+s,r = |r + s〉〈r | as

Πr+s,r =√

r !(r + s)!(–1)sr∑

p=0

1p!(s + p)!(r – p)!

U p, p+s , (12.25)

and so its upper symbol is given by the distribution supported by the origin:

f r+s,r (z, z) =√

r !(r + s)!(–1)sr∑

p=0

1p!(s + p)!(r – p)!

[∂ p+s

∂z p+s

∂ p

∂z p δ(z, z)

].

(12.26)

Note that this distribution, as is well known, can be approached, in the sense of thetopology on D′(R2), by smooth functions, such as linear combinations of deriva-tives of Gaussians. The projectors Πr ,r are then obtained trivially by setting s = 0 in(12.25) to get

Πr ,r =r∑

p=0

1p!

(rp

)U p, p . (12.27)


This “phase space formulation” of quantum mechanics enables us to mimic atthe level of functions and distributions the algebraic manipulations on operatorswithin the quantum context. By carrying out the coherent state quantization ofCartesian powers of planes, we could obtain an interesting “functional portrait” interms of a “star” product on distributions for the quantum logic based on manipu-lations of tensor products of quantum states.

12.6Finite-Dimensional Canonical Case

The idea of exploring various aspects of quantum mechanics by restricting theHilbertian framework to finite-dimensional space is not new, and has been inten-sively used in the last decade, mainly in the context of quantum optics [165–167],but also in the perspective of noncommutative geometry and “fuzzy” geometricalobjects [168], or in matrix model approaches in problems such as the quantumHall effect [169]. For quantum optics, a comprehensive review (mainly devoted tothe Wigner function) is provided in [170]. In [166, 167], the authors defined nor-malized finite-dimensional coherent states by truncating the Fock expansion of thestandard coherent states. Let us see through the approach presented in this chap-ter how we recover their coherent states [171]. We just restrict the choice of theorthonormal set {φn} to a finite subset of it, more precisely to the first N elements:

φn(z) = e–|z|22 zn√

n!, n = 0, 1, . . . N – 1 <∞ . (12.28)

The coherent states then read

|z〉 =e– |z|

2

2√N (|z|2)

N –1∑n=0

zn√

n!|n〉 , (12.29)

with

N (|z|2) = e–|z|2N –1∑n=0

|z|2n

n!. (12.30)

The quantization procedure (12.3) yields for the classical observables q and p theposition operator, QN , and the momentum operator, P N , or phase quadratures inquantum optics, acting in N-dimensional Hermitian space. They read as the N ~ Nmatrices

QN =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1√2

0 . . . 01√2

0 1 . . . 0

0 1. . .

. . ....

... . . .. . . 0

√N –1

2

0 0 . . .

√N –1

2 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠, (12.31)

12.6 Finite-Dimensional Canonical Case 203

P N = –i

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1√2

0 . . . 0– 1√

20 1 . . . 0

0 –1. . .

. . ....

... . . .. . . 0

√N –1

2

0 0 . . . –√

N –12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠. (12.32)

Their commutator is “almost” canonical:

[QN , P N ] = iIN – iN E N , (12.33)

where E N is the orthogonal projector on the last basis element:

E N =

⎛⎜⎝ 0 . . . 0...

. . ....

0 . . . 1

⎞⎟⎠ .

The presence of such a projector in (12.33) is clearly a consequence of the trunca-tion at the Nth level.

A very interesting finding comes from this finite-dimensional quantization of theclassical phase space. Let us examine the spectral values of the position operator,that is, the allowed or experimentally measurable quantum positions. They are justthe zeros of the Hermite polynomials HN (λ), and the same result holds for themomentum operator. We know that HN (0) = 0 if and only if N is odd and thatthe other zeros form a set symmetric with respect to the origin. Let us order thenonnull roots of the Hermite polynomial HN (λ) as{

–λ� N2 �(N ), –λ� N

2 �–1(N ), . . . , –λ1(N ), λ1(N ), . . . , λ� N2 �–1(N ), λ� N

2 �(N )}

.

(12.34)

It is a well-known property of the Hermite polynomials that λi+1(N )–λi (N ) > λ1(N )for all i v 1 if N is odd, whereas λi+1(N ) – λi (N ) > 2λ1(N ) for all i v 1 if N is even,and that the zeros of the Hermite polynomials HN and HN +1 intertwine.

Let us denote by λM (N ) = λ� N2 �(N ) (or λm (N ) = λ1(N )) the largest root (or the

smallest nonzero root in absolute value) of HN . A numerical study of the product

πN = λm (N )λM (N ) (12.35)

was carried out in [171]. It was found that πN goes asymptotically to π for largeeven N and to 2π for large odd N.

This result (which can be rigorously proved by using the Wigner semicircle lawfor the asymptotic distribution of zeros of the Hermite polynomials [172, 173]) is


Table 12.1 Values of σN = δN (Q)ΔN (Q) up to N = 106 . Compare with the value of 2π.

N δN(Q)ΔN(Q) 2π

10 4.713 054

55 5.774 856100 5.941 534

551 6.173 778

1000 6.209 6705555 6.259 760

10 000 6.267 356

55 255 6.278 122100 000 6.279 776

500 555 6.282 0201 000 000 6.282 450 6.283 185 3

important with regard to its physical implications in terms of correlation betweensmall and large distances. Define by

• ΔN (Q ) = 2λM (N ) the “size” of the “universe” accessible to exploration bythe quantum system

• δN (Q ) = λm (N ) (resp. δN (Q ) = 2λm (N )) for odd (or even) N, the “size” ofthe smallest “cell” for which exploration is forbidden by the same system,

and introduce the product

δN (Q )ΔN (Q ) == σN =

{4λm (N )λM (N ) for N even,

2λm (N )λM (N ) for N odd,, (12.36)

then σN , as a function of N, is strictly increasing and goes asymptotically to 2π(Table 12.1).

Hence, we can assert the new inequalities concerning the quantum position andmomentum:

δN (Q )ΔN (Q ) u 2π , δN (P )ΔN (P ) u 2π ∀N . (12.37)

In order to fully perceive the physical meaning of such inequalities, it is neces-sary to reintegrate into them physical constants or scales appropriate for the physi-cal system considered, that is, characteristic length l c and momentum pc :

δN (Q )ΔN (Q ) u 2πl2c , δN (P )ΔN (P ) u 2π p2

c ∀N , (12.38)

where δN (Q ) and ΔN (Q ) are now expressed in unit l c .Realistically, in any physical situation N cannot be infinite: there is an obvious

limitation on the frequencies or energies accessible to observation/experimenta-tion. So it is natural to work with a finite although large value of N, which need not

12.6 Finite-Dimensional Canonical Case 205

be determinate. In consequence, there exist irreducible limitations, namely, δN (Q )and ΔN (Q ), in the exploration of small and large distances, and both limitationshave the correlation δN (Q )ΔN (Q ) u 2πl2

c .Suppose that there exists, for theoretical reasons, a fundamental or “universal”

minimal length, say, lm , which could be the Planck length lPl =√

�G/c3 W 10–35 mor something else, depending on the experimental or observational context, or,equivalently, a universal ratio ρu = l c/lm v 1. Then, from δN (Q ) v lm , we inferthat there exists a universal maximal length lM given by

lM W 2πρulc . (12.39)

Of course, if we choose lm = l c , then the size of the “universe” is lM W 2πlm ,a fact that leaves no room for the observer and observed things! Now, if we choosea characteristic length appropriate for atomic physics, such as the Bohr radius, l c W10–10 m, and for the minimal length the Planck length, lm = lPl W 10–35 m, we findfor the maximal size the astronomical quantity lM W 1016 m W 1 light year, which isalso of the order of one parsec. On the other hand, if we consider the (controversial)estimated size of our present universe Lu = cT u , with T u W 13 ~ 109 years, we getfrom l p Lu W 2πl2

c a characteristic length l c W 10–5 m, that is, a wavelength in theinfrared region of the electromagnetic spectrum.

Another interesting outcome of the monotonic increase of the product

lm lM<→

N→∞2πl2

c

is that the reasoning leading to (12.39) can be reversed. Suppose that there existsan absolute confinement, of size lM , for the system considered. Then, at large N,there exists as well an impassable “core” of size

lm W 2πl2c

lM. (12.40)

Then, the allowed range of values of the characteristic length l c is lm u l c u lM/√

πsince, for the upper limit l c = lM/

√π, we have lm W lM , whereas at the lowest limit

we recover lM W 2πlm .Let us turn to another example, which might be viewed as more concrete, namely,

the quantum Hall effect in its matrix model version [169]. The planar coordinatesX 1 and X 2 of quantum particles in the lowest Landau level of a constant magneticfield do not commute:

[X 1, X 2] = iθ , (12.41)

where θ represents a minimal area. We recall that the average density of N → ∞electrons is related to θ by ρo = 1/2πθ and the filling fraction is ν = 2πρ0/B . Thequantity lm =

√θ can be considered as a minimal length. The model deals with

a finite number N of electrons:

[X 1,N , X 2,N ] = iθ(1 – N |N – 1〉〈N – 1|) . (12.42)


In this context, our inequalities read as

δN (X i )ΔN (X i ) u 2πl2c , i = 1, 2 , (12.43)

where l c corresponds to a choice of experimental unit. Since lm =√

θ affords anirreducible lower limit in this problem, we can assert that the maximal linear sizeLM of the sample should satisfy

lM u 2πl c√θ

l c (12.44)

for any finite N.The experimental interpretation of such a result certainly deserves a deeper in-

vestigation.As a final comment concerning the inequalities (12.37), we would like to insist on

the fact they are not just an outcome of finite approximations QN and P N (or X 1,N

and X 2,N ) to the canonical position and momentum operators (or to X 1 and X 2)in infinite-dimensional Hilbert space of quantum states. They hold however largethe dimension N is, as long as it is finite. Furthermore, let us advocate the idea thata quantization of the classical phase space results from the choice of a specific (re-producing) Hilbert subspace H in L2(C, μ(dz dz)) in which coherent states providea frame resolving the identity. This frame corresponds to a certain point of view indealing with the classical phase space, and this point of view yields the quantumversions QN and P N (or X 1,N and X 2,N ) of the classical coordinates q and p (or x1

and x2).

207

13Coherent State or Frame Quantization

13.1Introduction

Physics is part of the natural sciences and its prime object is what we call “na-ture”, or rather, in a more restrictive sense “time”, “space”, “matter”, “energy”, and“interaction”, which appear at a certain moment of the process in the form of “sig-nificant” data, such as position, speed, and frequencies. So the question arises howto process those data, and this raises the question of a selected point of view orframe. Faced with a set of “raw” collected data encoded into a certain mathemati-cal form and provided by a measure, that is, a function which attributes a weightof importance to subsets of data, we give in addition more or less importance todifferent aspects of those data by choosing, opportunistically, the most appropriateframe of analysis.

We include in this general scheme the quantum processing, that is, the way ofconsidering objects from a quantum point of view, exactly like we quantize theclassical phase space in quantum mechanics. To a certain extent, quantization per-tains to a larger discipline than just restricting ourselves to specific domains ofphysics such as mechanics or field theory. The aim of this chapter is to providea generalization of the Berezin–Klauder–Toeplitz quantization illustrated in Chap-ter 12, precisely the quantization of a measure space X once we are given a familyof coherent states or frame constructed along the lines given in Chapter 5. We alsodevelop the notion of lower and upper symbols resulting from such a quantizationscheme, and finally discuss the probabilistic content of the construction. A quiteelementary example, the quantization of the circle with 2 ~ 2 matrices, is presentedas an immediate illustration of the formalism.

13.2Some Ideas on Quantization

Many reviews exist on the quantization problem and the variety of approaches forsolving it; see, for instance, the recent ones by Ali and Englis [155] and by Lands-man [174]. For simplification, let us say that a quantization is a procedure that


208 13 Coherent State or Frame Quantization

associates with an algebra Acl of classical observables an algebra Aq of quantumobservables. The algebra Acl is usually realized as a commutative Poisson alge-bra13) of derivable functions on a symplectic (or phase) space X. The algebra Aq is,however, noncommutative in general and the quantization procedure must providea correspondence Acl �→ Aq : f �→ A f . Various procedures of quantization exist,and minimally require the following conditions, which loosely parallel those listedin Section 12.3:

• With the constant function 1 is associated the unity of Aq,

• The commutation relations of Aq reproduce the Poisson relations of Acl.Moreover, they offer a realization of the Heisenberg algebra.

• Aq is realized as an algebra of operators acting in some Hilbert space.

Most physical quantum theories may be obtained as the result of a canonical quan-tization procedure. However, the prescriptions for the latter appear quite arbitrary.Moreover, it is difficult, if not impossible, to implement it covariantly. It is thus dif-ficult to generalize this procedure to many systems. Geometrical quantization [175]exploits fully the symplectic structure of the phase space, but generally requiresmore structure, such as a symplectic potential, for example, the Legendre formon the cotangent bundle of a configuration space. In this regard, the deformationquantization appears more general in the sense that it is based on the symplecticstructure only and it preserves symmetries (symplectomorphisms) [176, 177].

The coherent state quantization that is presented here is by far more universalsince it does not even require a symplectic or Poisson structure. The only structurethat a space X must possess is a measure. This procedure can be considered fromdifferent viewpoints:

• It is mostly genuine in the sense that it verifies all the requirements above,including those relative to the Poisson structure when the latter is present.

• It may also be viewed as a “fuzzyfication” of X: an algebra Acl of functionson X is replaced by an algebra Aq of operators, which may be consideredas the “coordinates” of a fuzzy version of X, even though the original X isnot equipped with a manifold structure. The term “fuzzy” manifold, wherepoints have noncommutative coordinates, was proposed by Madore [178] inhis presentation of noncommutative geometry of simple manifolds such asthe sphere. As a matter of fact, ordinary quantum mechanics may also beviewed as a noncommutative version of the geometry of the phase space,where position and momentum operators do not commute [179]. In thisregard, the quantization of a “set of data” makes a fuzzy or noncommutativegeometry emerge.

13) A Poisson algebra A is an associative algebratogether with a Lie bracket X , Y �→ {X , Y }that also satisfies the Leibniz law: for any

X ∈ A the action DX (Y ) def= {X , Y } on A is aderivation {X , Y Z} = {X , Y }Z + Y {X , Z}.

As a matter of fact, the space of real-valuedsmooth functions over a symplectic manifold,when equipped with the Poisson bracket,forms a Poisson algebra.

13.3 One more Coherent State Construction 209

• Finally, this procedure is, to a certain extent, a change of point of view in con-sidering the “system” X, not necessarily a path to quantum physics. In thissense, it could be called a discretization or a regularization [180]. It showsa similitude with standard procedures pertaining to signal processing, forinstance, those involving wavelets, which are coherent states for the affinegroup transforming the time-scale half-plane into itself. In many respects,the choice of a quantization appears as the choice of a resolution in lookingat the system.

13.3One more Coherent State Construction

In Section 5.3 we drew a parallel between quantum mechanics and signal analysis:the object considered is an observation set X of data or parameters equipped witha measure μ defined on a σ-algebra F . For both theories, the natural framework ofinvestigation is the Hilbert space L2

K(X , μ) (K = R or C) of square-integrable real orcomplex functions f (x) on the observation set X:

∫X | f (x)|2 μ(dx) < ∞. We then

observed that “quantum processing” of X differs from signal processing since, inparticular, not all square-integrable functions are eligible as quantum states. Wemeet, precisely at this point, the quantization problem: how do we select quantumstates among simple signals? In other words, how do we select the true (projec-tive) Hilbert space of quantum states, denoted by K, that is, a closed subspace ofL2

K(X , μ), or equivalently the corresponding orthogonal projector IK?The only requirements onK, in addition to being a Hilbert space, amount to the

following technical conditions:

Condition 13.1 (Quantum states)

(i) For all φ ∈ K and all x, φ(x) is well defined (this is, of course, the case wheneverX is a topological space and the elements of K are continuous functions).

(ii) The linear map (“evaluation map”)

δx : K → K

φ �→ φ(x)(13.1)

is continuous with respect to the topology of K, for almost all x.

The last condition is realized as soon as the space K is finite-dimensional sinceall the linear forms are continuous in this case. We see below that some otherexamples can be found.


As a consequence, using the Riesz theorem14), there exists, for almost all x, a uniqueelement px ∈ K (a function) such that

〈px |φ〉 = φ(x) . (13.2)

We define the coherent states as the normalized vectors corresponding to px , writtenin Dirac notation as

|x〉 ==|px 〉

[N (x)]12

where N (x) == 〈px |px 〉 . (13.3)

One can see at once that for any φ ∈ K

φ(x) =[N (x)

] 12 〈x |φ〉. (13.4)

One obtains the following resolution of the identity of K that is at the basis of thewhole construction:

IK =∫

Xμ(dx)N (x) |x〉〈x | . (13.5)

This equation is a direct consequence of the following equalities:

〈φ1|∫

Xμ(dx)N (x) |x〉〈x ||φ2〉 =

∫X

μ(dx)N (x)〈φ1|x〉〈x |φ2〉

=∫

Xμ(dx) φ1(x)φ2(x) = 〈φ1|φ2〉 ,

which hold for any φ1, φ2 ∈ K.Note that

φ(x) =∫

Xμ(dx ′)

√N (x)N (x ′)〈x |x ′〉φ(x ′) , ∀φ ∈ K . (13.6)

Hence, K is a reproducing Hilbert space with kernel

K (x , x ′) =√N (x)N (x ′)〈x |x ′〉, (13.7)

and the latter assumes finite diagonal values (almost everywhere), K (x , x) = N (x),by construction. Note that this construction yields an embedding of X into K andone could interpret |x〉 as a state localized at x once a notion of localization hasbeen properly defined on X.

In view of (13.5), the set {|x〉} is called a frame for K. This frame is said to beovercomplete when the vectors {|x〉} are not linearly independent.

14) In a Hilbert spaceH with inner product 〈v1|v2〉,to any continuous linear form ϕ : H �→ K therecorresponds one and only one vector vϕ such thatϕ(v ) = 〈vϕ |v〉 for all v ∈ H.

13.4 Coherent State Quantization 211

The technical conditions and the definition of coherent states can be easily ex-pressed when we have an orthonormal basis of K. Let {φn , n ∈ I} be such a basis;the technical condition is equivalent to∑

n

|φn(x)|2 <∞ a.e. (13.8)

The coherent state is then defined by

|x〉 =1(

N (x)) 1

2

∑n

φn(x)φn with N (x) =∑

n

|φn(x)|2 .

We thus recover the original construction of coherent states described in Chap-ter 5. We have just adopted here a more “functional” approach. We also rememberthat the Hilbert space of “quantum states” can be chosen a priori, independentlyof the Hilbert space L2

K(X , μ) of “signals.” Given a separable Hilbert space H andan orthonormal basis (|en〉)n∈N in one-to-one correspondence with the φn ’s, onedefines in H the family of coherent states

|x〉 ==1√N (x)

∑n

φn (x)|en〉 . (13.9)

Consistently,

φn(x) ==√N (x)〈x |en〉 . (13.10)

These states resolve the unity inH∫X|x〉〈x | ν(dx) = IH , (13.11)

where ν(dx) = N (x) μ(dx).

13.4Coherent State Quantization

A classical observable is a function f (x) on X having specific properties with respectto some supplementary structure allocated to X, such as topology, geometry, orsomething else. It could be a distribution if the topological structure assigned to Xallows the existence of such an object. More precisely, within the framework of ourapproach, the actual definition of classical observable will depend on the coherentstate quantization scheme. Inspired by Chapter 12, we define the quantization ofa classical observable f (x) with respect to the family of states {|x〉, x ∈ X } as thelinear map associating with the function f (x) the operator inH given by

f �→ A f =∫

Xf (x)|x〉〈x | ν(dx) . (13.12)


Note that the function f (x) = 1 goes to the identity operator IH. Borrowing fromBerezin and Lieb their terminology, one names the function f (x) == A f (x) thecontravariant (Berezin) or the upper symbol (Lieb) of the operator A f , whereas onenames the mean value 〈x |A f |x〉 == A f (x) the covariant or the lower symbol of A f .

The terminology “lower” and “upper” comes from the following inequalities in-volving symbols of self-adjoint operators. Let g (·) be a real-valued convex function

g

(∑i

ai αi

)u

∑i

ai g (αi ) for all set {ai v 0} with∑

i

ai = 1

(13.13)

and let A be a self-adjoint operator on H that has well-defined lower and uppersymbols with respect to the coherent state frame {|x〉}x∈X . Then the Berezin–Liebinequalities∫

Xg (A(x)) ν(dx) u Tr g (A) u

∫X

g (A(x)) ν(dx) (13.14)

hold depending on suitable properties of A. A particular case of these inequalitieswas proved and applied to quantum spin systems in Chapter 7.

Now, in agreement with Chapter 12, we should make more precise, although stillloose, our definition of a classical observable in terms of lower symbols.

Definition 13.1 Given a frame {|x〉, x ∈ X }, a function X � x �→ f (x) ∈ K,possibly understood in a distributional sense, is a coherent state quantizable classicalobservable along the map f �→ A f defined by (13.12)

(i) if the map X � x �→ 〈x |A f |x〉 is a regular function with respect to someadditional structure allocated to X.

(ii) we must get the right classical limit, which means that 〈x |A f |x〉 W f (x) asa certain parameter goes to 0.

One can say that, according to this approach, a quantization of the observationset X is in one-to-one correspondence with the choice of a frame in the senseof (13.11). This term of frame [61] is more appropriate for designating the total fam-ily {|x〉}x∈X . To a certain extent, the coherent state quantization scheme consistsin adopting a certain point of view in dealing with X. This frame can be discrete orcontinuous, depending on the topology additionally allocated to the set X, and it canbe overcomplete, of course. The validity of a precise frame choice is determined bycomparing spectral characteristics of quantum observables A f with experimentaldata. Of course, a quantization scheme associated with a specific frame is intrin-sically limited to all those classical observables for which the expansion (13.12) ismathematically justified within the theory of operators in Hilbert space (e.g., weakconvergence). However, it is well known that limitations hold for any quantizationscheme.

13.4 Coherent State Quantization 213

The advantage of the coherent state quantization illustrated here is that it requiresa minimal significant structure on X, namely, the existence of a measure μ(dx),together with a σ-algebra of measurable subsets, and some additional structure tobe defined according to the context. The construction of the Hilbert space H (orK) is equivalent to the choice of a class of eligible quantum states, together witha technical condition of continuity. A correspondence between classical and quan-tum observables is then provided through a suitable generalization of the standardcoherent states.

An interesting question to be addressed concerns the emergence or not of non-commutativity through the quantum processing of the set X. Suppose the “large”signal space L2

K(X , μ) has an orthonormal basis {φn (x), n ∈ N} that satisfies therequired condition

N (x) ==∑

n

|φn(x)|2 <∞ a.e.

Then the coherent state quantization with the corresponding coherent states

|x〉 ==1√N (x)

∑n

φn (x)|φn〉

provides a commutative algebra of operators! Indeed, in this case, to f (x) therecorresponds the multiplication operator

A f φ(x) = f (x)φ(x) a.e.

This means that we reach through that coherent state quantization only this class ofoperators, and other operators cannot be expressed in such a “diagonal form” withrespect to the coherent state family. The essence of the noncommutative “quan-tum” reading of the observation set X lies in a strict inclusion of the space K ofquantum states into the space L2

K(X , μ) of “signals.”Another possible approach to quantization would consist first in proceeding to

the most demanding sampling, namely, in choosing the “continuous orthogonal”basis of L2

K(X , μ) made up of the “Dirac functions” {δy (x) , y ∈ X }, operationallydefined on a suitable dense subspace V of functions in L2

K(X , μ) by∫X

μ(dx) δy (x) v (x) == 〈δy |v〉 = v (y ) ∀ v ∈ V .

Here too we have a resolution of the identity at least on V :∫X

μ(dx) |δx 〉〈δx | = Id .

Then a quantization with this Dirac basis would also provide a commutative al-gebra of operators. Indeed, in this case, to a function f (x) there corresponds themultiplication operator

A f φ(x) = f (x)φ(x) a.e.


13.5A Quantization of the Circle by 2 ~ 2 Real Matrices

In Section 3.2.3, we illustrated the notion of resolution of the unity by consideringthe set of unit vectors as a continuous frame for the Euclidean plane:

1π

∫ 2π

0dθ |θ〉〈θ| = Id . (13.15)

Let us now employ this set of “coherent states” to describe the corresponding quan-tization along the lines illustrated in the previous section.

13.5.1Quantization and Symbol Calculus

The existence of the set {|θ〉} offers the possibility of proceeding to the followingquantization of the circle that associates with a function f (θ) the linear operator A f

in the plane:

f �→ A f =1π

∫ 2π

0dθ f (θ)|θ〉〈θ| . (13.16)

For instance, let us choose the angle function f (θ) = θ. Its quantized version isequal to the matrix

Aθ =

(π – 1

2– 1

2 π

), (13.17)

with eigenvalues π ± 12 .

The frame {|θ〉} allows one to carry out a symbol calculus à la Berezin and Lieb.With any self-adjoint linear operator A, that is, a real symmetric matrix with respectto some orthonormal basis, one associates the two types of symbol functions A(θ)and A(θ), respectively, defined on the unit circle by

A(θ) = 〈θ|A|θ〉 : lower or covariant symbol (13.18)

and

A =1π

∫ 2π

0dθ A(θ)|θ〉〈θ| . (13.19)

The upper or contravariant symbol A(θ) that appears in this operator-valued inte-gral is highly nonunique, but will be chosen as the simplest one.

Three basic matrices generate the Jordan algebra15) of all real symmetric 2 ~2 matrices. They are the identity matrix Id , the symbol of which is trivially the

15) A Jordan algebra is an algebra (not necessarilyassociative) over a field whose multiplicationis commutative and satisfies the Jordanidentity: (x y )x2 = x(yx2).

13.5 A Quantization of the Circle by 2 ~ 2 Real Matrices 215

function 1, and the two real Pauli matrices,

σ1 =

(0 11 0

), σ3 =

(1 00 –1

). (13.20)

Any element A of the algebra decomposes as

A ==

(a bb d

)=

a + d2

Id +a – d

2σ3 + bσ1 == αId + δσ3 + �σ1 . (13.21)

The product in this algebra is defined by

O′′ = A $ A′ =12

(AA′ + A′A

), (13.22)

which entails on the level of components α, δ, � the relations:

α′′ = αα′ + δδ′ + ��′, δ′′ = αδ′ + α′δ, �′′ = α�′ + α′� . (13.23)

The simplest upper symbols and the lower symbols of nontrivial basic elementsare, respectively, given by

cos 2θ = σ3(θ) =12

σ3(θ), sin 2θ = σ1(θ) =12

σ1(θ) . (13.24)

There follows for the symmetric matrix (13.21) the two symbols

A(θ) =a + d

2+

a – d2

cos 2θ + b sin 2θ = α + δ cos 2θ + � sin 2θ , (13.25)

A(θ) = α + 2δ cos 2θ + 2� sin 2θ = 2A(θ) – 12 Tr A . (13.26)

For instance, the lower symbol of the “quantum angle” (13.17) is equal to

〈θ|Aθ|θ〉 = π – 12 sin 2θ , (13.27)

a π-periodic function that smoothly varies between the two eigenvalues of Aθ, asshown in Figure 13.1.

One should notice that all these symbols belong to the subspace VA of real Fouri-er series that is the closure of the linear span of the three functions 1, cos 2θ, andsin 2θ. Also note that A(θ) is defined up to the addition of any function N(θ) thatmakes (13.19) vanish. Such a function lives in the orthogonal complement of VA .

The Jordan multiplication law (13.22) is commutative but not associative and itscounterpart on the level of symbols is the so-called �-product. For instance, we havefor the upper symbols

A $ A′(θ) == A(θ) � A′(θ) = αA′(θ) + α′A(θ) + δδ′ + ��′ – αα′ , (13.28)

and the formula for lower symbols is the same.We recall that the terminology of lower/upper is justified by two Berezin–Lieb

inequalities, which follow from the symbol formalism. Let us make explicit this


0 1 2 3 4 5 6 72.6

2.8

3

3.2

3.4

3.6

3.8

4Quantum angle in the plane

θ

<θ

|Aθ| θ

>

Fig. 13.1 The lower symbol of the angle operator in the two-di-mensional quantization of the circle is the π-periodic functionπ – 1

2 sin 2θ that varies between the two eigenvalues of Aθ .

inequality in the present context. Let g be a convex function. Denoting by λ± theeigenvalues of the symmetric matrix A, we have

1π

∫ 2π

0g (A(θ)) dθ u Tr g (A) = g (λ+) + g (λ–) u

1π

∫ 2π

0g (A(θ)) dθ . (13.29)

This double inequality is not trivial. Independently of the Euclidean context it readsas

〈g (t + r cos θ)〉 u 12

[g (t + r) + g (t – r)

]u 〈g (t + 2r cos θ)〉 , (13.30)

where t ∈ R, r v 0 and 〈 · 〉 denotes the mean value on a period. If we apply (13.30)to the exponential function g (t) = et , we get an intertwining of inequalities involv-ing Bessel functions of the second kind and the hyperbolic cosine:

· · · u I0(x) u cosh x u I0(2x) u cosh 2x u · · · ∀x ∈ R . (13.31)

13.5.2Probabilistic Aspects

Behind the resolution of the identity (13.15) lies an interesting interpretation interms of geometrical probability. Let us consider a Borel subset Δ of the interval

13.5 A Quantization of the Circle by 2 ~ 2 Real Matrices 217

[0, 2π) and the restriction to Δ of the integral (13.15):

a(Δ) =1π

∫Δ

dθ|θ〉〈θ| . (13.32)

One easily verifies the following properties:

a(∅) = 0 , a([0, 2π)) = Id ,

a(∪i∈ J Δi ) =∑i∈ J

a(Δi ) , if Δi ∩ Δ j = ∅ for all i �= j . (13.33)

The application Δ �→ a(Δ) defines a normalized measure on the σ-algebra of theBorel sets in the interval [0, 2π), assuming its values in the set of positive linearoperators on the Euclidean plane. Denoting the measure density (1/π)|θ〉〈θ| dθ bya(dθ), we shall also write

a(Δ) =∫

Δa(dθ) . (13.34)

Let us now put into evidence the probabilistic nature of the measure a(Δ). Let|φ〉 be a unit vector. The application

Δ �→ 〈φ|a(Δ)|φ〉 =1π

∫Δ

cos2(θ – φ) dθ (13.35)

is clearly a probability measure. It is positive, of total mass 1, and it inherits σ-addi-tivity from a(Δ). Now, the quantity 〈φ|a(Δ)|φ〉means that direction |φ〉 is examinedfrom the point of view of the family of vectors {|θ〉, θ ∈ Δ}. As a matter of fact, ithas a geometrical probability interpretation in the plane [181]. With no loss of gener-ality, let us choose φ = 0. Recall here the canonical equation describing a straightline D θ, p in the plane:

〈θ|u〉 == cos θ x + sin θ y = p , (13.36)

where |θ 〉 is the direction normal to D θ, p and the parameter p is equal to the dis-tance of D θ, p to the origin. It follows that dp dθ is the (nonnormalized) probabilitymeasure element on the set

{D θ, p

}of the lines randomly chosen in the plane.

Picking a certain θ, we consider the set {D θ, p} of the lines normal to |θ〉 that in-tersect the segment with origin O and length | cos θ| equal to the projection of |θ〉onto |0〉 as shown in Figure 13.2

The measure of this set is equal to(∫ cos2 θ

0d p

)dθ = cos2 θ dθ . (13.37)

Integrating (13.37) over all directions |θ〉 gives the area of the unit circle. Hence, wecan construe 〈φ|a(Δ)|φ〉 as the probability of a straight line in the plane belongingto the set of secants of segments that are projections 〈φ|θ〉 of the unit vectors |θ〉,θ ∈ Δ onto the unit vector |φ〉. One could think in terms of polarizer 〈θ| andanalyzer |θ〉 “sandwiching” the directional signal |φ〉.


O

|0〉

|θ〉

��

�

| {z }

cos θ

Fig. 13.2 Set {Dθ, p} of straight lines normal to |θ〉 that inter-sect the segment with origin O and length | cos θ | equal to theprojection of |θ〉 onto |0〉.

13.5.2.1Remark: A Two-Dimensional Quantization of the Interval [0, π) through a ContinuousFrame for the Half-PlaneFrom a strictly quantal point of view, we should have made equivalent any vector|v〉 of the Euclidean plane with its symmetric –|v〉, which amounts to dealing withthe half-plane viewed as the coset R2/Z2. Following the same procedure as above,we start from the Hilbert space L2

([0, π), 2

π dθ)

and we choose the same subset{cos θ, sin θ}, which is still orthonormal. Note that 2

π cos2 θ and 2π sin2 θ are now

exactly Wigner semicircle distributions. We then consider the continuous family(3.8) of coherent states |θ〉. They are normalized and they solve the identity exactlylike in (3.10) (just change the factor 1

π into 2π ). The previous material can be repeat-

ed in extenso but with the restriction that θ ∈ [0, π).

13.6Quantization with k-Fermionic Coherent States

As another illustration of the coherent state quantization, let us sketch an applica-tion to a more elaborate mathematical structure, namely, a para-Grassmann algebrafor which coherent states can be constructed [182–184].

Let us first proceed with the construction of these coherent states by followingthe scheme described in Chapter 5. The observation set X is the para-Grassmannalgebra16)Σk . The latter is defined as the linear span of

{1, θ, . . . , θk–1

}and their

16) A Grassmann (or exterior) algebra of a givenvector space V over a field is the algebragenerated by the exterior (or wedge) productfor which all elements are nilpotent, θ2 = 0.

Para-Grassmann algebras are generalizationsfor which, given an integer k > 2, all elementsobey θk = 0.

13.6 Quantization with k-Fermionic Coherent States 219

respective conjugates θi ; here, θ is a Grassmann variable satisfying θk = 0. A mea-sure on X is defined as

μ(dθdθ) = dθ w (θ, θ) dθ . (13.38)

Here, the integral over dθ and dθ should be understood in the sense of Berezin–Majid–Rodríguez-Plaza integrals [185]:∫

dθ θn = 0 =∫

dθ θn , for n = 0, 1, . . . , k – 2, (13.39)

with ∫dθ θk–1 = 1 =

∫dθ θk–1 . (13.40)

The “weight” w (θ, θ) is given by the q-deformed polynomial

w (θ, θ) =k–1∑n=0

([n]q ![n]q !

) 12 θk–1–nθk–1–n . (13.41)

The q-deformed numbers are defined by

[x ]q :=1 – qx

1 – q, [n]q! = [1]q[2]q · · · [n]q , . (13.42)

In the present case, q = e2πik is a kth root of unity and θθ = qθθ.

The (nonnormalized) para-Grassmann or k-fermionic coherent states should beunderstood as elements of Ck ⊗ Σk . They read as

|θ) =k–1∑n=0

θn

([n]q !)12

|n〉 , (13.43)

where {|n〉} is an orthonormal basis of the Hermitian space Ck .The coherent state quantization of the “spinorial” para-Grassmann algebra rests

upon the resolution of the unity Ik in Ck :∫∫dθ|θ) w (θ, θ) (θ| dθ = Ik . (13.44)

The quantization of a para-Grassmann-valued function f (θ, θ) maps f to the linearoperator A f on Ck :

A f =∫∫

dθ|θ) : f (θ, θ)w (θ, θ) : (θ| dθ . (13.45)

Here we actually recover the k ~ k-matrix realization of the so-called k-fermionicalgebra F k [184]. For instance, we have for the simplest functions


Aθ =k–1∑n=0

([n + 1]q

) 12 |n〉〈n + 1| , Aθ =

k–1∑n=0

([n + 1]q

) 12 |n + 1〉〈n| = A†θ .

(13.46)

Their (anti-)commutator reads as

[Aθ, Aθ] =k–1∑n=0

cos π 2n+12k

cos π2k|n〉〈n| , {Aθ, Aθ} =

k–1∑n=0

sin π 2n+12k

sin π2k|n〉〈n| . (13.47)

In the purely fermionic case, k = 2, we recover the canonical anticommutation rule{Aθ , Aθ} = I2.

13.7Final Comments

Let us propose in this conclusion, on an elementary level, some hints for under-standing better the probabilistic duality lying at the heart of the coherent statequantization procedure. Let X be an observation set equipped with a measure νand let a real-valued function X � x �→ a(x) have the status of “observable.” Thismeans that there exists an experimental device, Δa , giving access to a set or “spec-trum” of numerical outcomes Σa = {a j , j ∈ J } ⊂ R, commonly interpreted as theset of all possible measured values of a(x). To the set Σa are attached two probabilitydistributions defined by the set of functions Πa = {p j (x), j ∈ J }:

1. A family of probability distributions on the set J , J � j �→ p j (x),∑j∈J p j (x) = 1, indexed by the observation set X. This probability en-

codes what is precisely expected for this pair (observable a(x), device Δa ).2. A family of probability distributions on the measure space (X , ν), X � x �→

p j (x),∫

X p j (x) ν(dx) = 1, indexed by the set J .

Now, the exclusive character of the possible outcomes a j of the measurement ofa(x) implies the existence of a set of “conjugate” functions X � x �→ α j (x), j ∈ J ,playing the role of phases, and making the set of complex-valued functions

φ j (x) def=√

p j (x) eiα j (x) (13.48)

an orthonormal set in the Hilbert space L2K(X , ν),

∫X φ j (x)φ j ′ (x) ν(dx) = δ j j ′ .

There follows the existence of the family of coherent states or frame in the Hilber-tian closure K of the linear span of the φ j ’s:

X � x �→ |x〉 =∑j∈J

φ j (x) |φ j 〉, 〈x |x〉 = 1 ,∫

X|x〉〈x | ν(dx) = IK . (13.49)

Consistency conditions have to be satisfied together with this material: they followfrom the quantization scheme resulting from the frame (13.49).

13.7 Final Comments 221

Condition 13.2 (Frame consistency conditions)

(i) Spectral condition

The frame quantization of the observable a(x) produces an essentially self-adjointoperator Aa in K that is diagonal in the basis {φ j , j ∈ J }, with spectrumprecisely equal to

∑a:

Aadef=

∫X

a(x) |x〉〈x | ν(dx) =∑j∈J

a j |φ j 〉〈φ j | . (13.50)

(ii) Classical limit condition

The frame {|x〉} depends on a parameter κ ∈ [0,∞), |x〉 = |x , κ〉 such that thelimit

Aa(x , κ) def= 〈x , κ|Aa (κ)|x , κ〉 →κ→0

a(x) (13.51)

holds for a certain topology Tcl assigned to the set of classical observables.

Once these conditions have been verified, one can start the frame quantizationof other observables, for instance, the quantization of the conjugate observablesα j (x), and check whether the observational or experimental consequences or con-straints due to this mathematical formalism are effectively in agreement with our“reality”.Many examples will be presented in the next chapters. For some of them, we havein view possible connections with objects of noncommutative geometry (such asfuzzy spheres or pseudospheres). These examples show the extreme freedom wehave in analyzing a set X of data or possibilities just equipped with a measure, byjust following our coherent state quantization procedure. The crucial step lies inthe choice of a countable orthonormal subset in L2(X , μ) obeying the finitude con-dition (13.8). A CN (or l2 if N =∞) unitary transform of this original subset wouldactually lead to the same specific quantization, and the latter could also be ob-tained by using unitarily equivalent continuous orthonormal distributions definedwithin the framework of some Gel’fand triplet. Of course, further structure such asa symplectic manifold combined with spectral constraints imposed on some spe-cific observables will considerably restrict that freedom and should lead, hopefully,to a unique solution, such as Weyl quantization, deformation quantization, andgeometrical quantization are able to achieve in specific situations. Nevertheless,we believe that the generalization of the Berezin–Klauder–Toeplitz quantizationthat has been described here, and that goes far beyond the context of classical andquantum mechanics, not only sheds light on the specific nature of the latter, butwill also help to solve in a simpler way some quantization problems.

223

14Coherent State Quantization of Finite Set,Unit Interval, and Circle

14.1Introduction

The examples that are presented in this chapter are, although elementary, ratherunusual. In particular, we deal with observation sets X that are not phase space,and such sets are far from having any physical meaning in the common sense.We first explore the coherent state quantization of finite sets and the unit interval.Then, we return to the unit circle, already considered in the previous chapter: wewill fully exploit the complex Fourier series in order to propose a satisfying solutionto the quantum phase problem.

14.2Coherent State Quantization of a Finite Set with Complex 2 ~ 2 Matrices

An elementary (but not trivial!) exercise for illustrating the quantization schemeintroduced in the previous chapter concerns an arbitrary N-element set X = {xi ∈X } viewed as an observation set. An arbitrary nondegenerate measure on it is givenby a sum of Dirac measures:

μ(dx) =N∑

i=1

ai δxi , ai > 0 . (14.1)

The Hilbert space L2C(X , μ), denoted by L2(X , μ) in the sequel, is simply isomorphic

to the Hermitian space CN . An obvious orthonormal basis is given by{1√ai

�{xi }(x) , i = 1, . . . , N

}, (14.2)

where �{a} is the characteristic function of the singleton {a}.Let us consider the two-element orthonormal set {φ1 == φα == |α〉, φ2 == φ� == |�〉}

defined in the most generic way by

φα(x) =N∑

i=1

αi1√ai

�{xi }(x) , φ�(x) =N∑

i=1

�i1√ai

�{xi }(x) , (14.3)


224 14 Coherent State Quantization of Finite Set, Unit Interval, and Circle

where the complex coefficients αi and �i obey

N∑i=1

|αi |2 = 1 =N∑

i=1

|�i |2 ,N∑

i=1

αi �i = 0 . (14.4)

In a Hermitian geometry language, our choice of {φα , φ�} amounts to selecting inCN the two orthonormal vectors α = {αi}, � = {�i}, and this justifies our notationfor the indices. These vectors span a two-dimensional subspace ~= C2 in CN .

The expression for the coherent states is

|x〉 =1√N (x)

[φα(x)|α〉 + φ�(x)|�〉

], (14.5)

in whichN (x) is given by

N (x) =N∑

i=1

|αi |2 + |�i |2ai

�{xi }(x) . (14.6)

The resulting resolution of unity reads as

Id =N∑

i=1

(|α j |2 + |� j |2

)|xi 〉〈xi | . (14.7)

The overlap between two coherent states is given by the following kernel:

〈xi |x j 〉 =αi α j + �i � j√

|αi |2 + |�i |2√|α j |2 + |� j |2

. (14.8)

To any real-valued function f (x) on X, that is, to any vector f == ( f (xi )) in RN , therecorresponds the following Hermitian operator A f in C2, expressed in matrix formwith respect to the orthonormal basis (14.3):

A f =∫

Xμ(dx)N (x) f (x)|x〉〈x |

=

(∑Ni=1 |αi |2 f (xi )

∑Ni=1 αi �i f (xi )∑N

i=1 αi �i f (xi )∑N

i=1 |�i |2 f (xi )

)==

(〈F〉α 〈�|F|α〉〈α|F|�〉〈F〉�

),

(14.9)

where F holds for the diagonal matrix {( f (xi ))}. It is clear that, for a generic choiceof the complex αi ’s and �i ’s, all possible Hermitian 2 ~ 2 matrices can be obtainedin this way if N v 4. By generic we mean that the following 4 ~ N real matrix hasrank equal to 4:

C =

⎛⎜⎜⎝|α1|2 |α2|2 · · · |αN |2|�1|2 |�2|2 · · · |�N |2

R(α1�1) R(α2�2) · · · R(αN �N )I(α1�1) I(α2�2) · · · I(αN �N ) .

⎞⎟⎟⎠ (14.10)

14.2 Coherent State Quantization of a Finite Set with Complex 2 ~ 2 Matrices 225

The case N = 4 with det C �= 0 is particularly interesting since then one has unique-ness of upper symbols of Pauli matrices σ1 =

(0 11 0

), σ2 =

(0 –ii 0

), σ3 =

(1 00 –1

), σ0 =

Id , which form a basis of the four-dimensional Lie algebra of complex Hermitian2 ~ 2 matrices. As a matter of fact, the operator (14.9) decomposes with respect tothis basis as

A f = 〈 f 〉+σ0 + 〈 f 〉–σ3 + R (〈�|F|α〉) σ1 – I (〈�|F|α〉) σ2 , (14.11)

where the symbols 〈 f 〉± stand for the following averagings:

〈 f 〉± = 12

N∑i=1

(|αi |2 ± |�i |2

)f (xi ) = 1

2

(〈F〉α ± 〈F〉�

). (14.12)

Here the quantity 〈 f 〉+ alone has a mean value status, precisely with respect to theprobability distribution

pi = 12

(|αi |2 + |�i |2

). (14.13)

These average values also appear in the spectral values of the quantum observableA f :

Sp( f ) =

{〈 f 〉+ ±

√(〈 f 〉–

)2+ |〈�|F|α〉|2

}. (14.14)

Note that if the vector α = (1, 0, . . . , 0) is part of the canonical basis and � =(0, �2, . . . , �n) is a unit vector orthogonal to α, then A f is diagonal and Sp( f ) istrivially reduced to

(f (x1), 〈F〉�

). The simplest upper symbols for Pauli matrices

read in vector form as

σ0 =

⎛⎜⎜⎝1111

⎞⎟⎟⎠ , σ1 = C–1

⎛⎜⎜⎝0010

⎞⎟⎟⎠ , σ2 = C–1

⎛⎜⎜⎝000–1

⎞⎟⎟⎠ , σ3 = C–1

⎛⎜⎜⎝1

–100

⎞⎟⎟⎠ .

(14.15)

On the other hand, and for any N, components of the lower symbol of A f are givenin terms of another probability distribution, for which the value of each one is twicethe value of its counterpart in (14.13):

〈xl |A f |xl〉 = A f (xl ) =N∑

i=1

πli f (xi ) , (14.16)

with

πl l = |αl |2 + |�l |2 = 2 pl , πli =|αl αi + �l �i |2|αl |2 + |�l |2

, i �= l . (14.17)


Note that the matrix (πli ) is stochastic. As a matter of fact, components of lowersymbols of Pauli matrices are given by

σ0(xl ) = 1 , σ1(xl ) =2R (αl �l )|αl |2 + |�l |2

, (14.18)

σ2(xl ) =2I (αl�l )|αl |2 + |�l |2

, σ3(xl ) =|αl |2 – |�l |2|αl |2 + |�l |2

. (14.19)

Let us attempt to interpret these formal manipulations in terms of localization inthe set X. Consider X = {xi ∈ X } as a set of N real numbers. One can then viewthe real-valued function f defined by f (xi ) = xi as the position observable. On the“quantum” level determined by the choice of α = {αi}, � = {�i}, the measurementof this position observable has the two possible outcomes given by (14.14). More-over, the position xl is privileged to a certain (quantitative) extent in the expressionof the average value of the position operator when computed in state |xl〉.

Before ending this section, let us examine the lower-dimensional cases N = 2and N = 3. When N = 2 the basis change (14.3) reduces to a U (2) transformationwith SU (2) parameters α = α1, � = –�1, |α|2 – |�|2 = 1, and some global phasefactor. The operator (14.9) now reads as

A f = f +Id + f –

(|α|2 – |�|2 –2α�

–2α� |�|2 – |α|2

), (14.20)

with f ± := ( f (x1) ± f (x2))/2. We thus obtain a two-dimensional commutativealgebra of “observables” A f , as is expected since we consider the quantization ofC2 yielded by an orthonormal basis of C2! This algebra is generated by the identitymatrix I2~2 = σ0 and the SU (2) transform of σ3: σ3 → gσ3 g † with g =

( α �–� α

)∈

SU (2). As is easily expected in this case, lower symbols reduce to components:

〈xl |A f |xl〉 = A f (xl ) = f (xl ) , l = 1, 2 . (14.21)

Finally, it is interesting to consider the N = 3 case when all vector spaces consideredare real. The basis change (14.3) involves four real independent parameters, say, α1,α2, �1, and �2, all with modulus < 1. The counterpart of (14.10) reads here as

C3 =

⎛⎝(α1)2 (α2)2 1 – (α1)2 – (α2)2

(�1)2 (�2)2 1 – (�1)2 – (�2)2

α1�1 α2�2 –α1�1 – α2�2

⎞⎠ . (14.22)

If det C3 = (α1�2–α2�1)(�1�2–α1α2) �= 0, then one has uniqueness of upper symbolsof Pauli matrices σ1, σ3, and σ0 = I2~2, which form a basis of the three-dimensionalJordan algebra of real symmetric 2 ~ 2 matrices. These upper symbols read invector form as

σ0 =

⎛⎝111

⎞⎠ , σ1 = C3–1

⎛⎝001

⎞⎠ , σ3 = C3–1

⎛⎝ 1–10

⎞⎠ . (14.23)

Finally, the extension of this quantization formalism to N ′-dimensional sub-spaces of the original L2(X , μ) � CN appears to be straightforward on a technical ifnot interpretational level (see [186] for more examples).

14.3 Coherent State Quantization of the Unit Interval 227

14.3Coherent State Quantization of the Unit Interval

We explore here two-dimensional (and higher-dimensional) quantizations of theunit segment.

14.3.1Quantization with Finite Subfamilies of Haar Wavelets

Let us consider the unit interval X = [0, 1] of the real line, equipped with theLebesgue measure dx , and its associated Hilbert space denoted by L2[0, 1].

We start out the study by simply selecting the first two elements of the orthonor-mal Haar basis [56], namely, the characteristic function 1(x) of the unit interval andthe Haar wavelet:

φ1(x) = 1(x), φ2(x) = 1(2x) – 1(2x – 1) . (14.24)

Then we have

N (x) =2∑

n=1

|φn(x)|2 = 2 a.e. (14.25)

The corresponding coherent states read as

|x〉 =1√2

[φ1(x)|1〉 + φ2(x)|2〉

]. (14.26)

To any integrable function f (x) on the interval there corresponds the linear opera-tor A f on R2 or C2:

A f = 2∫ 1

0dx f (x)|x〉〈x |

=

[∫ 1

0dx f (x)

][|1〉〈1| + |2〉〈2|

]+

[∫ 1

0dx f (x)φ2(x)

] [|1〉〈2| + |2〉〈1|

],

(14.27)

or, in matrix form with respect to the orthonormal basis (14.24),

A f =

( ∫ 10 dx f (x)

∫ 10 dx f (x)φ2(x)∫ 1

0 dx f (x)φ2(x)∫ 1

0 dx f (x)

)

=

(〈 f 〉[0,1] 2

(〈 f 〉[0,1/2] – 〈 f 〉[1/2,1]

)2(〈 f 〉[0,1/2] – 〈 f 〉[1/2,1]

)〈 f 〉[0,1]

), (14.28)

where 〈 f 〉I = (1/ Vol I)∫

I f (x) dx denotes the average of f on the set I. In particular,with the choice f = φ1 we recover the identity, whereas for f = φ2 we get Aφ2 =

228 14 Coherent State Quantization of Finite Set, Unit Interval, and Circle(0 11 0

)= σ1, the first Pauli matrix. With the choice f (x) = x p , Re p > –1,

Ax p =1

p + 1

(1 2– p – 1

2– p – 1 1

). (14.29)

For an arbitrary coherent state |x0〉, x0 ∈ [0, 1], it is interesting to evaluate thelower symbol of Ax p . This gives

〈x0|Ax p |x0〉 =

{2–p

p+1 0 u x0 u 12 ,

2–2–p

p+112 u x0 u 1 ,

(14.30)

the two possible values being precisely the eigenvalues of the above matrix. Notethe average values of the “position” operator: 〈x0|Ax |x0〉 = 1/4 if 0 u x0 u 1/2 and3/4 if 1/2 u x0 u 1.

Clearly, like in the N = 2 case in the previous section, all operators A f commute,since they are linear combinations of the identity matrix and the Pauli matrix σ1.The procedure is easily generalized to higher dimensions. Let us add to the previ-ous set {φ1, φ2} other elements of the Haar basis, say, up to the “scale” J:

{φ1(x), φ2(x), φ3(x) =√

2φ2(2x), φ4(x) =√

2φ2(2x – 1) ,

. . . , φs (x) = 2 j/2φ2(2x – k), φN (x) = 2 J/2φ2(2x – 2 J + 1)} , (14.31)

where, at given j = 1, 2, . . . , J , the integer k assumes its values in the range 0 uk u 2 j – 1. The total number of elements of this orthonormal system is N = 2 J+1.The normalization function is equal to N (x) =

∑Nn=1 = |φn (x)|2 = 2 J+1, and this

clearly diverges at the limit J → ∞. Then, it is remarkable if not expected thatspectral values as well as average values of the “position” operator are given by〈x0|Ax |x0〉 = (2k + 1)/2 J+1 for k/2 J u x0 u (k + 1)/2 J where 0 u k u 2 J – 1. Hence,our quantization scheme yields a dyadic discretization of the localization in theunit interval.

14.3.2A Two-Dimensional Noncommutative Quantization of the Unit Interval

Now we choose another orthonormal system, namely, the first two elements of thetrigonometric Fourier basis,

φ1(x) = 1(x), φ2(x) =√

2 sin 2πx . (14.32)

Then we have

N (x) =2∑

n=1

|φn (x)|2 = 1 + 2 sin2 2πx . (14.33)

The corresponding coherent states read as

|x〉 =1√

1 + 2 sin2 2πx

[|1〉 +

√2 sin 2πx |2〉

]. (14.34)

14.4 Coherent State Quantization of the Unit Circle and the Quantum Phase Operator 229

To any integrable function f (x) on the interval there corresponds the linear opera-tor A f on R2 or C2 (in its matrix form):

A f =

( ∫ 10 dx f (x)

√2∫ 1

0 dx f (x) sin 2πx√2∫ 1

0 dx f (x) sin 2πx 2∫ 1

0 dx f (x) sin2 2πx

). (14.35)

Like in the previous case, with the choice f = φ1 we recover the identity, whereasfor f = φ2 we get Aφ2 = σ1, the first Pauli matrix.

We now have to deal with a Jordan algebra of operators A f , like in the N = 3 realcase in the previous section. It is generated by the identity matrix and the two realPauli matrices σ1 and σ3.

In this context, the position operator is given by

Ax =

(12 – 1√

2π– 1√

2π12

),

with eigenvalues 12 ±

1√2π

Note its average values as a function of the coherent stateparameter x0 ∈ [0, 1]:

〈x0|Ax |x0〉 =12

–2π

sin 2πx0

1 + 2 sin2 2πx0.

In Figure 14.1 we give the curve of 〈x0|Ax |x0 as a function of x0. It is interestingto compare with the two-dimensional Haar quantization presented in the previoussubsection.

14.4Coherent State Quantization of the Unit Circle and the Quantum Phase Operator

We now come back to the unit circle. We gave an example of quantization in Sec-tion 13.5, resulting from the orthonormal system {cos θ, sin θ} in L2(S1, dθ/π).Here, we explore the coherent state quantization based on an arbitrary number ofelements of the Fourier exponential basis [187]. As an interesting byproduct of this“fuzzy circle,” we give an expression for the phase or angle operator, and we dis-cuss its relevance in comparison with various phase operators proposed by otherauthors.

14.4.1A Retrospective of Various Approaches

In classical physics, angle and action are conjugated canonical variables. It washoped to get the exact correspondence for their quantum counterpart after somequantization procedure. Taking for granted (true for the harmonic oscillator) thecorrespondence I → N , N being the number operator, and θ → θ, we should get


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 14.1 Average value 〈x0|Ax |x0〉 of position operator Ax versus x0 (cf. eigenvalues of Ax ).

the canonical commutation relation

[N , θ] = iId , (14.36)

and from this the Heisenberg inequality ΔN Δθ v 12 .

Since the first attempt by Dirac in 1927 [163] various definitions of angle or phaseoperator have been proposed with more or less satisfactory success in terms of con-sistency with regard to the requirement that phase operators and number operatorsform a conjugate Heisenberg pair obeying (14.36).

To obtain this quantum-mechanical analog, the polar decomposition of raisingand lowering operators

a = exp(i θ)N 1/2 , a† = N 1/2 exp(–i θ) (14.37)

was originally proposed by Dirac, with the corresponding uncertainty relation

Δθ ΔN v 12 . (14.38)

But the relation between operators (14.36) is misleading. The construction of a uni-tary operator is a delicate procedure and there are three main problems with it.First, we have that for a well-defined number state the uncertainty on the phasewould be greater than 2π. This inconvenience, also present in the quantization of


the angular momentum–angle pair, adds to the well-known contradiction, namely,“1 = 0”, lying in the matrix elements of the commutator

–iδnn′ = 〈n′|[N , θ]|n〉 = (n – n′)〈n′|θ|n〉 . (14.39)

In the angular momentum case, this contradiction is avoided to a certain extent byintroducing a proper periodical variable Φ(φ) [188]. If Φ is just a sawtooth func-tion, its discontinuities at φ = (2n + 1)π, n ∈ Z, give rise to Dirac peaks in thecommutation relation:

[Lz , Φ] = –i

{1 – 2π

∞∑n=–∞

δ(φ – (2n + 1)π)

}. (14.40)

The singularities in (14.40) can be excluded, as proposed by Louisell [189], takingsine and cosine functions of θ to recover a valid uncertainty relation. Louisell in-troduced operators cos θ and sin θ and imposed the commutation rule [cos θ, N ] =i sin θ and [sin θ, N ] = –i cos θ.

Nevertheless, the problem is harder in the number-phase case because, as shownby Susskind and Glogower [190], the decomposition (14.37) itself leads to the defi-nition of nonunitary operators:

exp(–i θ) =∞∑

n=0

|n〉〈n + 1|{+|ψ〉〈0|}, and h.c. , (14.41)

and this nonunitarity explains the inconsistency revealed in (14.39). To overcomethis handicap, a different polar decomposition was suggested in [190]:

a = (N + 1)12 E –, a† = E +(N + 1)

12 , (14.42)

where the operators E± are still nonunitary because of their action on the extremalstate of the semibounded number basis [188]. Nevertheless, the addition of therestriction

E –|0〉 = 0 (14.43)

permits us to define Hermitian operators

C = 12 (E – + E +) = C † ,

S =12i

(E – – E +) = S† .(14.44)

These operators are named “cosine” and “sine” because they reproduce the samealgebraic structure as the projections of the classical state in the phase space of theoscillator problem.

In an attempt to determine a Hermitian phase operator θ fitting (14.36) in theclassical limit, and for which constraints such as (14.43) would be avoided, Popov


and Yarunin [191, 192] and, later, Barnett and Pegg [193] used an orthonormal setof eigenstates of θ defined on the number state basis as

|θm〉 =1√N

N –1∑n=0

einθm |n〉 , (14.45)

where, for a given N, the following equidistant subset of the angle parameter isselected:

θm = θ0 +2πm

N, m = 0, 1, . . . , N – 1 , (14.46)

with θ0 as a reference phase. Orthonormality stems from the well-known proper-ties of the roots of the unity as happens with the discrete Fourier transform basis:

N –1∑n=0

ein(θm–θm′ ) =N –1∑n=0

ei2π(m–m′ ) nN = N δmm′ . (14.47)

Indeed, z = e2πi qN for any q ∈ Z is a solution of

(1 – zN ) = (1 – z)(1 + z + · · · + zN –1) = 0 .

The inverse (unitary!) transform from the Pegg–Barnett orthonormal basis{|θm〉} to the number basis {|n〉} is given by

|n〉 =1√N

N –1∑m=0

e–in θm |θm〉 . (14.48)

Hence, a system in a number state is equally likely to be found in any phase state,and the reciprocal relationship is also true.

The (Pegg–Barnett) phase operator on CN is then defined by the spectral condi-tions

θbp|θm〉 = θm |θm〉 , (14.49)

or, equivalently, constructed through the spectral resolution

θbp =N –1∑m=0

θm |θm〉〈θm | . (14.50)

Its expression in the number state basis reads as

θbp =

(θ0 + π

N – 1N

)IN +

2πN

N –1∑n=/n′

ei(n–n′)θ0

e2πi n–n′N – 1

|n〉〈n′| . (14.51)

On the other hand, the expression of the number operator N in the phase basisreads as

N =N –1∑n=0

|n〉〈n| = N – 12

IN +N –1∑

m=/m′

1

e–2πi m′ –mN – 1

|θm〉〈θm′ | . (14.52)


Note the useful summation formula valid for any z = e2πi qN , q ∈ Z, q =/ 0:

N –1∑n=0

nzn =N

z – 1.

The commutation rule with the number operator expressed in the number statebasis results:

[N , θbp] =2πN

N –1∑n=/n′

(n – n′)ei(n–n′)θ0

e2πi n–n′N – 1

|n〉〈n′| . (14.53)

In the phase state basis it is

[N , θbp] =2πN

N –1∑m=/m′

(m – m′)1

1 – e–2πi m–m′N

|θm〉〈θm′ | . (14.54)

Now, from the asymptotic behavior at large N >> m, m′, n, n′,

〈θm|[N , θbp]|θm′ 〉 W i (1 – δmm′ ) ,

〈n|[N , θbp]|n′〉 W –i (1 – δnn′ )ei(n–n′)θ0 ,(14.55)

it seems hopeless to get something like canonical commutation rules with thisphase operator resulting from the unitary change of basis [number basis→ phasebasis].

Let us turn our attention to the unitary ei θbp . Since θbp is Hermitian, the operatorei θbp is unitary. Its expression in the phase state basis is immediate:

ei θbp =N –1∑m=0

eiθm |θm〉〈θm | = eiθ0

N –1∑m=0

e2πi mN |θm〉〈θm| . (14.56)

On the other hand, its expression in number state basis is off-diagonal and “circu-lar”:

ei θbp =N –2∑n=0

|n〉〈n + 1| + eiN θ0 |N – 1〉〈0| . (14.57)

One can form from ei θbp Hermitian cosine and sine combinations with the expectedrelations:

cos2 θbp + sin2 θbp = 1 , [cos θbp, sin θbp] = 0 ,

〈n| cos2 θbp|n〉 = 12 = 〈n| sin2 θbp|n〉 .

Finally, annihilation and creation operators acting on CN can be defined throughthe natural factorization

a = ei θbp N 1/2 =N –2∑n=0

√n + 1|n〉〈n + 1| , a† =

N –1∑n=1

√n|n〉〈n – 1| .


We then recover the finite-dimensional version of the canonical quantization:

[a, a†] = IN – N |N – 1〉〈N – 1| .

The Pegg–Barnett construction, which amounts to an adequate change of or-thornormal basis in CN , gives for the ground number state |0〉 a random phasethat avoids some of the drawbacks encountered in previous developments. Notethat taking the limit N → ∞ is questionable within a Hilbertian framework; thisprocess must be understood in terms of mean values restricted to some suitablesubspace and the limit has to be taken afterwards. In [193] the pertinence of thestates (14.45) was proved by the expected value of the commutator with the num-ber operator. The problem appears when the limit is taken since it leads to anapproximate result.

More recently an interesting approach to the construction of a phase operatorwas used by Busch, Lahti, and their collaborators within the framework of measure-ment theory [194, 195, 197] (see also [196]). Phase observables are characterized asphase-shift-covariant positive operator measures, with the number operator play-ing the part of the shift generator.

14.4.2Pegg–Barnett Phase Operator and Coherent State Quantization

Let us now reformulate the construction of the Pegg–Barnett phase operator alongthe lines of coherent state quantization. The observation set X is the set of Nequidistant angles on the unit circle:

X ={

θm = θ0 +2πm

N, m = 0, 1, . . . , N – 1

}. (14.58)

We equip X with the discrete equally weighted normalized measure:∫X

f (x) μ(dx) def=1N

N –1∑m=0

f (θm) . (14.59)

The Hilbert space L2(X , μ), equipped with the scalar product

〈 f |g〉 =1N

N –1∑m=0

f (θm)g (θm) , (14.60)

is thus isomorphic to CN . We now just choose as an orthonormal set the discreteFourier basis:

φn(θm) = e–inθm , withN –1∑n=0

|φn (θm)|2 = N . (14.61)

The coherent states emerging from this orthonormal set are just the Pegg–Barnettphase states:

|θm〉 =1√N

N –1∑n=0

einθm |φn〉 , (14.62)


where the kets |φn〉 can be identified with the number states |n〉. Of course, wetrivially have normalization and resolution of the unity in CN :

〈θm|θm〉 = 1 ,∫

X|θm〉〈θm|N μ(dx) = IN . (14.63)

The resolution of the unity immediately allows the construction of diagonal opera-tors in CN :

f (x) �→∫

Xf (x)|θm〉〈θm|N μ(dx) =

N –1∑m=0

f (θm)|θm〉〈θm| = A f . (14.64)

As expected from the discussion in Chapter 13, this quantization yields a commu-tative algebra of operators. The Pegg–Barnett phase operator is nothing other thanthe quantization of the function angle f (x) = x ∈ X .

14.4.3A Phase Operator from Two Finite-Dimensional Vector Spaces

As suggested in [193], the commutation relation will approximate better the canon-ical one (14.36) if one enlarges enough the Hilbert space of states. Hence, we followa construction of the phase operator based on a coherent state quantization schemethat involves two finite-dimensional vector spaces. This will produce a suitable com-mutation relation at the infinite-dimensional limit, still at the level of mean values.

The observation set X is now the set of M equidistant angles on the unit circle:

X ={

θm = θ0 +2πmM

, m = 0, 1, . . . , M – 1}

. (14.65)

Set X is equipped with the discrete equally weighted normalized measure:∫X

f (x) μ(dx) def=1M

M–1∑m=0

f (θm) . (14.66)

The Hilbert space L2(X , μ), with the scalar product

〈 f |g〉 =1M

M–1∑m=0

f (θm)g (θm) , (14.67)

is isomorphic to CM . Let us choose as an orthonormal set the Fourier set with Nelements, N < M :

φn(θm) = e–inθm , withN –1∑n=0

|φn (θm)|2 = N . (14.68)

The coherent states now read

|θm〉 =1√N

N –1∑n=0

einθm |φn〉 , (14.69)


where the kets |φn〉 can be identified with the number states |n〉. These states,different from (14.62), are normalized and resolve the unity in CN :

〈θm|θm〉 = 1,∫

X|θm〉〈θm|N μ(dx) = IN . (14.70)

The important difference from the case M = N is that we have lost orthonormality:

〈θm|θm′ 〉 =1N

N –1∑n=0

e2πin m′–mM

=

⎧⎪⎨⎪⎩1 if m = m′ ,

1N

eiπ(m′–m) N–1M

sin π N (m′–m)M

sin π m′–mM

if m =/ m′ .(14.71)

The coherent state quantization reads as

f (x) �→∫

Xf (θm)|θm〉〈θm|N μ(dx) =

M–1∑m=0

f (θm)|θm〉〈θm| = A f , (14.72)

and provides a noncommutative algebra of operators as soon as M > N . The opera-tors that emerge from this coherent state quantization are now different from thePegg–Barnett operators. In particular, we obtain as a phase operator

Aθ =NM

M–1∑m=0

θm|θm〉〈θm|

=

(θ0 + π

M – 1M

)IN +

2πM

N –1∑n=/n′

ei(n–n′)θ0

e2πi n–n′M – 1

|φn〉〈φn′ | . (14.73)

The coherent states are not phase eigenstates any more, Aθ|θm〉 =/ θm|θm〉, and theexpression of the lower symbols is quite involved:

〈θm|Aθ|θm〉 =NM

θm +1M

M–1∑m′=0,m′=/m

θm′

(sin π N (m′–m)

M

sin π m′–mM

)2

. (14.74)

Let us examine its commutation relation with the number operator expressed inthe number state basis:

[N , Aθ] =2πM

N –1∑n=/n′

(n – n′)ei(n–n′)θ0


|φn〉〈φn′ | . (14.75)

Its lower symbol reads as

〈θm|[N , Aθ]|θm〉 =2π

N M

N –1∑n=/n′

(n – n′)e2πi(n–n′) m

M


. (14.76)


Its asymptotic behavior at large M >> N ,

〈θm|[N , Aθ]|θm′ 〉 W –iN

N –1∑n=/n′

e2πi(n–n′) mM

= i –iN

(sin π N (m)

M

sin π mM

)2

, (14.77)

shows that we have, to some extent, improved the situation.

14.4.4A Phase Operator from the Interplay Between Finite and Infinite Dimensions

We show now that there is no need to discretize the angle variable as in [193] torecover a suitable commutation relation. We adopt instead the Hilbert space ofsquare-integrable functions on the circle as the natural framework for defining anappropriate phase operator in a finite-dimensional subspace.

So we take as an observation set X the unit circle S1 provided with the measureμ(dθ) = dθ/2π. The Hilbert space is L2(X , μ) = L2(S1, dθ/2π) and has the innerproduct

〈 f |g〉 =∫ 2π

0f (θ)g (θ)

dθ2π

. (14.78)

In this space we choose as an orthonormal set the N first Fourier exponentials withnegative frequencies:

φn(θ) = e–inθ , with N (θ) =N –1∑n=0

|φn(θ)|2 = N . (14.79)

The phase states are now defined as the corresponding “coherent states”:

|θ) =1√N

N –1∑n=0

einθ|φn〉 , (14.80)

where the kets |φn〉 can be directly identified with the number states |n〉, and theround bracket denotes the continuous labeling of this family. We have, by construc-tion, normalization and resolution of the unity inHN ~= CN :

(θ|θ) = 1,∫ 2π

0|θ)(θ|N μ(dθ) = IN . (14.81)

Unlike in (14.45), the states (14.80) are not orthogonal but overlap as

(θ′|θ) =ei N–1

2 (θ–θ′)

Nsin N

2 (θ – θ′)

sin 12 (θ – θ′)

. (14.82)


Note that for N large enough these states contain all the Pegg–Barnett phase statesand besides they form a continuous family labeled by the points of the circle. Thecoherent state quantization of a particular function f (θ) with respect to the contin-uous set (14.80) yields the operator A f defined by

f (θ) �→∫

Xf (θ)|θ)(θ|N μ(dθ) def= A f . (14.83)

An analogous procedure has been used within the framework of positive-opera-tor-valued measures [194, 195]: the phase states are expanded over an infinite or-thogonal basis with the known drawback of defining the convergence of the series|φ〉 =

∑n einθ|n〉 from the Hilbertian arena and the related questions concerning

operator domains. When it is expressed in terms of the number states, the operator(14.83) takes the form

A f =N –1∑

n,n′=0

cn′–n( f )|n〉〈n′| , (14.84)

where cn( f ) is the Fourier coefficient of the function f (θ),

cn( f ) =∫ 2π

0f (θ)e–inθ dθ

2π.

Therefore, the existence of the quantum version of f is ruled by the existence ofits Fourier transform. Note that A f is self-adjoint only when f (θ) is real-valued.In particular, a self-adjoint phase operator of the Toeplitz matrix type is readilyobtained by choosing f (θ) = θ:

Aθ = –iN –1∑n=/n′

n,n′=0

1n – n′

|n〉〈n′|, (14.85)

an expression that has to be compared with (12.13). Its lower symbol or expectationvalue in a coherent state is given by

(θ|Aθ|θ) =iN

N –1∑n,n′=0n=/n′

ei(n–n′)θ

n′ – n. (14.86)

Owing to the continuous nature of the set of |θ), all operators produced by thisquantization are different from the Pegg–Barnett operators. As a matter of fact, thecommutator [N , Aθ] expressed in terms of the number basis reads as

[N , Aθ] = –iN –1∑

n,n′=0n=/n′

|n〉〈n′| = iIN + (–i )IN , (14.87)


and has all diagonal elements equal to 0. Here IN =∑N –1

n,n′=0 |n〉〈n′| is the N ~ Nmatrix with all entries equal to 1. The spectrum of this matrix is reduced to thevalues 0 (degenerate N –1 times) and N, that is, (1/N ) IN is an orthogonal projectoron one vector. More precisely, the normalized eigenvector corresponding to theeigenvalue N is along the diagonal in the first quadrant in CN :

|vN 〉 = |θ = 0) =1√N

N –1∑n=0

|n〉 . (14.88)

Other eigenvectors span the hyperplane orthogonal to |vN 〉. We can choose themas the set with N – 1 elements:{

|vn〉 def=1√2

(|n + 1〉 – |n〉) , n = 0, 1, . . . , N – 2

}. (14.89)

The matrix IN is just N times the projector |vN 〉〈vN |. Hence, the commutation rulereads as

[N , Aθ] = –iN –1∑n=/n′

n,n′=0

|n〉〈n′| = i (Id – N |vN 〉〈vN |) . (14.90)

A further analysis of this relation through its lower symbol yields, for the matrixIN , the function

(θ|IN |θ) =1N

N –1∑n,n′=0

ei(n–n′)θ =1N

sin2 N θ2

sin2 θ2

. (14.91)

In the limit at large N this function is the Dirac comb (a well-known result indiffraction theory):

limN→∞

1N

sin2 N θ2

sin2 θ2

=∑k∈Z

δ(θ – 2kπ) . (14.92)

Recombining this with (14.90) allows us to recover the canonical commutation rulethrough its lower symbol and with the addition of a Dirac comb:

(θ|[N , Aθ]|θ) WN→∞ i – i∑k∈Z

δ(θ – 2kπ) . (14.93)

This expression is the expected one for any periodical variable as was seen in(14.40).

Note that (14.93) is found through the expected value over phase coherent statesand not in any physical state like in [193]. This shows that states (14.80), as standardcoherent states, are the closest to classical behavior. Another main feature is thatany of these states is equally weighted over the number basis that confirms a totalindeterminacy on the eigenstates of the number operator. The opposite is also true:


the number state is equally weighted over all the family (14.80) and in particularthis coincides with the results in [193].

The creation and annihilation operators are obtained using first the quantization(14.83) with f (θ) = e±iθ,

Ae±iθ =∫ 2π

0e±iθN |θ)(θ|dθ

2π, (14.94)

and then including the number operator as Aeiθ N12 == a in a similar way to that

in [193], where the authors used instead ei θP B N12 . The commutation relation be-

tween both operators is

[a, a†] = 1 – N |N – 1〉〈N – 1| , (14.95)

which converges to the common result only when the expectation value is taken onstates where extremal state components vanish as N tends to infinity.

As the phase operator is not built from a spectral decomposition, it is clear thatAθ2 =/ A2

θ and the link with an uncertainty relation is not straightforward as in [193].Instead, as suggested in [195], a different definition for the variance should be used.

The phase operator constructed here has most of the advantages of the Pegg–Barnett operator but allows more freedom within the Hilbertian framework. It isclear that a well-defined phase operator must be parameterized by all points in thecircle to have convergence to the commutation relation consistent with the classicallimit. It is also clear that the inconveniences due to the nonperiodicity of the phasepointed out in [190] are avoided from the very beginning by the choice of X == S1.

241

15Coherent State Quantization of Motionson the Circle, in an Interval, and Others

15.1Introduction

In most of the introductory textbooks devoted to quantum mechanics the quan-tum versions of two simple models are presented, namely, the motion of a particleon the circle and in an interval (the infinite square well potential). In this chapter,we revisit these examples in the light of coherent state quantization of the cor-responding phase spaces, namely, the cylinder [198] and an infinite strip in theplane [199, 200]. We also apply our method to motion on the 1 + 1 de Sitter space-time, since the corresponding phase space is topologically a cylinder. We finallyconsider under the same angle a more exotic example, the motion on a discreteset of points as was presented in [201] under the name of “shadow” Schrödingerquantum mechanics.

15.2Motion on the Circle

15.2.1The Cylinder as an Observation Set

Quantization of the motion of a particle on the circle (like the quantization of polarcoordinates in the plane) is an old question with so far mildly evasive answers.A large body of literature exists concerning the subject, more specifically devotedto the problem of angular localization and related Heisenberg inequalities; see, forinstance, [202].

Let us apply our scheme of coherent state quantization to this particular prob-lem. Let us first recall the material presented in the last section of Chapter 5. Theobservation set X is the phase space of a particle moving on the circle, precisely thecylinder S1 ~ R = {x == (�, J), |0 u � < 2π, J ∈ R}, equipped with the measureμ(dx) = 1

2π d J d�. The functions φn(x) forming the orthonormal system needed to


242 15 Coherent State Quantization of Motions on the Circle, in an Interval, and Others

construct coherent states are suitably weighted Fourier exponentials:

φn(x) =( ε

π

)1/4e– ε

2 ( J–n)2ein� , n ∈ Z , (15.1)

where ε > 0 is a regularization parameter that can be arbitrarily small. The coherentstates [62, 63, 67] read as

|x〉 == | J, �〉 =1√N ( J)

( επ

)1/4 ∑n∈Z

e– ε2 ( J–n)2

e–in�|en〉 , (15.2)

where the states |en〉’s, in one-to-one correspondence with the φn ’s, form an or-thonormal basis of some separable Hilbert space H. For instance, they can be con-sidered as Fourier exponentials ein� forming the orthonormal basis of the Hilbertspace L2(S1, dθ/2π) ~= H. They are the spatial modes in this representation. The nor-malization factor is a periodic train of normalized Gaussians and is proportional toan elliptic theta function,

N ( J) =

√επ

∑n∈Z

e–ε( J–n)2=

Poisson

∑n∈Z

e2πin J e– π2ε n2

, (15.3)

and satisfies limε→0N ( J) = 1.It should be noted that the quantization of the observation set follows the selec-

tion (or polarization) in the (modified) Hilbert space

L2

(S1 ~ R,

√επ

12π

e–ε J2d J d�

)of all Laurent series in the complex variable z = eε J–i�.

15.2.2Quantization of Classical Observables

By virtue of (13.12), the quantum operator (acting onH) associated with the classi-cal observable f (x) is obtained through

A f :=∫

Xf (x)|x〉〈x |N (x)μ(dx) =

∑n,n′

(A f

)nn′|en〉〈en′ | , (15.4)

where (A f

)nn′

=

√επ

e– ε8 (n–n′)2

∫ +∞

–∞d J e– ε

2

(J– n+n′

2

)2 12π

∫ 2π

0d� e–i(n–n′)� f ( J , �) .

(15.5)

If f is J-dependent only, then A f is diagonal with matrix elements that are Gaus-sian transforms of f ( J):(

A f ( J))

nn′= δnn′

√επ

∫ +∞

–∞d J e– ε

2 ( J–n)2f ( J) . (15.6)

15.2 Motion on the Circle 243

For the most basic one, associated with the classical observable J, this yields

A J =∫

Xμ(dx)N ( J) J | J , �〉〈 J , �| =

∑n∈Z

n |en〉〈en |, (15.7)

and this is nothing but the angular momentum operator, which reads in angularposition representation (Fourier series) as A J = –i∂/∂�.

If f is �-dependent only, f (x) = f (�), we have

A f (�) =∫

Xμ(dx)N ( J) f (�) | J, �〉〈 J , �| (15.8)

=∑

n,n′∈Z

e– ε4 (n–n′)2

cn–n′ ( f )|en〉〈en′ | , (15.9)

where cn( f ) is the nth Fourier coefficient of f. In particular, we have for

• the operator “angle,”

A� = πIH +∑n=/n′

ie– ε

4 (n–n′)2

n – n′|en〉〈en′ | , (15.10)

• the operator “Fourier fundamental harmonic,”

Aei� = e– ε4

∑n

|en+1〉〈en| . (15.11)

In the isomorphic realization of H in which the kets |en〉 are the Fourier exponen-tials ei n�, Aei� is a multiplication operator by ei� up to the factor e– ε

4 (which can bemade arbitrarily close to 1).

15.2.3Did You Say Canonical?

The “canonical” commutation rule

[A J , Aei� ] = Aei� (15.12)

is canonical in the sense that it is in exact correspondence with the classical Poissonbracket{

J , ei�}

= iei� . (15.13)

For other nontrivial commutators having this exact correspondence, see [203].There could be interpretational difficulties with commutators of the type

[A J , A f (�) ] =∑n,n′

(n – n′)e– ε4 (n–n′)2

cn–n′( f ) |en〉〈en′ | , (15.14)


and, in particular, for the angle operator itself,

[A J , A�] = i∑n=/n′

e– ε4 (n–n′)2 |en〉〈en′ | , (15.15)

to be compared with the classical { J, �} = 1.We have already encountered such difficulties in Chapters 12 and 14, and we

guess they are only apparent. They are due to the discontinuity of the 2π-periodicsaw function B (�) which is equal to � on [0, 2π). They are circumvented if we exam-ine, like we did in those chapters, the behavior of the corresponding lower symbolsat the limit ε→ 0. For the angle operator,

〈 J0, �0|A�| J0, �0〉 =π +12

(1 +N ( J0 – 1

2 )N ( J0)

) ∑n=/0

ie– ε

2 n2+in�0

n

~ε→0

π +∑n=/0

iein�0

n, (15.16)

where we recognize at the limit the Fourier series of B (�0). For the commutator,

〈 J0, �0|[A J , A�]| J0, �0〉 =12

(1 +N ( J0 – 1

2 )N ( J0)

)(–i +

∑n∈Z

ie– ε2 n2+in�0

)~

ε→0–i + i

∑n

δ(�0 – 2πn) .

(15.17)

So we (almost) recover the canonical commutation rule except for the singularityat the origin mod 2π.

15.3From the Motion of the Circle to the Motion on 1 + 1 de Sitter Space-Time

The material in the previous section is now used to describe the quantum motionof a massive particle on a 1 + 1 de Sitter background, which means a one-sheetedhyperboloid embedded in a 2 + 1 Minkowski space. Here, we just summarize thecontent of [198]. The phase space X is also a one-sheeted hyperboloid,

J21 + J2

2 – J20 = κ2 > 0 , (15.18)

with (local) canonical coordinates ( J, �), as for the motion on the circle. Phase spacecoordinates are now viewed as basic classical observables,

J0 = J , J1 = J cos � – κ sin � , J2 = J sin � + κ cos � , (15.19)

and obey the Poisson bracket relations

{ J0, J1} = – J2 , { J0, J2} = J1 , { J1, J2} = J0 . (15.20)

15.4 Coherent State Quantization of the Motion in an Infinite-Well Potential 245

They are, as expected, the commutation relations of so(1, 2) � sl(2, R), which is thekinematical symmetry algebra of the system. Applying the coherent state quantiza-tion (15.4) at ε =/ 0 produces the basic quantum observables:

A J0 =∑

n

n|en〉〈en| , (15.21a)

AεJ1

=12

e– ε4

∑n

(n +

12

+ iκ

)|en+1〉〈en | + c.c. , (15.21b)

AεJ2

=12i

e– ε4

∑n

(n +

12

+ iκ

)|en+1〉〈en| – cc . (15.21c)

The quantization is asymptotically exact for these basic observables since

[A J0 , AεJ1

] = iAεJ2

, [A J0 , AεJ2

] = –iAεJ1

, [AεJ1

, AεJ2

] = –ie– ε2 A J0 . (15.22)

Moreover, the quadratic operator

C ε = (AεJ1

)2 + (AεJ2

)2 – e– ε2 (A J0 )2 (15.23)

commutes with the Lie algebra generated by the operators A J0 , AεJ1

, AεJ1

, that is,it is the Casimir operator for this algebra. In the representation given in (15.21a)–(15.21c), its value is fixed to e– ε

2 (κ2 + 14 )I and so admits the limit

C ε ~ε→0

(κ2 +

14

)Id .

Hence, we have produced a coherent state quantization that leads asymptotically tothe principal series of representations of SO0(1, 2).

15.4Coherent State Quantization of the Motion in an Infinite-Well Potential

15.4.1Introduction

Even though the quantum dynamics in an infinite square well potential representsa rather unphysical limit situation, it is a familiar textbook problem and a simpletractable model for the confinement of a quantum particle. On the other hand, thismodel has a serious drawback when it is analyzed in more detail. Namely, whenone proceeds to a canonical standard quantization, the definition of a momentumoperator with the usual form –i�d/dx has a doubtful meaning. This subject hasbeen discussed in many places (see, e.g., [134]), and the attempts to circumventthis anomaly range from self-adjoint extensions [134–137] to PT symmetry ap-proaches [204].

First of all, the canonical quantization assumes the existence of a momentum op-erator (essentially) self-adjoint in L2(R) that respects some boundary conditions at


the boundaries of the well. These conditions cannot be fulfilled by the usual deriva-tive form without the consequence of losing self-adjointness. Moreover, there ex-ists an uncountable set of self-adjoint extensions of such a derivative operator thatmakes truly delicate the question of a precise choice based on physical require-ments [135–137, 200].

When the classical particle is trapped in an infinite-well interval Δ, the Hilbertspace of quantum states is L2(Δ, dx) and the quantization problem becomes simi-lar, to a certain extent, to the quantization of the motion on the circle S1. Notwith-standing the fact that boundary conditions are not periodic but impose the condi-tion instead that the wave functions in position representation vanish at the bound-ary, the momentum operator P for the motion in the infinite well should be thecounterpart of the angular momentum operator Jop for the motion on the circle.Since the energy spectrum for the infinite square well is like {n2, n ∈ N∗}, weshould expect that the spectrum of P should be Z∗, like the one for Jop withoutthe null eigenvalue. We profit from this similarity between the two problems byadapting the coherent states for the motion on the circle to the present situation.More precisely, we introduce two-component vector coherent states, in the spiritof [205], as infinite superpositions of spinors that are eigenvectors of P. We thenexamine the consequences of our choice after coherent state quantization of basicquantum observables, such as position, energy, and a quantum version of the prob-lematic momentum. In particular we focus on their mean values in coherent states(“lower symbols”) and quantum dispersions. As will be shown, the classical limitis recovered after choosing appropriate limit values for some parameters presentin the expression of our coherent states.

15.4.2The Standard Quantum Context

As already indicated in Chapter 9, the wave function Ψ(q, t) of a particle of mass mtrapped inside the interval q ∈ [0, L] (here we use the notation L = πa) has to obeythe conditions

ψ ∈ L2([0, L], dq), Ψ(0, t) = Ψ(L, t) = 0 ∀ t . (15.24)

Its Hamiltonian reads as

H == Hw = –�2

2md2

dx2 , (15.25)

and is self-adjoint [206] on an appropriate dense domain in (15.24). FactorizingΨ as Ψ(q, t) = e– i

� Ht Ψ(q, 0), we have for Ψ(q, 0) == ψ(q) the eigenvalue equationHψ(q) = E ψ(q), together with the boundary conditions (15.24). Normalized eigen-states and corresponding eigenvalues are then given by

ψn(q) =

√2L

sin(

nπqL

), 0 � q � L ,

Hψn = E nψn , n = 1, 2, . . . ,

(15.26)


with

E n =�2π2

2mL2 n2 == �ωn2 , ω =�E π2

2mL2==

2πT r

, (15.27)

where T r is the revival time to be compared with the purely classical round-triptime.

15.4.3Two-Component Coherent States

The classical phase space of the motion of the particle is the infinite strip X =[0, L] ~ R = {x = (q, p)|q ∈ [0, L], p ∈ R} equipped with the measure μ(dx) = dq dp.Typically, we have two phases in the periodic particle motion with a given energy:one corresponds to positive values of the momentum, p = mv , while the other oneis for negative values, p = –mv . This observation naturally leads us to introducethe Hilbert space of two-component complex-valued functions (or spinors), square-integrable with respect to μ(dx):

L2C2 (X , μ(dx)) � C2 ⊗ L2

C(X , μ(dx))

=

{Φ(x) =

(φ+(x)φ–(x)

), φ± ∈ L2

C(X , μ(dx))

}. (15.28)

Inspired by the coherent states for the motion on the circle, we choose our or-thonormal system as formed of the following vector-valued functions Φn,κ(x), κ =±,

Φn,+(x) =

(φn,+(x)

0

), Φn,–(x) =

(0

φn,–(x)

),

φn,κ(x) =√

c exp(

–1

2ρ2 ( p – κpn)2)

sin(

nπqL

), κ = ± , n = 1, 2, . . . ,

(15.29)

where

c =2

ρL√

π, pn =

√2mE n =

�πL

n . (15.30)

The half-width ρ > 0 in the Gaussians is a parameter which has the dimensionof a momentum, say, ρ = �πϑ/L, with ϑ > 0 a dimensionless parameter. Thisparameter can be arbitrarily small, like for the classical limit. It can be arbitrarilylarge, for instance, in the case of a very narrow well. The functions Φn,κ(x) arecontinuous, vanish at the boundaries q = 0 and q = L of the phase space, and obey


the finiteness condition:

0 < N (x) == N (q, p) == N+(x) +N–(x) =∑κ=±

∞∑n=1

Φ†n,κ(x)Φn,κ(x)

= c∞∑

n=1

[exp

(–

1ρ2 ( p – pn)2

)+ exp

(–

1ρ2 ( p + pn)2

)]~ sin2

(nπ

qL

)<∞ . (15.31)

The expression of N (q, p) == c S(q, p) can be simplified to

S(q, p) = R

{12

∞∑n=–∞

[1 – exp

(i2πn

qL

)]exp

(–

1ρ2 ( p – pn)2

)}. (15.32)

Series N and S can be expressed in terms of elliptic theta functions. Function Shas no physical dimension, whereas N has the same dimension as c, that is, theinverse of an action.

We are now in a position to define our vector coherent states along the linesof [205]. We set up a one-to-one correspondence between the functions Φn,κ’s andtwo-component states

|en ,±〉 def= |±〉 ⊗ |en〉 , |+〉 =

(10

), |–〉 =

(01

), (15.33)

forming an orthonormal basis of some separable Hilbert space of the form K =C2 ⊗H. The latter can be viewed also as the subspace of L2

C2 (X , μ(dx)) equal to theclosure of the linear span of the set of Φn,κ’s. We choose the following set of 2 ~ 2diagonal real matrices for our construction of vectorial coherent states:

Fn(x) =

(φn,+(q, p) 0

0 φn,–(q, p)

). (15.34)

Note that N (x) =∑∞

n=1 tr(Fn(x)2). Vector coherent states, |x , �〉 ∈ C2 ⊗H = K, arenow defined for each x ∈ X and � ∈ C2 by the relation

|x , �〉 =1√N (x)

∞∑n=1

Fn(x) |�〉 ⊗ |en〉 . (15.35)

In particular, we single out the two orthogonal coherent states

|x , κ〉 =1√N (x)

∞∑n=1

Fn(x)|en , κ〉 , κ = ± . (15.36)


By construction, these states also satisfy the infinite square well boundary condi-tions, namely, |x , κ〉q=0 = |x , κ〉q=L = 0. Furthermore they fulfill the normalizations

〈x , κ|x , κ〉 =Nκ(x)N (x)

,∑κ=±

〈x , κ|x , κ〉 = 1 (15.37)

and the resolution of the identity in K:∑κ=±

∫X|x , κ〉〈x , κ|N (x)μ(dx)

=∑κ=±

∞∑n,n′=1

∫ ∞

–∞

∫ L

0Fn(q, p)Fn′ (q, p)|en , κ〉〈en′ , κ|dqd p

=∑κ=±

∞∑n=1

|en , κ〉〈en , κ| = σ0 ⊗ IH = IK , (15.38)

where σ0 denotes the 2 ~ 2 identity matrix consistently with the Pauli matrix nota-tion σμ to be used in the sequel.

15.4.4Quantization of Classical Observables

The quantization of a generic function f (q, p) on the phase space is given by theexpression (15.4), which is, for our particular choice of coherent state

f �→ A f =∑κ=±

∫ ∞

–∞

∫ L

0f (q, p)|x , κ〉〈x , κ|N (q, p)dqd p

=∞∑

n,n′=1

|en〉〈en′ | ⊗(

( f +)nn′ 00 ( f –)nn′

), (15.39)

where

( f ±)nn′ =∫ ∞

–∞d p

∫ L

0dq φn,±(q, p) f (q, p) φn′,±(q, p) . (15.40)

For the particular case in which f is a function of p only, f ( p), the operator is givenby

A f =∑κ=±

∫ ∞

–∞

∫ L

0f ( p)|x , κ〉〈x , κ|N (q, p)dqd p

=1

ρ√

π

∞∑n=1

|en〉〈en| ⊗(

( f +)nn′ 00 ( f –)nn′

), (15.41)

with

( f ±)nn′ =∫ ∞

–∞d p f ( p) exp

(–

1ρ2 ( p ∓ pn)2

). (15.42)

Note that this operator is diagonal on the |n, κ〉 basis.


15.4.4.1Momentum and EnergyIn particular, using f ( p) = p , one gets the operator

P == A p =∞∑

n=1

pn σ3 ⊗ |en〉〈en| , (15.43)

where σ3 =(

1 00 –1

).

For f ( p) = p2, which is proportional to the Hamiltonian, the quantum counter-part reads as

A p2 =ρ2

2IK +

∞∑n=1

p2n σ0 ⊗ |en〉〈en | =

ρ2

2IK + ( p )2 . (15.44)

Note that this implies that the operator for the square of the momentum does notcoincide with the square of the momentum operator. Actually they coincide up toO(�2).

15.4.4.2PositionFor a general function of the position f (q) our quantization procedure yields thefollowing operator:

Aq =∞∑

n,n′=1

exp

(–

14ρ2 ( pn – pn′ )2

)[dn–n′ ( f ) – dn+n′ ( f )

]σ0 ⊗ |en〉〈en′ | ,

(15.45)

where

dm ( f ) ==1L

∫ L

0f (q) cos

(mπ

qL

)dq . (15.46)

In particular, for f (q) = q we get the “position” operator

Q == Aq =L2

IK –2Lπ2

∞∑n,n′v1,

n+n′=2k+1

exp

(–

14ρ2 ( pn – pn′ )2

)[1

(n – n′)2 –1

(n + n′)2

]

~ σ0 ⊗ |en〉〈en′ | ,(15.47)

with k ∈ N. Note the appearance of the classical mean value for the position on thediagonal.


15.4.4.3Commutation RulesNow, to see to what extent these momentum and position operators differ fromtheir classical (canonical) counterparts, let us consider their commutator:

[Q , P ] =2�

π

∞∑n=/n′

n+n′=2k+1

C n,n′ σ3 ⊗ |en〉〈en′ | (15.48)

Fig. 15.1 Eigenvalues of Q, here q (a), P, herep (b), and [Q , P ] (c), for increasing values ofthe characteristic momentum ρ = �πϑ/L ofthe system, and computed for N ~ N approxi-mation matrices. Units were chosen such that� = 1 and L = π so that ρ = ϑ and pn = n.Note that for Q with ρ small, the eigenvaluesadjust to the classical mean value L/2. The

spectrum of P is independent of ρ as is shownin (15.43). For the commutator, the valuesare purely imaginary (reprinted from [Garciade Leon, P., Gazeau, J.P., and Quéva, J.: Theinfinite well revisited: coherent states andquantization, Phys. Lett., 372, p. 3597, 2008]with permission from Elsevier).


C n,n′ = exp

(–

14ρ2 ( pn – pn′ )2

)(n – n′)

[1

(n – n′)2 –1

(n + n′)2

]. (15.49)

This is an infinite antisymmetric real matrix. The respective spectra of the finitematrix approximations of this operator and of position and momentum opera-tors are compared in Figures 15.1 and 15.2 for various values of the regulatorρ = �πϑ/L = ϑ in units � = 1 and L = π. When ρ assumes large values, theeigenvalues of [Q , P ] accumulate around±i, that is, they become almost canonical.

Fig. 15.2 Eigenvalues of Q, P, and [Q , P ] ofN ~ N approximation matrices for increas-ingly larger values of ρ = �πϑ/L = ϑ in units� = 1 and L = π. The spectrum of P is in-dependent of ρ as is shown in (15.43). Forthe commutator, the eigenvalues are purely

imaginary and tend to accumulate around i�and –i� as ρ increases (reprinted from [Gar-cia de Leon, P., Gazeau, J.P., and Quéva, J.:The infinite well revisited: coherent states andquantization, Phys. Lett., 372, p. 3597, 2008]with permission from Elsevier).


Conversely, when ρ → 0 all eigenvalues tend to 0, and this behavior correspondsto the classical limit.

15.4.5Quantum Behavior through Lower Symbols

Lower symbols are computed with normalized coherent states. The latter are de-noted as follows:

|x〉 = |x , +〉 + |x , –〉 . (15.50)

Hence, the lower symbol of a quantum observable A should be computed as

A(x) = 〈x |A|x〉 == A++(x) + A+–(x) + A–+(x) + A––(x) .

This gives the following results for the observables previously considered.

15.4.5.1PositionIn the same way, the mean value of the position operator in a vector coherent state|x〉 is given by

〈x |Q |x〉 =L2

–Q(q, p) , (15.51)

where we can distinguish the classical mean value for the position corrected by thefunction

Q(q, p) =2Lπ2

1S

∞∑n,n′=1,n=/n′

n+n′=2k+1

exp

(–

14ρ2 ( pn – pn′ )2

)[1

(n – n′)2 –1

(n + n′)2

]

~

[exp

(–

12ρ2 [( p – pn)2 + ( p – pn′ )2]

)+

+ exp

(–

12ρ2 [( p + pn)2 + ( p + pn′ )2]

)]sin

(nπ

qL

)sin

(n′π

qL

).

(15.52)

15.4.5.2MomentumThe mean value of the momentum operator in a vector coherent state |x〉 is givenby the affine combination:

〈x |P |x〉 =M(x)N (x)

,

M(x) = c∞∑

n=1

pn

[exp

(–

1ρ2 ( p – pn)2

)– exp

(–

1ρ2 ( p + pn)2

)]~ sin2

(nπ

qL

).

(15.53)


15.4.5.3Position–Momentum CommutatorThe mean value of the commutator in a normalized state Ψ =

( φ+φ–

)is the pure

imaginary expression

〈Ψ|[Q , P ]|Ψ〉 =2i�π

∞∑n=/n′

n+n′=2k+1

exp

(–

14ρ2 ( pn – pn′ )2

)(n – n′) ~

~

[1

(n – n′)2 –1

(n + n′)2

]I(〈φ+|n〉〈n′|φ+〉 – 〈φ–|n〉〈n′|φ–〉

).

(15.54)

Given the symmetry and the real-valuedness of states (15.36), the mean value of thecommutator when Ψ is one of our coherent states vanishes, even if the operatordoes not. This result is due to the symmetry of the commutator spectrum withrespect to 0. As is shown on right in Figures 15.1 and 15.2, the eigenvalues of thecommutator tend to ±i� as ρ, or equivalently ϑ, increases. Still, there are somepoints with modulus less than �. This leads to dispersions ΔQΔP in coherentstates |x〉 that are no longer bounded from below by �/2. Actually, the lower boundof this product, for a region in the phase space as large as we wish, decreases as ϑ

diminishes. A numerical approximation is shown in Figure 15.3.

15.4.6Discussion

From the mean values of the operators obtained here, we verify that the coher-ent state quantization gives well-behaved momentum and position operators. Theclassical limit is reached once the appropriate limit for the parameter ϑ is found.If we consider the behavior of the observables as a function of the dimensionlessquantity ϑ = ρL/�π, in the limit ϑ → 0 and when the Gaussian functions forthe momentum become very narrow, the lower symbol of the position operator isQ(q, p) ~ L/2. This corresponds to the classical average value position in the well.On the other hand, at the limit ϑ→∞, for which the Gaussians involved spread toconstant functions, the function Q (q, p) converges numerically to the function q.In other words, the position operator obtained through this coherent state quan-tization yields a fair quantitative description for the quantum localization withinthe well. Clearly, if a classical behavior is sought, the values of ϑ have to be chosennear 0. This gives localized values for the observables. Consistent with these obser-vations, the behavior of the product ΔQΔP for low values of ϑ shows uncorrelatedobservables at any point in the phase space, whereas for large values of this pa-rameter the product is constant and almost equal to the canonical quantum lowestlimit �/2. This is shown in Figure 15.3.

It is interesting to note that if we replace the Gaussian distribution, used herefor the p variable in the construction of the coherent states, by any positive evenprobability distribution R ∈ p �→ π( p) such that

∑n π( p – n) < ∞ the results are


Fig. 15.3 Product ΔQΔP for various valuesof ρ = �πϑ/L = ϑ in units � = 1 and L = π.Note the modification of the vertical scalefrom one picture to another. Again, the po-sition–momentum pair tends to decorrelateat low values of the parameter, as it shoulddo in the classical limit. On the other hand

it approaches the usual quantum-conjugatepair at high values of ρ (reprinted from [Gar-cia de Leon, P., Gazeau, J.P., and Quéva, J.:The infinite well revisited: coherent states andquantization, Phys. Lett., 372, p. 3597, 2008]with permission from Elsevier).


not so different! The momentum spectrum is still Z and the energy spectrum hasthe form {n2 + constant}.

15.5Motion on a Discrete Set of Points

Now let us consider a problem inspired by modern quantum geometry, where geo-metrical entities are treated as quantum observables, as they have to be for them tobe promoted to the status of objects and not to be simply considered as a substantialarena in which physical objects “live”.

The content and terminology of this section were mainly inspired by [201], inwhich a toy model of quantum geometry is described. The game is to rebuild a“shadow” Schrödinger quantum mechanics on all possible discretizations of thereal line. A simple way to do this is to adapt the previous frame quantization of themotion on the circle to an arbitrary discretization of the line. Actually, we deal withthe same phase space as for the motion of the particle on the line, that is, the plane,but provided with a different measure, to get an arbitrarily discretized quantumposition. Let us enumerate the successive steps of the coherent state quantizationprocedure applied to this specific situation.

• The observation set X is the plane R2 = {x == (q, p)}.• The measure (actually a functional) on X is partly of the “Bohr type”:

μ( f ) =∫ +∞

–∞dq lim

T→+∞

1T

∫ T2

– T2

d p f (q, p) . (15.55)

• We choose as an orthonormal system of functions φn(x) suitably weightedFourier exponentials associated with a discrete subset γ = {an} of the realline:

φn(x) =( ε

π

) 14

e– ε2 (q–an )2

e–ian p . (15.56)

Notice again that the continuous distribution x �→ |φn(x)|2 is the normallaw centered at an for the position variable q.

• The “graph” (in the Ashtekar language) γ is supposed to be uniformlydiscrete (there exists a nonzero minimal distance between successive ele-ments) in such a way that the aperiodic train of normalized Gaussians or“generalized” theta function

N (x) == N (q) =

√επ

∑n

e–ε(q–an )2(15.57)

converges. Poisson summation formulas can exist, depending on the struc-ture of the graph γ [207].

15.5 Motion on a Discrete Set of Points 257

• Accordingly, the coherent states read as

|x〉 = |q, p〉 =1√N (q)

∑n

φn (x)|φn〉 . (15.58)

• Quantum operators acting onH are yielded by using

A f =∫

Xf (x)|x〉〈x |N (x)μ(dx) . (15.59)

What are the issues of such a quantization scheme in terms of elementary observ-ables such as position and momentum? Concerning position, the algorithm thatfollows illustrates well the auberge espagnole17) character of our quantization proce-dure (and actually of any quantization procedure), since the outcome is preciselythat space itself is quantized, as expected from our choice (15.56). We indeed obtainfor the position∫

Xμ(dx)N (q) q |q, p〉〈q, p| =

∑n

an |φn〉〈φn| . (15.60)

Hence, the graph γis the “quantization” of space when the latter is viewed from thepoint of view of coherent states precisely based on γ (. . . ). On the other hand, thispredetermination of the accessible space on the quantum level has dramatic conse-quences for the quantum momentum. Indeed, the latter experiences the following“discrete” catastrophe∫

Xμ(dx)N (q) p|q, p〉〈q, p|

=∑n,n′

limT→+∞

[sin

((an – an′ ) T

2

)(an – an′ )

–2T

sin((an – an′ ) T

2

)(an – an′ )2

]~ e– ε

4 (an –an′ )2 |φn〉〈φn′ | , (15.61)

and the matrix elements do not exist for an =/ an′ and are ∞ for an = an′ . Never-theless, a means of circumventing the problem is to deal with Fourier exponentialseiλ p as classical observables, instead of the mere function p. This choice is naturallyjustified by our original option of a measure adapted to the almost-periodic struc-ture of the space generated by the functions (15.56). For a single “frequency” λ weget ∫

Xμ(dx)N (q) eiλ p |q, p〉〈q, p| =

∑n,n′

e– ε4 (an –an′ )

2δλ,an′ –an |φn〉〈φn′ |. (15.62)

17) An auberge espagnole (Spanish inn) is a Frenchmetaphor for a place where you get what youbring.


The generalization to superpositions of Fourier exponentials eiλ p is straightfor-ward. In particular, one can define discretized versions of the momentum by con-sidering coherent state quantized versions of finite differences of Fourier exponen-tials

1i

eiλ′ p – eiλ p

λ′ – λ, (15.63)

in which “allowed” frequencies should belong to the set of “interpositions” γ – γ′

in the graph γ.As a final comment, let us repeat here the remark we made at the end of the

previous section: the Gaussian distribution in (15.56) is not the unique choice wehave at our disposal. Any even probability distribution q �→ π(q) such that

∑n π(q –

an) <∞ will yield similar results.

259

16Quantizations of the Motion on the Torus

16.1Introduction

This chapter is devoted to the coherent states associated with the discrete Weyl–Heisenberg group and to their utilization for the quantization of the chaotic motionon the torus. The material presented here is part of a work by Bouzouina and DeBièvre [208] on the problem of the equipartition of the eigenfunctions of quantizedergodic maps on the torus. We describe two quantization procedures. The firstone is based on those coherent states, along the lines of the book. The secondone is an adaptation of the Weyl quantization to the underlying discrete symmetry.We then examine the classical limit of certain area-preserving ergodic maps onthe two-dimensional torus T2, viewed as a phase space, with canonical coordinates(q, p) ∈ [0, a) ~ [0, b). We will see how the desired equipartition result can be easilyproved for a large class of models.

The content of this chapter, although more mathematically involved, was selectedto show the wide range of applications of coherent states in connection with chaoticdynamics on both quantum and classical levels.

16.2The Torus as a Phase Space

The two-dimensional torus T2 is the Cartesian product of two circles, T2 ~= S1 ~ S1.It is considered here as the Cartesian product of two cosets:

T2 = R/aZ ~ R/bZ = R2/Γ , a > 0 , b > 0 , (16.1)

where Γ is the lattice Γ = {(ma, nb) , (m, n) ∈ Z2}. Its topology is that of the Carte-sian product of two semiopen intervals: T2 = [0, a) ~ [0, b) = {(q, p), 0 u q < a, 0 up < b}. The variable q could be considered as the position (or angle), whereas pwould be the momentum. In this sense, the torus is a phase space or symplec-tic manifold equipped with the 2-form ω = dq ∧ dp and the resulting normalizedmeasure reads as μ = dq dp/(ab).


260 16 Quantizations of the Motion on the Torus

Let us consider invertible maps Φ : T2 �→ T2 preserving the measure in thesense that μ

(Φ–1E

)= μ(E ) for all measurable set E ⊂ T2. A first example of such

transformations is provided by the irrational translations of the torus. Given twonumbers, α1 and α2, that are incommensurable to the periods a and b, respectively,that is, α1 /∈ a Q and α2 /∈ b Q, the transformation

τα : T2 � (q, p) �→ (q + α1, p + α2) ∈ T2 , α == (α1, α2) (16.2)

clearly leaves invariant the measure μ.Another elementary transformation, which we call skew translation (or momen-

tum affine) and denote by Φ�,k , is also measure-preserving:

Φ�,k = τ(0,�) ◦ K , � /∈ bQ , K =

(1 k0 1

), k ∈ Z . (16.3)

The matrix K is an element of the group SL(2, Z) and acts on the pairs (q, p) viewedas 2-vectors.

On a more general level, we consider the hyperbolic automorphisms of the torus.They are defined by matrices of the type

A =

(α � a

bγ b

a δ

), A

(qp

)=

(αq + � a

b pγ b

a q + δ p

), α , � , γ , δ ∈ Z , (16.4)

with |Tr A| = |α + δ| > 2 and det A = αδ – �γ = 1. As solutions of λ2 – λ Tr A + 1 = 0,

the eigenvalues λ± = 12

(Tr A ±

√(Tr A)2 – 4

)are quadratic algebraic integers such

that |λ+| > 1 and λ– = 1/λ+. This action on the torus, known as Anosov diffeomor-phism, stretches in the eigendirection corresponding to λ+, whereas it contracts inthe direction determined by λ–. It gives rise to deformations of figures such as thefamous Arnold cat. The set of such transformations forms the subgroup of SL(2, R)leaving invariant the lattice Γ . This subgroup is isomorphic to SL(2, Z). There existperiodic points in the torus for such hyperbolic actions. They are given by

x =( q

a,

pb

)∈ Q2/Z2 .

We have shown three examples of invertible transformations Φ of the torus that

are measure-preserving. The triplet σ def= (T2, Φ, μ) is a dynamical system. It is saidto be ergodic if for all integrable functions on the torus, f ∈ L1(T2, μ), its “temporal”average is equal to its “spatial” average:

μ( f ) def=∫

T2

f (x) μ(dx)

“spatial”

= limT→∞

1T

T –1∑k=0

f ◦Φk(x)

“temporal”

a.e. (16.5)

Now, there holds a general result for characterizing ergodicity of a dynamicalsystem (X , Φ, μ), where X is a measure space with measure μ, here X = T2, and Φis an invertible and measure-preserving map.

16.3 Quantum States on the Torus 261

Theorem 16.1

The dynamical system σ = (T2, Φ, μ) is ergodic if and only if at least one of thefollowing conditions is fulfilled:

(i) Any measurable set E ⊂ T2, which is Φ-invariant (i.e., Φ–1E = E ), is suchthat μ(E ) = 0 or μ(T2 \ E ) = 0.

(ii) If f ∈ L∞(T2, μ), that is, is bounded almost everywhere with respect to μ, isΦ-invariant, that is, f ◦ Φ = f , then it is constant almost everywhere withrespect to μ.

If σ is ergodic, then the set of periodic orbits of the map Φ is μ-negligible.Within this ergodicity context, one notices that the hyperbolic automorphims (16.4)are mixing. Generally, a map Φ is mixing if for all f , g ∈ L2(T2, μ) the followingcondition is fulfilled:

limk→∞

∫T2

f (Φk(x)) g (x) μ(dx) = μ( f ) μ(g ) . (16.6)

In particular, by choosing for f and g the characteristic functions f = �E 1 and g =�E 2 of two measurable sets E 1, E 2 ⊂ T2, a mixing map verifies

limk→∞

μ((

ΦkE 1)∩ E 2

)= μ(E 1) μ(E 2) . (16.7)

In other words, the part of the set ΦkE 1 intersecting E 2 has a measure that isasymptotically proportional to the measure of E 2, which means that E 1 (if μ(E 1) >0) spreads uniformly in T2.

16.3Quantum States on the Torus

To set up the quantum states on the torus, let us first recall two main features ofquantum states on the plane viewed as a phase space:

(i) The (pure) quantum states of a particle having the line R as a configurationspace and so the plane R2 as a phase space are most generally tempereddistributions ψ ∈ S′(R), or simply S′, for example, plane waves or Diracdistributions, and, when normalizable, are elements of the Hilbert spaceL2(R), the latter being viewed as a subspace of S′(R).

(ii) The translations acting on the classical phase space realize, on the quan-tum level, as transformations of L2(R) that belong to a unitary irreduciblerepresentation of the Weyl–Heisenberg group

Weyl–Heisenberg group � (φ, q, p) �→ U (φ, q, p) = e– i� φ e

i� pQ–qP︸︷︷︸==D (z)

,

(16.8)


with the notational modification φ == –s with regard to Chapter 3. The com-position law here reads as

U (φ, q, p) U (φ′, q′, p ′) = U(φ + φ′ + 1

2 (q p ′ – pq′), q + q′, p + p ′)

.

Such a representation is extendable to tempered distributions.

Let us adapt this Weyl–Heisenberg formalism to the torus. Since the phase spaceis now T2 = R2 \ Γ , with Γ = {(ma, nb), (m, n) ∈ Z2}, it is justified to impose thefollowing periodic conditions on the quantum states ψ:

U (a, 0) ψ = e–iκ1a ψ , U (0, b) ψ = eiκ2b ψ , (16.9)

where, for simplification, U (q, p) stands for U (0, q, p). The pairs

κ = (κ1, κ2) ∈[

0,2πa

)~

[0,

2πb

)(16.10)

are elements of what we naturally call the reciprocal torus T 2.A crucial point of the construction is that, with the periodicity conditions (16.9),

the quantization is not trivial if and only if the torus is “quantized” in the sense thatthere exists N ∈ N such that

a b = 2π�N . (16.11)

Then, the states ψ assume in “position” representation the following form:

ψ =

√aN

∑m∈Z

cm(ψ) δxm , with xm =ab

2πNκ2 + m

aN

, (16.12)

where the expansion coefficients are subjected to the consistent periodicity condi-tion cm+N (ψ) = eiκ1acm(ψ).

The representation (16.12) is unique. Let us prove this statement.Condition (16.9) means that ψ is a common eigenfunction of U (a, 0) and

U (0, b)). Therefore,

U (0, b) U (a, 0)ψ = U (a, 0) U (0, b)ψ = e– i� ab U (0, b) U (a, 0)ψ ,

the latter equality resulting from the Weyl–Heisenberg commutation rule on thegroup level. It implies the quantization of the torus: ab

2π� = N ∈ N.Next, we have for ψ ∈ S′

U (0, b)ψ(x) = ei� bQ ψ(x) = e

i� bxψ(x) = eiκ2b ψ(x) ,

and thus(e

ib� (x–�κ2) – 1

)ψ(x) = 0 .

16.3 Quantum States on the Torus 263

The solutions to this equation in the space of tempered distributions S′ read astranslated Dirac distributions:

ψ ∝ δxm , with xm = �κ2 + 2πm�

b=

ab2πN

κ2 + maN

.

Hence, by superposition,

ψ =

√aN

∑m∈Z

cm (ψ)δxm ,

where we have included the global normalization factor√

aN . Finally, from the

eigenvalue equation U (a, 0)φ = e–iκ1aψ and the action U (a, 0)δxm (x) = δxm (x + a) =δxm–a(x) = δxm–N (x), we easily derive the periodicity condition on the coefficientsthemselves:

cm+N (ψ) = eiκ1acm(ψ) .

�The linear span of these solutions is a vector space of dimension N, which we

denote by S′(κ, N ). Let us say more about it. First, it can be obtained from theSchwartz space S(R) of rapidly decreasing smooth functions through the sym-metrization operator Pκ defined as follows:

S(R) � ψ Pκ�→ Pκψ =∑

m,n∈Z

(–1)N mnei(κ1ma–κ2 nb) U (ma, nb)ψ ∈ S′(κ, N ) .

(16.13)

The proof of S′(κ, N ) = Pκ S(R) is based on the Poisson summation formula.As a Hilbert space, S′(κ, N ) is equipped with the scalar product

〈ψ|ψ′〉κ,N =N –1∑j=0

c j (ψ)c j (ψ′) . (16.14)

This Hilbert space will be denoted by H�(κ). With respect to this inner product,the operators U

(n1

aN , n2

bN

), (n1, n2) ∈ Z2 are unitary. They represent elements of

the form(2π� s, n1

aN , n2

bN

), with s ∈ Z. Such elements form, thanks to (16.11),

a discrete subgroup of the Weyl–Heisenberg group, sometimes called the discreteWeyl–Heisenberg group. Note that the first element of the triplet has no effectiveaction since it is represented as a unit phase, and so can be omitted in the notation.Now, it is a general result of group representation theory that this unitary repre-sentation decomposes into a direct sum of unitary irreducible representation Uκ

of the discrete Weyl–Heisenberg group:

U

(n1

aN

, n2bN

)=

∫ 2πa

0

∫ 2πb

0ν(dκ) Uκ

(n1

aN

, n2bN

), (16.15)


where

ν(dκ) def=ab

(2π)2 d2κ . (16.16)

Consistently, we have the Hilbertian decomposition:

L2(R) ~=

∫ 2πa

0

∫ 2πb

0ν(dκ)H�(κ) , (16.17)

where the Hilbert space H�(κ) carries the representation Uκ. This decompositionactually reflects the well-known Bloch theorem for wave functions in a periodicpotential. It is proven that the representations Uκ and Uκ′ are not equivalent ifκ =/ κ′. We thus obtain a continuous family of nonequivalent quantizations labeledby the pairs κ = (κ1, κ2).

Let us turn our attention to the momentum representation F�S′(κ, N ), obtainedby Fourier transform,

ψ( p) = (F�ψ) ( p) =1√2π�

∫ +∞

–∞e– i

� px ψ(x) dx ,

from the above representation. Such a Fourier transform is just the restriction toS′(κ, N ) of the Fourier transform of tempered distributions, F� : S′ �→ S′. Theelements of F�S′(κ, N ) have the following form:

ψκ =

√bN

∑n∈Z

dn (ψ) δ�n with �n =ab

2πNκ1 + n

bN

. (16.18)

Similarly to the position representation, the coefficients obey the periodicity condi-tion

dn+N (ψ) = e–iκ2b dn(ψ) . (16.19)

The relation between respective coefficients in position and momentum represen-tations is given by

dn(ψ) =1√N

e–i κ2bN ( aκ1

2π +n)N –1∑m=0

cm(ψ)e–im( 2πN n+ aκ1

N ) . (16.20)

Since we are dealing with N-dimensional Hilbert spaces (i.e., Hermitian spaces)H�(κ), it is useful to establish the one-to-one correspondence between the abovefunctional representations and the Hermitian space CN . This correspondence isdefined through the following set of N “orthogonal” Dirac combs in S′ put in one-to-one correspondence with the canonical basis of CN :

eκj =

√aN

∑m

e–imaκ1 δx jm

, 0 u j u N – 1 , (16.21a)

f κk =

√bN

∑n

e–inbκ2 δ�kn

, 0 u k u N – 1 , (16.21b)

with x jm = abκ2

2πN + j aN – ma and �k

n = abκ12πN + k b

N – nb .

16.4 Coherent States for the Torus 265

16.4Coherent States for the Torus

The coherent states for the torus, here denoted by ψZ(q, p), are obtained by applyingthe operator Pκ to the standard coherent states. However, as a preliminary, we mod-ify the definitions and notation of the latter to adapt them to the present context.First, we modify the group-theoretical construction à la Perelomov of the standardcoherent states by choosing as a ground state in position representation a phase-modulated Gaussian function:

ψZ(0,0)(x) def=

(IZπ�

) 14

ei RZ2� x2

e– IZ2� x2

, (16.22)

whereZ is a complex parameter with IZ > 0. We then define coherent states as theunitary transport of this ground state by the displacement operator. In the positionrepresentation they read as

ψZ(q, p)(x) def= ei� ( pQ–qP ) ψZ(0,0)(x) =

(IZπ�

) 14

e– i pq2� e

i� px ei Z2� (x–q)2

. (16.23)

These functions clearly belong the Schwartz space S . We recall the resolution ofthe unity in L2(R):∫

R2

dq d p2π�

∣∣ψZ(q, p)

⟩ ⟨ψZ(q, p)

∣∣ = Id . (16.24)

The coherent states on the torus are then defined through the operator Pκ:

ψZ,κ(q, p)

def= Pκ ψZ(q, p) =N –1∑j=0

c j (q, p) eκj , (16.25)

where the parameters (q, p) are now restricted to the torus, (q, p) ∈ [0, a) ~ [0, b),and the coefficients in the expansion formula are given by

c j (q, p) =

(IZπ�

) 14

√aN

e– i pq2�

∑m∈Z

eiκ1ma e– i� x j

m e– iZ2�

(x j

m –q)2

,

with x jm = j

aN

+abκ2

2πN– ma . (16.26)

The convergence of the series holds because of the condition IZ > 0. These co-efficients are suitably expressed in terms of the elliptic theta function of the thirdkind, ϑ3 [209], which is defined as

ϑ3(x ; τ) =∑n∈Z

τn2e2inx , |τ| < 1 . (16.27)

Therefore,

c j (q, p) =

(IZπ�

) 14

√aN

e– i pq2� e

i� p( j a

N + ab2πN κ2–q) ϑ3(X j (q, p); τ) , (16.28)


with X j =aκ1

2–

a p2�

–aZ2�

(j

aN

+ab

2πNκ2 – q

),

and τ = ei Za22� .

(16.29)

The norm of the coherent state (16.25) is computed in accordance with (16.14):

‖ψZ,κ(q, p)‖2

H� (κ) =

(IZπ�

) 12 a

N

N –1∑j=1

∣∣ϑ3(X j ; τ)∣∣2 == N (q, p) . (16.30)

In concordance with the general construction, illustrated in Chapter 3, of coher-ent states on a measure space,

(T2, μ(dx = dq d p/(2π�))

)in the present case, these

coefficients form a finite orthogonal set in the Hilbert space L2(T2, dq d p/(2π�)

)and an orthogonal basis for the N-dimensional subspace isometric to H�(κ). Thedirect demonstration is rather cumbersome and we will omit it here. Furthermore,since one is working here with a representation of the states as tempered distribu-tions, the question of normalization is not pertinent and it is preferable to appeal tothe Schur lemma and discrete Weyl–Heisenberg unitary irreducible representationarguments to prove that the following resolution of the identity holds inH�(κ):∫

T2

dq d p2π�

∣∣ψZ,κ(q, p)

⟩ ⟨ψZ,κ

(q, p)

∣∣ = IH�(κ) . (16.31)

The isometric map W (κ,Z) : H�(κ) �→ W (κ,Z)H�(κ) ⊂ L2(

T2, dq d p2π�

)is pre-

cisely defined by

H�(κ) � ψ �→W (κ,Z)ψ def=⟨

ψZ,κ(q, p) | ψ

⟩. (16.32)

The range of W (κ,Z) is a reproducing kernel Hilbert subspace of L2(

T2, dq d p2π�

).

Note two important properties of these coherent states, proved in [208], whichwill be used for deriving some estimates in the sequel.

An estimate The overlap of two coherent states obeys the following inequality:

∃C > 0 such that ∃α > 0∣∣∣⟨ψZ,κ

(q, p)|ψZ′ ,κ(q′ , p ′)

⟩∣∣∣ u C e–αN , (16.33)

where α is a function of |q – q′| and |p – p ′|.

A “classical” limit In the limit of large N and by fixing the product N�, we get thefollowing limit for the norm of coherent states:

limN→∞ ,N�=cst

‖ψZ,κ(q, p)‖H� (κ) = 1 . (16.34)

16.5 Coherent States and Weyl Quantizations of the Torus 267

16.5Coherent States and Weyl Quantizations of the Torus

16.5.1Coherent States (or Anti-Wick) Quantization of the Torus

Once the resolution of the identity (16.31) on the torus has been proved, it be-comes possible to proceed to the coherent state (anti-Wick) quantization. With anyessentially bounded observable f (q, p), that is, f (q, p) ∈ L∞(T2, μ), we associatethe operator

OpC S�,κ( f ) def=

∫T2

dq dp2π�

f (q, p)∣∣ψZ,κ

(q, p)

⟩ ⟨ψZ,κ

(q, p)

∣∣ . (16.35)

This operator is defined for all κ ∈ T 2 and for all � > 0. Let us be more preciseby stating the following properties of the coherent state quantization in the presentcontext:

(i) OpC S�,κ†( f ) = OpC S

�,κ( f ) ,(ii) If f ∈ C∞(T2), then we can extend the quantization to the whole Hilbert

space (16.17) as follows [208]:

OpC S� ( f ) =

∫ 2πa

0

∫ 2πb

0ν(dκ) OpC S

�,κ( f ) ,

where the operator OpC S� ( f ) results from the standard coherent state quan-

tization,

OpC S� ( f ) def=

∫R2

dq d p2π�

f (q, p)∣∣ψZ(q, p)

⟩ ⟨ψZ(q, p)

∣∣ . (16.36)

(iii) For all f ∈ C∞(T2) and κ ∈ T 2, we have the estimate∥∥OpC S�,κ( f )

∥∥L(H�(κ))

u ‖ f ‖∞ .

(iv) The following semiclassical behavior for the lower symbols holds:

limN→∞

⟨ψZ,κ

(q, p)|OpC S�,κ( f ) ψZ,κ

(q, p)

⟩= f (q, p) ∀ (q, p) ∈ T2 and f ∈ C∞(T2) .

(16.37)

16.5.2Weyl Quantization of the Torus

The Weyl quantization for the motion of a particle on the line consists in mappingthe space S0 of smooth functions that have bounded derivatives at any order,

S0 def={

f ∈ C∞(R2) , all derivatives are bounded}

, (16.38)


into the space L(S(R)) of bounded operators in the Schwartz space. It is defined asfollows:

S0 � f �→ OpW� ( f ) ∈ L(S(R)) ,

S � ψ �→ OpW� ( f ) ψ == ψ′ , ψ′(x) =

∫R

d y dp2π�

ei� (x–y ) p f

(x + y2

, p)

ψ(y ) .

(16.39)

Equivalently,

OpW� ( f ) =

∫R2

dq dp2π�

f (q, p)ei� ( pQ–qP ) , (16.40)

where f stands for the symplectic Fourier transform of f:

f (q, p) def=∫

R2

ei� (q�– px) f (x , �) . (16.41)

It has been proven [210] that the map OpW� has a unique extension to the tempered

distributions S′(R) and yields bounded operators in L2(R).The Weyl quantization on the torus T2 is just the particularization of (16.40) to

the discrete Weyl–Heisenberg group:

S′(R2) ⊃ S0(R2) ⊃ C∞(T2) � f �→ OpW� ( f ) ,

OpW� ( f ) def=

∑m,n

f m,n ei( 2πa nQ– 2π

b mP ) =∑m,n

f m,n U

(maN

,nbN

),

(16.42)

with coefficients f m,n defined by f (q, p) =∑

m,n f m,nei( 2πa nq– 2π

b m p).Like for the coherent state quantization, let us list the main properties of the

Weyl quantization of the torus:

(i) The restriction of OpW� ( f ) to the Hilbert space H�(κ) is well defined from

the fact that OpW� ( f )H�(κ) ⊂ H�(κ) for all f ∈ C∞(T2).

(ii) The following decomposition results:

OpW� ( f ) =

∫ 2πa

0

∫ 2πb

0ν(dκ) OpW

�,κ( f ) .

(iii) The following estimate for the difference between the Weyl quantization ofthe product of two functions and the product of the two respective Weylquantizations of the functions holds:

For all f , g ∈ C∞(T2) there exists C > 0 such that

∀N v 1 ,∥∥∥OpW

�,κ( f ) OpW�,κ†(g ) – OpW

�,κ( f g)∥∥∥L(H�(κ))

uCN

.

16.6 Quantization of Motions on the Torus 269

(iv) In the limit N →∞, we have for the trace

limN→∞

1N

tr(OpW

�,κ( f ))

=∫

T2

dq dpab

f (q, p) .

(v) There exists the following estimate for the difference between Weyl and co-herent state quantizations:∥∥OpW

�,κ( f ) – OpC S�,κ( f )

∥∥L(H�(κ))

= O(N –1

). (16.43)

In consequence, it can be asserted that the Weyl quantization and the coher-ent state quantization are identical up to �:∥∥OpW

� ( f ) – OpC S� ( f )

∥∥ = O (�) . (16.44)

16.6Quantization of Motions on the Torus

We are now equipped to proceed with the quantization of the symplectic transfor-mations of the torus introduced in the first section of this chapter.

16.6.1Quantization of Irrational and Skew Translations

These transformations are proved to be ergodic and uniquely ergodic in the sensethat there exists one and only one Φ-invariant probability measure.

Any irrational translation τα, α = (α1, α2), α1/α2 /∈ Q, is unitarily represented bythe operator Mκ (τα) onH�(κ) defined as

Mκ (τα) = ei α1 α22� Mκ

(τ(α1,0)

)Mκ

(τ(0,α2)

), (16.45)

with

Mκ

(τ(0,α2)

)eκ

jdef= ei(κ2+ 2π

b j)α1 eκj ,

and

Mκ

(τ(α1,0)

)f κ

jdef= e–i(κ1+ 2π

a j)α1 f κj .

The decomposition formula and the eigenequations are consistent with U (α1, α2) =ei α1α2

2� U (α1, 0) U (0, α2) and the fact that U (α1, 0) (resp. U (0, α2)) is diagonal in themomentum (or spatial) representation.

Concerning the skew translations, we have the representation

Φ�,k �→ Mκ

(Φ�,k

)= Mκ

(τ(0,�)

)◦Mκ(K ) , (16.46)

where Mκ(K ), for

K =

(1 k0 1

)realizes a quantization of the group SL(2, Z) that is explained in the next paragraph.


16.6.2Quantization of the Hyperbolic Automorphisms of the Torus

The task now is to build a unitary representative of the hyperbolic transformationof the torus:

A =

(α � a

bγ b

a δ

)�→ unitary M(A) .

It is proven that for all � > 0 and for all κ ∈ T 2, there exists κ′ ∈ T 2 such thatM(A)H�(κ) ⊂ H�(κ′), where(

κ′1κ′2

)= A

(κ1

κ2

)+ πN

( α �b

γ δa

)mod

(2πb

2πa

),

and where the map

A =

(a1 a2

a3 a4

)�→ M(A)

is, in general, defined on any function ψ ∈ S(R) by

(M(A) ψ)(x) =

(i

2π�

)1/2 ∫ +∞

–∞e

i� S(x ,y ) ψ(y ) d y ,

S(x , y ) def=12

a4

a2x2 –

1a2

x y +12

a1

a2y 2 .

(16.47)

The operator M(A) is the quantum propagator associated with the classical discretedynamics defined by the matrix A. It is extended by duality to the space of tempereddistributions S′(R). It fulfills the important intertwining property

M(A) U (q, p) (M(A))† = U

(A

(qp

)). (16.48)

Let us now restrict the transformations A to be the element of the group SL(2, Z)and such that |Tr A| > 2. The following was proved in [208]:

Proposition 16.1

For all � > 0, there exists κ ∈ T 2 such that the unitary representatives M(A)stabilize the Hilbert space H�(κ) of quantum states on the torus,

M(A)H�(κ) ⊂ H�(κ) ,

if and only if A ∈ SL(2, Z) has the form

A =

(even odd ~ a

bodd ~ b

a even

)or A =

(odd even ~ a

beven ~ b

a odd

).

Note that the 2-component momentum κ can be chosen independently of �.

16.6 Quantization of Motions on the Torus 271

16.6.3Main Results

The applications of the previous results concern particularly the control of the per-turbations of the hyperbolic automorphisms of the torus T2, the demonstration ofa semiclassical Egorov theorem on the relation between quantization and temporalevolution, the propagation of coherent states on the torus, and the semiclassicalbehavior of the spectra of evolution operators of the type Mκ(Φ) ∈ U

(H�(κ)

).

Let us say more about this semiclassical Egorov theorem. It means that quanti-zation and temporal evolution commute up to �. More precisely,

Theorem 16.2

Let H be a smooth observable on the torus, that is, H ∈ C∞(T2). For all f ∈C∞(T2), for all t ∈ R, and for all κ ∈ T 2, we have the following estimate:∥∥∥e–i t

� Op�,κ(H ) Op�,κ( f ) e–i t� Op�,κ(H ) – Op�,κ

(f ◦ψH

t

)∥∥∥L(H�(κ))

= O (N –1) ,

(16.49)

where ψHt is the classical flow associated with the observable H, and the notation

Op�,κ stands for both types of quantization (coherent state or Weyl).

This theorem is exact in the case of hyperbolic automorphisms of the torus and theWeyl quantization:

OpW�,κ( f ◦ A) = (M(A))†OpW

�,κ( f )M(A) . (16.50)

Concerning the spectral properties of the evolution operator, one gets preciseinformation on the semiclassical behavior of the eigenvalues eiθN

j , 1 u j u N 1,and the corresponding eigenfunctions φiθN

j , 1 u j u N 1, of the unitary operatorsMκ(Φ) representative of automorphisms Φ : T2 �→ T2.

Proposition 16.2

[[211]] Let Φ a hyperbolic automorphism of the torus. Then the eigenvalues of theunitary representative Mκ(Φ) are Lebesgue uniformly distributed on the unit circleat the semiclassical limit:

limN→∞

# { j , θNj ∈ [θ0, θ0 + η]}

N= λ[θ0, θ0 + η] , (16.51)

where λ is the Lebesgue measure.

This result also holds for the Hamiltonian perturbations of Φ.Finally, let us end this chapter by formulating the essential equipartition result in

regard to the initial purpose to examine ergodicity on the torus on a semiclassical


level. This amounts to examining the possible limits at N → ∞ of the matrixelements (and particularly the lower symbols)⟨

φNk |Op�,κ( f )|φN

j

⟩of the quantized versions of observables for both types of quantizations.

Theorem 16.3

Suppose that for all smooth observable f on the torus, that is, f ∈ C∞(T2), for allk ∈ N, there exists C > 0 so that for all N∥∥Mκ (Φ)–k Op�,κ( f ) Mκ (Φ)k – Op�,κ

(f ◦Φk

)∥∥L(H�(κ))

uCN

. (16.52)

Here Φ is an area-preserving map on the torus such that ∀N∃κ ∈[[0, 2π

a

)~[

[0, 2πb

)and the unitary representative Mκ(Φ) satisfies (16.52).

Write

Mκ(Φ) φNj = λN

j φNj

for the eigenvalues and eigenfunctions of Mκ(Φ).Then there exists a set of indices E (N ) ⊂ [1, N ] satisfying lim

N→∞# E (N )

N = 1 such

that, for all functions f ∈ C∞(T2) and for all maps J : N ∈ N∗ �→ J (N ) ∈ E (N ),we have the ergodicity property of the lower symbols:

limN→∞

⟨φNJ (N )

∣∣ Op�,κ( f )∣∣φNJ (N )

⟩=

∫T2

f (q, p)dq dp

ab, (16.53)

uniformly with respect to the map J .

273

17Fuzzy Geometries: Sphere and Hyperboloid

17.1Introduction

This is an extension to the sphere S2 and to the one-sheeted hyperboloid in R3, the“1 + 1 de Sitter space-time,” of the quantization of the unit circle as was describedin Chapter 14. We show explicitly how the coherent state quantization of thesemanifolds leads to fuzzy or noncommutative geometries. The quantization of thesphere rests upon the unitary irreducible representations D j of SO (3) or SU (2),j ∈ N/2. The quantization of the hyperboloid is carried out with unitary irreduciblerepresentations of SO0(1, 2) or SU (1, 1) in the principal series, namely, U– 1

2 +iρ,ρ ∈ R. The limit at infinite values of the representation parameters j and ρ, restorescommutativity for the geometries considered.

It should be stressed that we proceed to a “noncommutative” reading of a givengeometry and not of a given dynamical system. This means that we do not considerin our approach any time parameter and related evolution.

17.2Quantizations of the 2-Sphere

In this section, we proceed to the fuzzy quantization [212] of the sphere S2 by usingthe coherent states built in Chapter 6 from orthonormal families of spin sphericalharmonics

(σY jm

)– j�m� j

. We recall that for a given σ such that 2σ ∈ Z and jsuch that 2|σ| � 2 j ∈ N there corresponds the continuous family of coherentstates (6.9) living in a (2 j + 1)-dimensional Hermitian space. For a given j, we thusget 2 j +1 realizations, corresponding to the possible values of σ. These realizationsyield a family of 2 j + 1 nonequivalent quantizations of the sphere. We show inparticular that the case σ = 0 is singular in the sense that it maps the Cartesiancoordinates of the 2-sphere to null operators.

We then establish the link between this coherent state quantization approach tothe 2-sphere and the original Madore construction [178] of the fuzzy sphere andwe examine the question of equivalence between the two procedures. Note thata construction of the fuzzy sphere based on Gilmore–Perelomov–Radcliffe coher-


274 17 Fuzzy Geometries: Sphere and Hyperboloid

ent states (in the case σ = j ) was also carried out by Grosse and Prešnajder [213].They proceeded with a covariant symbol calculus à la Berezin with its correspond-ing �-product. However, their approach is different from the coherent state quanti-zation illustrated here.

Appendix C contains a set of formulas concerning the group SU (2), its unitaryrepresentations, and the spin spherical harmonics, specially needed for a completedescription of our coherent state approach to the 2-sphere.

17.2.1The 2-Sphere

We now apply our coherent state quantization to the unit sphere S2 viewed as anobservation set. Our approach should not be confused with the quantization ofthe phase space for the motion on the 2-sphere (i.e., quantum mechanics on the2-sphere; see, e.g., [214, 216]). A point of X is denoted by its spherical coordinates,x == r = (θ, φ). Through the usual embedding in R3, we may view x as a vectorr = (xi ) ∈ R3, the “ambient” space, obeying

∑3i=1(xi )2 = 1. We equip S2 with the

usual (nonnormalized) SO (3)-invariant measure μ(dx) = sin θdθdφ.

17.2.2The Hilbert Space and the Coherent States

At the basis of the coherent state quantization procedure is the choice of the Her-mitian space H = Hσ j , which is a subspace of L2(S2), and which carries a (2 j + 1)-dimensional unitary irreducible representation of the group SU (2). We recall thatH = Hσ j is the vector space spanned by the spin spherical harmonics σY jμ ∈L2(S2). With the usual inner product of L2(S2), the spin spherical harmonics pro-vide an orthonormal basis of Hσ j . We also recall that the special case σ = 0 corre-sponds to the ordinary spherical harmonics 0Y jm = Y jm .

The spin spherical harmonic basis allows us to identifyHσ j with C2 j+1:

σY jμ � |μ〉 ↪→ (0, . . . , 0, 1, 0, . . . , 0)t , with μ = – j , – j + 1, . . . , j ,

(17.1)

where the 1 is at position μ and the superscript t denotes the transpose.The normalized sigma-spin coherent states associated with the spin spherical

harmonics are given by

|x〉 == |r ; σ〉 = |θ, φ; σ〉 =1√N (r )

j∑μ=– j

σY jμ(r)|σ jμ〉 , |r ; σ〉 ∈ Hσ j , (17.2)

with

N (r) =j∑

μ=– j

|σY jμ(r)|2 =2 j + 1

4π.

In the sequel, we will keep the shortcup notation |x〉 for |r ; σ〉 as far as possible.

17.2 Quantizations of the 2-Sphere 275

17.2.3Operators

Let us denote by Oσ j == End(Hσ j ) the space of linear operators (endomorphisms)acting on Hσ j . This is a complex vector space of dimension (2 j + 1)2 and an alge-bra for the natural composition of endomorphisms. The spin spherical harmonicbasis allows us to write a linear endomorphism of Hσ j (i. e. an element of Oσ j ) ina matrix form. This provides the algebra isomorphism

Oσ j ~= M(2 j + 1, C) == M2 j+1 ,

the algebra of complex matrices of order 2 j + 1, equipped with the matrix product.The projector |x〉〈x | is a particular linear endomorphism of Hσ j , that is, an ele-

ment ofOσ j . Being Hermitian by construction, it may be seen as a Hermitian ma-trix of order 2 j + 1, that is, an element of Herm2 j+1 ⊂ M2 j+1. Note that Herm2 j+1

and M2 j+1 have respective (complex) dimensions ( j + 1)(2 j + 1) and (2 j + 1)2.

17.2.4Quantization of Observables

According to our now-familiar prescription, the coherent state quantization asso-ciates with the classical observable f : S2 �→ C the quantum observable

f �→ A f =∫

S2

μ(dx) f (x)N (x)|x〉〈x |

=j∑

μ,ν=– j

∫S2

μ(dx) f (x)σY jμ(x)σY jν(x)|μ〉〈ν| . (17.3)

This operator is an element of Oσ j . Of course its existence is subjected to the con-vergence of (17.3) in the weak sense as an operator integral. The expression abovegives directly its expression as a matrix in the spin spherical harmonic basis, withmatrix elements (A f )μν:

A f =j∑

μ,ν=– j

(A f )μν|μ〉〈ν|

with (A f )μν =∫

S2

μ(dx) f (x)σY jμ(x)σY jν(x) . (17.4)

When f is real-valued, the corresponding matrix belongs to Herm(2 j+1). Also, wehave A f = (A f )† (matrix transconjugate), where we have used the same notationfor the operator and the associated matrix.

Note the interesting expression of the lower symbol of A f , easily obtained fromthe expression (6.81) of the overlap of two sigma-spin coherent states, and viewed


as a sort of Jacobi transform on the sphere of the function f:

〈x |A f |x〉 =∫

S2

μ(dx ′) f (x ′)N (x ′)|〈x |x ′〉|2

=2 j + 1

4π

∫S2

sin θ dθ dφ(

1 + r · r ′2

)2σ ∣∣∣P (0,2σj–σ (r · r ′)

∣∣∣2f (r ′) .

(17.5)

Also note that the map (17.3) can be extended to a class of distributions on thesphere, in the spirit of Chapter 12.

17.2.5Spin Coherent State Quantization of Spin Spherical Harmonics

The quantization of an arbitrary spin harmonic νY kn yields an operator in Hσ j

whose (2 j + 1) ~ (2 j + 1) matrix elements are given by the following integral result-ing from (17.3):[

Aν Y kn

]μμ′

=∫

XσY ∗jμ(x)σY jμ′ (x)νY kn(x)μ(dx)

=∫

X(–1)σ–μ

–σY j–μ(x)σY jμ′ (x)νY kn(x) μ(dx) . (17.6)

17.2.6The Usual Spherical Harmonics as Classical Observables

As asserted in Appendix C (Section C.7), it is only when ν – σ + σ = 0, that is, whenν = 0, that the integral (17.6) is given in terms of a product of two 3 j symbols.Therefore, the matrix elements of AY �m in the spin spherical harmonic basis aregiven in terms of the 3 j symbols by

(AY �m

)μν

= (–1)σ–μ(2 j + 1)

√(2� + 1)

4π

(j j �

–μ ν m

)(j j �

–σ σ 0

). (17.7)

This generalizes formula (2.7) of [217]. This expression is a real quantity.

17.2.7Quantization in the Simplest Case: j = 1

In the simplest case j = 1, we find for the matrix elements (17.7)

[AY 10

]mn

= σ

√3

4π1

j ( j + 1)mδmn , (17.8)

[AY 11

]mn

= –σ

√3

4π1

j ( j + 1)

√( j – n)( j + n + 1)

2δmn+1 , (17.9)

17.2 Quantizations of the 2-Sphere 277

[AY 1–1

]mn

= σ

√3

4π1

j ( j + 1)

√( j + n)( j – n + 1)

2δmn–1 . (17.10)

From a comparison with the actions (6.75a–6.75c) of the spin angular momentumon the spin-σ spherical harmonics, we have the identification:

AY 10 = σ

√3

4π1

j ( j + 1)Λσ j

3 , (17.11)

AY 11 = –σ

√3

8π1

j ( j + 1)Λσ j

+ , (17.12)

AY 1–1 = σ

√3

8π1

j ( j + 1)Λσ j

– . (17.13)

The remarkable identification of the quantized versions of the components of r =(xi ) pointing to S1 with the components of the spin angular momentum operatorresults:

Axa = K Λσ ja , with K ==

σj ( j + 1)

. (17.14)

17.2.8Quantization of Functions

Any function f on the 2-sphere with reasonable properties (continuity, integrability,and so on) may be expanded in spherical harmonics as

f =∞∑�=0

�∑m=–�

f �mY �m , (17.15)

from which follows the corresponding expansion of A f . However, the 3 j symbolsare nonzero only when a triangular inequality is satisfied. This implies that theexpansion is truncated at a finite value, giving

A f =2 j∑�=0

�∑m=–�

f �mAY �m . (17.16)

This relation means that the (2 j + 1)2 observables (AY �m ), � � 2 j , –� � m � �

provide a second basis of Oσ j .The f �m are the components of the matrix A f ∈ Oσ j in this basis.

17.2.9The Spin Angular Momentum Operators

17.2.9.1Action on FunctionsThe Hermitian space Hσ j carries a unitary irreducible representation of the groupSU (2) with generators Λσ j

a defined in (6.71–6.73). The latter belong to Oσ j . Their


action is given in (6.75a–6.75c). and the above calculations have led to the crucialrelations (17.14). We see here the peculiarity of the ordinary spherical harmonics(σ = 0) as an orthonormal system for the quantization procedure: they would leadto a trivial result for the quantized version of the Cartesian coordinates! On theother hand, the quantization based on the Gilmore–Radcliffe spin coherent states,σ = j , yields the maximal value: K = 1/( j + 1). Hereafter we assume σ =/ 0.

17.2.9.2Action on OperatorsThe SU (2) action onHσ j induces the following canonical (infinitesimal) action onOσ j = End(Hσ j )

Lσ ja :�→ Lσ j

a A == [Λσ ja , A] (the commutator) , (17.17)

here expressed through the generators.We prove in Appendix D, (D4), that

Lσ ja AY �m = A Ja Y �m , (17.18)

from which we get

Lσ j3 AY �m = mAY �m and (Lσ j )2AY �m = �(� + 1)AY �m . (17.19)

We recall that the (AY �m )��2 j form a basis of Oσ j . The relations above make AY �m

appear as the unique (up to a constant) element ofOσ j that is common eigenvectorto Lσ j

3 and (Lσ j )2, with eigenvalues m and �(� + 1) respectively. This implies bylinearity that, for all f such that A f makes sense,

Lσ ja A f = A Ja f and (Lσ j )2A f = A J2 f . (17.20)

17.3Link with the Madore Fuzzy Sphere

17.3.1The Construction of the Fuzzy Sphere à la Madore

Let us first recall the Madore construction of the fuzzy sphere as it was originallypresented in [178] (p. 148), which we slightly modify to make the correspondencewith the coherent state quantization. It starts from the expansion of any smoothfunction f ∈ C∞(S2) in terms of spherical harmonics,

f =∞∑�=0

�∑m=–�

f �mY �m . (17.21)

Let us denote by V � the (2� + 1)-dimensional vector space generated by the Y �m , atfixed �.

17.3 Link with the Madore Fuzzy Sphere 279

Through the embedding of S2 into R3 any function in S2 can be considered asthe restriction of a function on R3 (which we write with the same notation), and,under some mild conditions, such functions are generated by the homogeneouspolynomials in R3. This allows us to express (17.21) in a polynomial form in R3:

f (x) = f (0) +∑(i1)

f (i)xi + . . . +∑

(i1i2...i � )

f (i1i2...i �)xi1 xi2 . . . xi � + . . . , (17.22)

where each subsum is restricted to a V � and involves all symmetric combinationsof the i k indices, each one varying from 1 to 3. This gives, for each fixed value of�, 2� + 1 coefficients f (i1i2 ...i �) (� fixed), which are those of a symmetric traceless3 ~ 3 ~ . . . ~ 3 (� times) tensor.

The fuzzy sphere with 2 j+1 cells is usually written as(S fuzzy, j

), with j an integer

or semi-integer. We here extend the Madore procedure and this leads to a σ-indexedfamily of fuzzy spheres

{(σS fuzzy, j

)}. Let us list the steps of this construction:

1. We consider a (2 j + 1)-dimensional irreducible unitary representation ofSU (2). The standard construction considers the vector space V j of dimen-sion 2 j + 1, on which the three generators of SU (2) are expressed as theusual (2 j +1) ~ (2 j +1) Hermitian matrices Ja . We make instead a differentchoice, namely, the three Λσ j

a , which correspond to the choice of the rep-resentation space Hσ j (instead of V j in the usual construction). Since theyobey the commutation relations of su(2),

[Λσ ja , Λσ j

b ] = iεabcΛσ jc , (17.23)

the usual procedure may be applied. As we have seen, Hσ j can be realizedas the Hilbert space spanned by the orthonormal basis of spin sphericalharmonics {σY jμ}μ=– j ... j , with the usual inner product.

Since the standard derivation of all properties of the fuzzy sphere rests onlyupon the abstract commutation rules (17.23), nothing but the representa-tion space changes if we adopt the representation space H instead of V.

2. The operators Λσ ja belong to Oσ j and have a Lie algebra structure through

the skew products defined by the commutators. But the symmetrized prod-ucts of operators provide a second algebra structure, which we write as Oσ j ,at the basis of the construction of the fuzzy sphere: these symmetrized prod-ucts of the Λσ j

a , up to power 2 j , generate the algebra Oσ j (of dimension(2 j + 1)2) of all linear endomorphisms ofHσ j , exactly like the ordinary Ja ’sdo in the original Madore construction. This is the analog of the standardconstruction of the fuzzy sphere, with the Ja and V j replaced by Λσ j

a andHσ j .

3. The construction of the fuzzy sphere of radius R is defined by associatingan operator f in Oσ j with any function f. Explicitly, this is done by firstreplacing each coordinate xi by the operator

xa == κΛσ ja , with κ =

R√j ( j + 1)

(17.24)


in the above expansion (17.22) of f (in the usual construction, this wouldbe Ja instead of Λσ j

a ). We immediately note that the three operators (17.24)obey the constraint

(xa)2 + (xa)2 + (xa)2 = R2 (17.25)

and the commutation rules[xa, xb

]= iεab

c x c . (17.26)

This is the “noncommutative sphere of radius R”.

4. Next, we replace in (17.22) the usual product by the symmetrized productof operators, and we truncate the sum at index � = 2 j . This associates withany function f an operator f ∈ Oσ j .

5. The vector space M2 j+1 of (2 j + 1) ~ (2 j + 1) matrices is linearly generatedby a number (2 j + 1)2 of independent matrices. According to the above con-struction, a basis of M2 j+1 can be selected as formed by all the products ofthe Λσ j

a up to power 2 j + 1 (which is necessary and sufficient to close thealgebra).

6. The commutative algebra limit is restored by letting j go to infinity whileparameter κ goes to zero and κ j is fixed to κ j = R.

The geometry of the fuzzy sphere(S fuzzy, j

)is thus constructed after making the

choice of the algebra of the matrices of the representation, with their matrix prod-uct. It is taken as the algebra of operators, which generalize the functions. Therank (2 j + 1) of the matrices allows us to view them as endomorphisms in a Her-mitian space of dimension (2 j + 1). This is exactly what allows the coherent statequantization introduced in the previous section.

17.3.2Operators

We have defined the action on Oσ j :

Lσ ja A == [Λσ j

a , A] .

The formula (17.22) expresses any function f of V � as the reduction to S2 of a ho-mogeneous polynomial of order �:

f =∑α,�,γ

f α,�,γ(x1)α(x2)�(x3)γ; α + � + γ = � .

The action of the ordinary momentum operators J3 and J2 is straightforward.Namely,

J3 f =∑α,�,γ

f α,�,γ(–i )[�(x1)α+1(x2)�–1(x3)γ – α(x1)α–1(x2)�+1(x3)γ

], (17.27)

and similarly for J1 and J2.

17.3 Link with the Madore Fuzzy Sphere 281

On the other hand, we have by definition

f =∑α,�,γ

f α,�,γS(

(x1)α(x2)�(x3)γ)

, (17.28)

where S(·) means symmetrization. Recalling xa = κΛσ ja , and using (17.23), we

apply the operator Lσ j3 to this expression:

Lσ j3 f ==

[Λσ j

3 , f]

=∑α,�,γ

f α,�,γ

[Λσ j

3 , S(

x1αx2�

x3γ)]

. (17.29)

We prove in Appendix E that the commutator of the symmetrized term is thesymmetrized commutator. Then, using the identity

[ J, AB · · ·M ] = [ J , A]B · · ·M + A[ J, B ] · · ·M + · · · + AB · · · [ J, M ] ,

which results easily (by induction) from [ J, AB ] = [ J , A]B + A[ J, B ], it follows that

Lσ j3 f ==

[Λσ j

3 , f]

=∑α,�,γ

f α,�,γ

(iαx1

α–1x2�+1

x3γ–i�x1

α+1x2�–1

x3γ)

. (17.30)

We thus have proven

Lσ j3 f = J3 f . (17.31)

Similar identities hold for Lσ j1 ,Lσ j

2 , and thus for (Lσ j )2.It follows that Y �m appears as an element of Oσ j that is a common eigenvector

of Lσ j3 , with eigenvalue m, and of (Lσ j )2, with eigenvalue �(� + 1). Since we proved

above that such an element is unique (up to a constant), it results that each Y �m ∝AY �m . Thus, the Y �m ’s, for � u j , – j u m u j , form a basis of Oσ j .

Then, the Wigner–Eckart theorem (see D) implies that AY �m = C (�)Y �m , wherethe proportionality constant C (�) does not depend on m (this can actually bechecked directly). These coefficients can be calculated directly, after noting that

Y �� ∝ (Λ+)� ∝(

x1 + i x2)�

.

In fact,

Y �� = a(�)(

x1 + i x2)�

; a(�) =

√(2� + 1)!

2�+1√

π�!.

We obtain

C (�) = 2� (–1) j+σ–2�(2 j + 1)κ�

√(2 j – �)!

(2 j + � + 1)!

(j j �

–σ σ 0

). (17.32)


17.4Summary

Two families of quantization of the sphere have been presented in the previoussections, namely,

(i) the usual construction of the fuzzy sphere, which depends on the parame-ter j. This parameter defines the “size” of the discrete cell.

(ii) the coherent state quantization approach, which depends on two parame-ters, j and σ =/ 0.

These two quantizations may be formulated as involving the same algebra of op-erators (quantum observables) O, acting on the same Hermitian space H. In Ta-ble 17.1 we compare these two types of “fuzzyfication” of the sphere. Note that HandO are not the Hermitian space and algebra appearing in the usual constructionof the fuzzy sphere (when we consider them as embedded in the space of functionson the sphere, and of operators acting on them), but they are isomorphic to them,and nothing is changed.

The difference lies in the fact that the quantum counterparts A f and f of a givenclassical observable f differ in both approaches. Thus, the coherent state quanti-zation really differs from the usual fuzzy sphere quantization. It follows from thecalculations above that all properties of the usual fuzzy sphere are shared by thecoherent state quantized version. The only point to be checked is whether it givesthe sphere manifold in some classical limit. The answer is positive as far as theclassical limit is correctly defined. Simple calculations show that it is obtained asthe limit j �→ ∞, σ �→ ∞, provided that the ratio σ/ j tends to a finite value. Thus,one may consider that the coherent state quantization leads to a one (discrete)-

Table 17.1 Coherent state quantization of the sphere iscompared with the Madore construction of the fuzzy spherethrough correspondence formulas.

Coherent states fuzzy sphere Madore-like fuzzy sphere

Hilbert space H = Hσ j = span(σY jμ) ⊂ L2(S2)

Endomorphisms O = Oσ j = EndHσ j

Spin angular momentumoperators

Λσ ja ∈ O

Observables A f ∈ Oσ j ; Axa = K Λσ ja f ∈ Oσ j ; xa = κΛσ j

a

Action of angular momentum Lσ ja A f ==

[Λσ j

a , A f

]= A Ja f Lσ j

a f ==[

Λσ ja , f

]= Ja f

Correspondence AY �m = C (�)Y �m

17.5 The Fuzzy Hyperboloid 283

parameter family of fuzzy spheres if we impose relations of the type σ = j – σ0, forfixed σ0 > 0, for instance.

17.5The Fuzzy Hyperboloid

In a continuation of the previous sections, we now turn our attention to the con-struction of a fuzzy version of the two-dimensional “de Sitter” hyperboloid by usinga coherent state quantization [218]. Two-dimensional de Sitter space-time can beviewed as a one-sheeted hyperboloid embedded in a three-dimensional Minkowskispace:

MH = {x ∈ R3 : x2 = ηα� xαx� = (x0)2 – (x1)2 – (x2)2 = –H–2} . (17.33)

In Figure 17.1 we show this model of space-time in its four-dimensional version.We recall (see, e.g., [219] and references therein) that the de Sitter space-time isthe unique maximally symmetric solution of the vacuum Einstein equations withpositive cosmological constant Λ. This constant is linked to the constant Ricci cur-vature 4Λ of this space-time. There exists a fundamental length H–1 :=

√3/(cΛ).

The isometry group of the de Sitter manifold is, in the four-dimensional case, the10-parameter de Sitter group SO0(1, 4). The latter is a deformation of the properorthochronous Poincaré group P↑+ .

In the case of our two-dimensional toy model, the isometry group is SO0(1, 2) orits double covering SU (1, 1) � SL(2, R), as already mentioned in Section 14.3. ItsLie algebra is spanned by the three Killing vectors K α� = xα∂� – x�∂α (K 12: com-

Fig. 17.1 1 + 3 (or 1 + 1) de Sitter space-time viewed as a one-sheeted hyperboloid embedded in a five (or three)-dimensionalMinkowski space (the bulk). Coordinate x0 plays, to someextent, the role of de Sitter time, and x4 (or x2) is the extradimension needed to embed the de Sitter hyperboloid into theambient 1 + 4 (or 1 + 2) Minkowski space.


pact, for “space translations”; K 02: noncompact, for “time translations”; K 01: non-compact, for Lorentz boosts). These Killing vectors are represented as (essentially)self-adjoint operators in a Hilbert space of functions on MH , square-integrable withrespect to some invariant inner (Klein–Gordon type) product.

The quadratic Casimir operator has eigenvalues that determine the unitary irre-ducible representations:

Q = – 12 Mα�Mα� = – j ( j + 1) Id = (ρ2 + 1

4 ) Id , (17.34)

where j = – 12 + iρ, ρ ∈ R+ for the principal series.

Comparing the geometrical constraint (17.33) with the group-theoretical one(17.34) (in the principal series) suggests the “fuzzy” correspondence [218]:

xα �→ xα =r2

εα�γ M�γ , i.e. , x0 = rM21 , x1 = rM02 , x2 = rM10 .

(17.35)

r being a constant with length dimension. The following commutation rules areexpected:

[x0, x1] = i r x2 , [x0, x2] = –i r x1 , [x1, x2] = i r x0 , (17.36)

with ηα�xαx� = –r2(ρ2 + 14 ) Id , and its “commutative classical limit”, r → 0, ρ →

∞, rρ = H–1.Let us now proceed to the coherent state quantization of the two-dimensional

de Sitter hyperboloid. The “observation” set X is the hyperboloid MH . Convenientintrinsic coordinates are those of the topologically equivalent cylindrical structure,(τ, θ), τ ∈ R, 0 u θ < 2π, through the parameterization x0 = rτ, x1 = rτ cos θ –H–1 sin θ, x2 = rτ sin θ+H–1 cos θ, with the invariant measure μ(dx) = dτ dθ/(2π).The functions φm (x) forming the orthonormal system needed to construct coher-ent states are like those already chosen in Chapter 15 for the quantization of themotion on the circle:

φm (x) =( ε

π

)1/4e– ε

2 (τ–m)2eimθ, m ∈ Z , (17.37)

where the parameter ε > 0 can be arbitrarily small and represents a necessaryregularization. Through the usual construction the coherent states read as

|τ, θ〉 =1√N (τ)

( επ

)1/4 ∑m∈Z

e– ε2 (τ–m)2

e–imθ|m〉 , (17.38)

where |φm〉 == |m〉. We recall that the normalization factor

N (τ) =

√επ

∑m∈Z

e–ε(τ–m)2<∞

is a periodic train of normalized Gaussians and is proportional to an elliptic thetafunction.

17.5 The Fuzzy Hyperboloid 285

The coherent state quantization scheme yields the quantum operator A f , actingon H and associated with the classical observable f (x). For the most basic one,associated with the coordinate τ, one gets

Aτ =∫

Xτ |τ, θ〉〈τ, θ|N (τ) μ(dx) =

∑m∈Z

m|m〉〈m| . (17.39)

This operator reads in angular position representation (Fourier series) as Aτ =–i∂/∂θ, and is easily identified as the compact representative M12 of the Killingvector K 12 in the principal series of SO0(1, 2) unitary irreducible representation.Thus, the “time” component x0 is naturally quantized, with spectrum rZ, throughx0 �→ x0 = –rM12. For the two other ambient coordinates one gets

x1 =re– ε

4

2

∑m∈Z

{ pm|m + 1〉〈m| + h .c} , x2 =re– ε

4

2i

∑m∈Z

{pm |m + 1〉〈m| – h .c} ,

(17.40)

with pm = (m + 12 + iρ). The commutation rules are those of so(1, 2), which are those

of (17.36) with a local modification to [x1, x2] = –i re– ε2 x0. The commutative limit at

r → 0 is apparent. It is proved that the same holds for higher-degree polynomialsin the ambient space coordinates.

287

18Conclusion and Outlook

What can we retain after such a journey through a (little) part of the coherent statelandscape, beyond the physical context? At the risk of imposing upon the readera quite partial view of this wide subject, let us propose a few insights into the deepsignificance of coherent states, once this notion has been cleared of its quantumphysics dressing. We have shown that at the heart of the existence of coherentstates there is a set X of parameters or data potentially measurable through someexperimental or observational protocol, say, E . The set X is equipped with a mea-sure so that we deal with a measure space (X , μE ), a minimal structure that is alsodetermined by E , along with a mixing of symmetry or conservation principles anddegree of confidence (see Section A.12).

The term (classical) “observable” designates a function f (x) on X, susceptible tobeing measured within the framework imposed by E . Ideally, once a function hasbeen assigned as an observable, all the values it assumes are accessible to measure-ment. For instance, if X represents the phase space of the motion of a particle onthe line, that is, the set {(q, p)} of possible initial positions and velocities, all thevalues assumed by a function f (q, p) acknowledged as an observable can be mea-sured, and they constitute the spectrum of the observable f. The word “spectrum”is usually employed to designate an image18) or a distribution, of components ofphysical quantities such as light, sound, and particles arranged according to char-acteristics such as wavelength, frequency, charge, and energy. An observable f canbe viewed as a diagonal continuous or discrete, infinite or finite matrix, with ele-ments the values found in its spectrum. A possible interpretation is to consider fas the multiplication operator M f in the Hilbert space L2(X , μE ) provided by themeasure space:

M f : φ �→ φ′ = M f φ (M f φ)(x) = f (x) φ(x) . (18.1)

It is clear that the set of such observables forms a commutative algebra, in abstrac-tion from operator domain restrictions.

Changing the experimental/observational protocol, substituting E with a newone, say, E′, can lead to discrepancies in this ideal scheme, in the sense that certain

18) The reader is invited to scrutinize the imageon the front cover of this book!


288 18 Conclusion and Outlook

functions f (x) (think of the combination ( p2 + q2)/2 in the example of one-dimen-sional motion) lose their status of observable in the previous sense: the spectrumis not the same as the set of all the values assumed by the function. It is then neces-sary to build a formalism able to account for this new evidence: the customary wayis to view f (x) as a nondiagonal matrix, a more general operator, acting on somelinear space, with spectrum the set of observed values in conformity with E′.

The question becomes finding the right mathematical framework in which f (x)is realized as an operator. Clearly, the adoption of the new protocol E′, possiblytogether with the new measure space (X , μE′ ), results in the emergence of a gener-ically noncommutative algebra of operators. We have observed that this noncom-mutativity occurs along with a reduction of the Hilbert space L2(X , μE′ ) to a closedsubspace, say, K. This reduction should be thought of as the action of a projectorPK on that Hilbert space:

K = PK L2(X , μE′ ) . (18.2)

Note that the restriction of PK to the subspace K is the identity operator IK onthe latter. It is precisely at this point that the existence of a family of normalizedvectors |x〉 resolving, as elements of L2(X , μE′ ), the projector PK, or, as elements ofK, resolving the unity IK,∫

X|x〉〈x | νE′ (dx) = PK or IK , (18.3)

allows us to implement the construction of the operator corresponding to an ob-servable f. Indeed, such a function, which was viewed as a multiplication operatorM f according to the former protocol E , becomes, under the projection PK, theoperator

A f = PKM f PK =∫

Xf (x)|x〉〈x | νE′ (dx) , (18.4)

the quantized version of f under the quantization provided by the family {|x〉 , x ∈X }. Actually, we have followed a mathematical procedure known in specific situa-tions as the Toeplitz quantization of the set X.

Of course, we could be faced with ambiguities or conflicts of the type: withinthe same protocol, two different observables could lead to different projections,and so to incompatible physics or to different interpretations of the original set X.A change of protocol is then necessary. These possibilities raise deep questions,beyond the scope of this book.

289

Appendix AThe Basic Formalism of Probability Theory

Many excellent textbooks exist on the subject. I recommend one of them, which isbased on a course by Sinai [220].

A.1Sigma-Algebra

Let X be a set. A family F of subsets of X is a σ-algebra if and only if it has thefollowing properties:

(i) The empty set ∅ is in F .

(ii) If A is in F , then so is the complement of A.

(iii) If A1, A2, A3, . . . is a sequence in F , then their (countable) union is also inF .

From (i) and (ii) it follows that X is in F ; from (ii) and (iii) it follows that the σ-al-gebra is also closed under countable intersections. σ-algebras are mainly used todefine measures on X, as are defined in the next section. An ordered pair (X ,F ),where X is a set and F is a σ-algebra over X, is called a measurable space.

A.1.1Examples

1. The family consisting only of the empty set ∅ and X is a σ-algebra over X,the so-called trivial σ-algebra. Another σ-algebra over X is given by the powerset of X, that is, the set P(X ) of all subsets of X.

2. If {Fa} is a family of σ-algebras over X, then the intersection of all Fa is alsoa σ-algebra over X.

3. If U is an arbitrary family of subsets of X, then we can form a special σ-alge-bra from U , called the σ-algebra generated by U . We denote it by σ(U ) anddefine it as follows. First note that there is a σ-algebra over X that containsU , namely, the power set of X. Let Φ be the family of all σ-algebras over Xthat contain U (i.e., a σ-algebra F over X is in Φ if and only if U is a subset


290 Appendix A The Basic Formalism of Probability Theory

ofF .) Then we define σ(U ) to be the intersection of all σ-algebras in Φ. σ(U )is then the smallest σ-algebra over X that contains U .

4. This leads to the most important example: the Borel algebra over any topo-logical space is the σ-algebra generated by the open sets (or, equivalently, bythe closed sets). Note that this σ-algebra is not, in general, the whole powerset.

5. On the Euclidean space Rn , another σ-algebra is of importance: that of allLebesgue measurable sets. This σ-algebra contains more sets than the Borelalgebra on Rn and is preferred in integration theory.

A.2Measure

A measure μ is a function defined on a σ-algebra F over a set X and taking valuesin [0,∞) such that the following properties are satisfied:

1. The empty set has measure zero:

μ(∅) = 0 . (A1)

2. Countable additivity or σ-additivity: if E 1, E 2, E 3, is a countable sequence ofpairwise disjoint sets in F , the measure of the union of all the E i is equalto the sum of the measures of each E i :

μ

(∞⋃i=1

E i

)=∞∑i=1

μ (E i ) . (A2)

The triple (X ,F , μ) is then called a measure space, and the members ofF are calledmeasurable sets.

A.3Measurable Function

If F is a σ-algebra over X and G is a σ-algebra over Y, then a function f : X �→ Y ismeasurable if the preimage of every set in G is in F .

By convention, if Y is some topological space, such as the space of real num-bers R or the complex numbers C, then the Borel σ-algebra generated by the opensets on Y is used, unless otherwise specified.

If a function from one topological space to another is measurable with respectto the Borel σ-algebras on the two spaces, the function is also known as a Borelfunction. Continuous functions are Borel; however, not all Borel functions are con-tinuous.

A.4 Probability Space 291

Given two measurable spaces (X 1,F1) and (X 2,F2), a measurable function f :X 1 �→ X 2 and a measure μ : F1 �→ [0, +∞], the pushforward of μ is defined to be themeasure f ∗ μ : F2 �→ [0, +∞] given by

f ∗ μ(B ) def= μ(

f –1(B ))

for B ∈ F2 . (A3)

A.4Probability Space

A probability space is a set Ω, together with a σ-algebra F on Ω and a measure Pon that σ-algebra, such that P (Ω) = 1. The set Ω is called the sample space and theelements of F are called the events. The measure P is called the probability measure,and P (E ) is the probability of the event E ∈ F .

Every set A with nonzero probability defines another probability

P (B |A) =P (A ∩ B )

P (A)(A4)

on the space. This is usually read as “probability of B given A.” If this conditionalprobability of B given A is the same as the probability of B, then B and A are saidto be independent.

In the case that the sample space is finite or countably infinite, a probabilityfunction can also be defined by its values on the elementary events {ω1}, {ω2}, . . .,where Ω = {ω1, ω2, . . .}.

A.5Probability Axioms

In a context proper to probability theory, the properties of a probability measureP : F �→ [0, 1] are imposed under the form of axioms known as the Kolmogorovaxioms:

K1 For any set E,

0 u P (E ) u 1 . (A5)

That is, the probability of an event set is represented by a real number between 0 and 1.

K2

P (Ω) = 1 . (A6)

That is, the probability that some elementary event in the entire sample set will occuris 1. More specifically, there are no elementary events outside the sample set. Thisis often overlooked in some mistaken probability calculations; if one cannot preciselydefine the whole sample set, then the probability of any subset cannot be defined either.


K3 Any countable sequence of mutually disjoint events E 1, E 2, . . . satisfies

P (E 1 ∪ E 2 ∪ · · · ) = P (E 1) + P (E 2) + · · · . (A7)

That is, the probability of an event set which is the union of other disjoint subsets isthe sum of the probabilities of those subsets. This is called σ-additivity. If there is anyoverlap among the subsets, this relation does not hold.

A probability distribution assigns to every interval of the real numbers a proba-bility, so that the probability axioms are satisfied. In technical terms, a probabilitydistribution is a probability measure whose domain is the Borel algebra on thereals.

A.6Lemmas in Probability

(i) From the Kolmogorov axioms one deduces other useful rules for calculatingprobabilities:

P (A ∪ B ) = P (A) + P (B ) – P (A ∩ B ) . (A8)

That is, the probability that A or B will happen is the sum of the probabilities thatA will happen and that B will happen, minus the probability that A and B willhappen. This can be extended to the inclusion–exclusion principle.

(ii)P (Ω – E ) = 1 – P (E ) . (A9)

That is, the probability that any event will not happen is 1 minus the probabilitythat it will.

(iii) Using conditional probability as defined above, it also follows immediatelythat

P (A ∩ B ) = P (A) P (B |A) . (A10)

That is, the probability that A and B will happen is the probability that A willhappen, times the probability that B will happen given that A happened; thisrelationship gives Bayes’s theorem (see next). It then follows that A and B areindependent if and only if

P (A ∩ B ) = P (A) P (B ) . (A11)

A.7Bayes’s Theorem

Bayes’s theorem relates the conditional and marginal probabilities of events A andB, where B has a nonvanishing probability:

P (A|B ) =P (B |A) P (A)

P (B ). (A12)

A.8 Random Variable 293

More generally, let Ω =⋃

i Ai , Ai ∩ A j = ∅ for i =/ j , be a partition of the eventspace. Then we have

P (Ai |B ) =P (B |Ai ) P (Ai )∑j P (B |A j ) P (A j )

, (A13)

for any Ai in the partition.

A.8Random Variable

Let (Ω,F , P ) be a probability space and (Y , Σ) be a measurable space. Then a ran-dom variable � is formally defined as a measurable function � : Ω �→ Y . In otherwords, the preimage of the “well-behaved” subsets of Y (the elements of Σ) areevents, and hence are assigned a probability by P.

When the measurable space is the measurable space over the real numbers, onespeaks of real-valued random variables. Then, the function � is a real-valued ran-dom variable if

{ω ∈ Ω | �(ω) u r} ∈ F , ∀ r ∈ R . (A14)

A.9Probability Distribution

The probability distribution of a real-valued variable � can be uniquely describedby its cumulative distribution function F �(x) (also called a distribution function),which is defined by

F �(x) = P (� u x) (A15)

for any x in R.More generally, given a random variable � : Ω �→ Y between a probability space

(Ω,F , P ), the sample space, and a measurable space (Y , Σ), called the state space,a probability distribution on (Y , Σ) is a probability measure � ∗ P : Σ �→ [0, 1] onthe state space, where � ∗ P is the push-forward measure of P.

A distribution is called discrete if its cumulative distribution function consistsof a sequence of finite jumps, which means that it belongs to a discrete randomvariable �, a variable which can only attain values from a certain finite or countableset. Discrete distributions are characterized by a probability mass function p suchthat P (� = x) = p(x).

A distribution is called continuous if its cumulative distribution function is con-tinuous. A random variable � is called continuous if its distribution function iscontinuous. In that case P (� = x) = 0 for all x ∈ R. Note also that there are proba-bility distribution functions which are neither discrete nor continuous.


The so-called absolutely continuous distributions (frequently and loosely namedcontinuous distributions) can be expressed by a probability density function: a non-negative Lebesgue integrable function f defined on the reals such that

F (x) = P (� u x) =∫ x

–∞f (u) du . (A16)

Discrete distributions do not admit such a density. There are continuous distribu-tions (e.g., the devil’s staircase) that also do not admit a density; they are calledsingular continuous distributions.

A.10Expected Value

If � is a random variable defined on a probability space (Ω,F , P ), then the expectedvalue of �, denoted E (�), or sometimes 〈�〉, is defined as

E (�) =∫

Ω� dP , (A17)

where the notation stands for the Lebesgue integral resulting from the measure P.Note that not all random variables have an expected value, since the integral maynot exist (e.g., Cauchy distribution).

If � is a discrete random variable with probability mass function p(x), then theexpected value becomes

E (�) =∑

i

x i p(xi ) . (A18)

If the probability distribution of � admits a probability density function f (x),then the expected value can be computed as

E (�) =∫ +∞

–∞x f (x) dx . (A19)

The expected value of an arbitrary function of �, g (�), with respect to the probabilitydensity function f (x) is given by

E (g (�)) =∫ +∞

–∞g (x) f (x) dx . (A20)

A.11Conditional Probability Densities

There is also a version of Bayes’s theorem for continuous distributions. It is some-what harder to derive, since probability densities, strictly speaking, are not proba-bilities, so Bayes’s theorem has to be established by a limit process. Bayes’s theorem

A.12 Bayesian Statistical Inference 295

for probability densities is formally similar to the theorem for probabilities:

f �(x |η = y ) =f �,η(x , y )

f η(y )=

f η(y |� = x) f �(x)f η(y )

=f η(y |� = x) f �(x)∫ +∞

–∞ f η(y |� = u) f �(u) du, (A21)

where the continuous analog of the law of total probability is used in the denomi-nator,

f η(y ) =∫ +∞

–∞f η(y |� = u) f �(u) du . (A22)

In (A21),• f �,η(x , y ) is the joint distribution of � and η,

• f �(x |η = y ) is the conditional density function for random variable �,

• and f �(x) and f η(y ) are the marginal density functions for the randomvariables � and η, respectively.

A.12Bayesian Statistical Inference

In cases of incomplete knowledge of the preparation, a probabilistic experi-ment may be modeled by a family of probability distributions, for example,{pθ(y ), θ ∈ Θ}, where pθ is a probability function in the discrete case and a prob-ability density function in the continuous case for random variable η. Here weconsider the case where the parameter θ which indexes the family is a continuousvariable over the set Θ. For one (unknown) value of θ, the model is assumed to bepredictive. For simplicity, we consider the case of a single parameter, but one mayextend the reasoning to a multiparameter model.

After the experiment has been performed, and pointer value y 0, say, has been ob-tained, interest might center upon an inferred probability distribution over the pa-rameter space Θ. In that case, we take Θ to be a measure space such as (Θ, μ(dθ)).Suppose that μ(dθ) can be written in terms of a density function μ(dθ) = Π(θ) dθ.Then Bayesian statistical inference consists of using Bayes’s formula for condition-al distributions in the following way:

f (θ|y 0) =pθ(y 0) Π(θ)∫

Θ pθ′ (y 0) Π(θ′) dθ′. (A23)

In statistical parlance,

• f (θ|y 0) is referred to as a posterior probability density function on the pa-rameter space.

• μ(dθ) is called an a priori measure on the parameter space.


• θ �→ pθ(y 0) for given y o , as a function of θ, is referred to as a likelihoodfunction.

Note that, in theory, we have a collection of possible likelihood functions in thatthere is one for each possible value of y.

A.13Some Important Distributions

A.13.1Degenerate Distribution

This is the probability distribution of a random variable which always has the samevalue. It is localized at a point x0 on the real line.

The cumulative distribution function of the degenerate distribution is then theHeaviside step function θ(x–x0), which is equal to 1 for x v x0 and to 0 if x < x0. Asa discrete distribution, the degenerate distribution does not have a proper density.Actually, within the framework of distribution theory, Dirac’s delta function canserve this purpose.

A.13.2Uniform Distribution

The discrete uniform distribution is a discrete probability distribution that can becharacterized by saying that all values of a finite set of possible values are equallyprobable. If the values of a random variable with a discrete uniform distributionare real, it is possible to express the cumulative distribution function in terms ofthe degenerate distribution:

F (x ; n) =1n

n∑i=1

θ(x – xi ) . (A24)

The continuous uniform distribution is a family of probability distributions suchthat for each member of the family all intervals of the same length on the distribu-tion’s support are equally probable. The probability density function of the contin-uous uniform distribution is

f (x) =

{1

b – afor a u x u b ,

0 for x < a or x > b=

θ(x – a) – θ(x – b)b – a

. (A25)

A.13.2.1Bernoulli DistributionThis is a discrete family of probability distributions indexed by parameter p, 0 up u 1. Put q = 1 – p . The random variable � has only two possible values, say, 0

A.13 Some Important Distributions 297

and 1. Then the probability mass function is given by

P (�|x) = px (1 – p)1–x =

⎧⎨⎩ p if x = 1 ,q if x = 0 ,0 otherwise .

(A26)

A.13.2.2Binomial DistributionThe family is indexed by parameters p and n, where 0 u p u 1 and n is a posi-tive integer. Put q = 1 – p . The binomial distribution is given by the terms in theexpansion of (q + p)n .

P (� = k) =

(nk

)pk qn–k for k = 0, 1, . . . , n . (A27)

A model leading to this distribution may be given by the following. If n indepen-dent trials are performed and in each there is a probability p that an outcome ωoccurs, then the number of trials in which ω occurs can be represented by a ran-dom variable � having the binomial distribution with parameters n and p.

When n = 1, the distribution is the Bernoulli distribution.

A.13.2.3Hypergeometrical DistributionThis is a discrete probability distribution that describes the number of successes ina sequence of n draws from a finite population without replacement. For a randomvariable � following the hypergeometrical distribution with parameters N, D, andn, the probability of getting exactly k successes is given by

P (� = k) =

(Dk

) (N –Dn–k

)(Nn

) . (A28)

A.13.2.4Negative Binomial DistributionThe family is indexed by parameters P > 0 and positive real number r. Put Q =1 + P . The negative binomial distribution is given by the terms in the expansion of(Q – P )–r . Thus,

P (� = k) =

(r + k – 1

r – 1

) (PQ

)r (1 –

PQ

)r

, (A29)

for k = 0, 1, 2, . . .. Unlike the binomial distribution, r need not be an integer. Whenr is an integer, the distribution is sometimes called the Pascal distribution.

When r = 1, we have P (� = k) = (P/Q )kQ–1, which is called the geometrical orBose–Einstein distribution.

A model leading to this distribution when r is a positive integer may be the fol-lowing. Put P = (1 – p)/p and r = m. Let random variable � represent the number


of independent trials necessary to obtain m occurrences of an event ω which hasconstant probability p of occurring at each trial. Then we have

P (� = m + k) =

(m + k – 1

m – 1

)pm (1 – p)k (A30)

for k = 1, 2, 3, . . ..

A.13.2.5Poisson DistributionThe family is indexed by parameter λ > 0. Then random variable � has the Poissondistribution when the probability distribution is given by

P (� = k) =e–λ λk

k!, (A31)

for k = 1, 2, 3, . . ..The Poisson distribution is the limit of a sequence of binomial distributions in

which n tends to infinity and p tends to zero such that n p (binomial mean) remainsequal to λ (Poisson mean) [221].

A.13.2.6Poisson Process [222]A stochastic process {N (t), t v 0} is said to be a counting process if N (t) repre-sents the total number of events ω that have occurred up to time t and satisfies thefollowing conditions:

(i) N (t) v 0.

(ii) N (t) is integer-valued.

(iii) If s < t , then N (s) u N (t).

(iv) For s < t , N (t) – N (s) equals the number of events that have occurred in theinterval (s, t).

A counting process possesses independent increments if the number of eventswhich occur in disjoint time intervals are independently distributed.

A counting process is a Poisson process with rate λ, λ > 0, if it satisfies thefollowing conditions:

(i) N (0) = 0.

(ii) The process has independent increments.

(iii) The number of events in any interval of length t has the Poisson distributionwith mean λ t .


A.13.2.7Mixtures of Discrete Probability Distributions [223]Consider a family of discrete distributions indexed by a parameter λ ∈ Λ. For ran-dom variable �, write P (� = x) = px (λ). Let G(dλ) represent a probability measureon Λ. Then the probability distribution

P r(� = x) =∫

Λpx (λ) G(dλ)

is described as px (λ) mixed on λ by G(dλ).For example, if px (λ) is the Poisson distribution and G(dλ) is the exponential

distribution, then we have the result that � has the geometrical distribution. Moregenerally, a Poisson variate mixed on λ by the gamma distribution (see below) hasthe negative binomial distribution.

A.13.2.8Beta DistributionThe beta distribution is a continuous probability distribution with the probabilitydensity function defined on the interval [0, 1]:

f (x) =1

B (a, b)xa–1 (1 – x)b–1 , (A32)

where a and b are parameters that must be greater than zero, and B (a, b) =Γ(a)Γ(b)/Γ(a + b) is the beta function expressed in terms of the gamma functions.

A.13.2.9Wigner Semicircle DistributionThe Wigner semicircle distribution is a continuous probability distribution withthe interval [–R, R] as support. The graph of its probability density function f isa semicircle of radius R centered at the origin (actually a semiellipse with suitablenormalization):

f (x) =2

π R2

√1 – x2 , (A33)

for R < x < R, and f (x) = 0 if x > R or x < –R.This distribution arises as the limiting distribution of eigenvalues of many ran-

dom symmetric matrices as the size of the matrix approaches infinity.

A.13.2.10Gamma DistributionThe gamma distribution is a continuous probability distribution. Its probabilitydensity function can be expressed in terms of the gamma function:

f (x ; k, θ) =e–x/θ

θk

xk–1

Γ(k), (A34)

where k > 0 is the shape parameter and θ > 0 is the scale parameter.


A.13.2.11Normal DistributionThe normal distribution is certainly the most popular among continuous probabil-ity distributions. It is also called the Gaussian distribution. It is actually a familyof distributions of the same general form, differing only in their location and scaleparameters: the mean and standard deviation. Its probability density function withmean μ and standard deviation σ (equivalently, variance σ2) is given by

f (x ; μ, σ) =1

σ√

2πe(x–μ)2/2σ2

. (A35)

The standard normal distribution is the normal distribution with a mean of zeroand a standard deviation of one. Because the graph of its probability density resem-bles a bell, it is often called a bell curve.

Approximately normal distributions occur in many situations, as a result of thecentral limit theorem. When there is reason to suspect the presence of a large num-ber of small effects acting additively and independently, it is reasonable to assumethat observations will be normal. There are statistical methods to empirically testthat assumption.

Two examples of the occurrence of the normal law in physics:

Photon counts Light intensity from a single source varies with time, and is usu-ally assumed to be normally distributed. However, quantum mechanics interpretsmeasurements of light intensity as photon counting. Ordinary light sources thatproduce light by thermal emission should follow a Poisson distribution or Bose–Einstein distribution on very short time scales. On longer time scales (longer thanthe coherence time), the addition of independent variables yields an approximatelynormal distribution. The intensity of laser light, which is a quantum phenomenon,has an exactly normal distribution.

Measurement errors Repeated measurements of the same quantity are expected toyield results which are clustered around a particular value. If all major sources oferrors have been taken into account, it is assumed that the remaining error mustbe the result of a large number of very small additive effects, and hence normal.Deviations from normality are interpreted as indications of systematic errors whichhave not been taken into account. Note that this is the central assumption of themathematical theory of errors.

A.13.2.12Cauchy DistributionThe Cauchy distribution is an example of a continuous distribution that does nothave an expected value or a variance. In physics it is usually called a Lorentzian, andit is the distribution of the energy of an unstable state in quantum mechanics. Inparticle physics, the extremely short lived particles associated with such unstablestates are called resonances. The Cauchy distribution has the probability density


function

f (x ; x0, γ) =1

π γ1

1 +(

x–x0γ

)2 , (A36)

where x0 is the location parameter, specifying the location of the peak of the dis-tribution, and γ is the scale parameter, which specifies the half-width at half-maxi-mum. The amplitude of the above Lorentzian function is given by 1/πγ.

303

Appendix BThe Basics of Lie Algebra, Lie Groups, and TheirRepresentations

The theory of groups and Lie groups, Lie algebras, and their representations iswidely known and many excellent books cover it, for instance, [74, 224, 225]. Werecall here a few basic facts, mainly extracted from [11].

B.1Group Transformations and Representations

A transformation of a set S is a one-to-one mapping of S onto itself. A group G isrealized as a transformation group of a set S if with each g ∈ G there is associateda transformation s �→ g · s of S where for any two elements g 1 and g2 of G ands ∈ S we have (g1 g2) · s = g1 · (g2 · s).

The set S is then called a G-space. A transformation group is transitive on S if,for each s1 and s2 in S, there is a g ∈ G such that s2 = g · s1. In that case, the set Sis called a homogeneous G-space.

A (linear) representation of a group G is a continuous function g �→ T (g ) whichtakes values in the group of nonsingular continuous linear transformations ofa vector space V , and which satisfies the functional equation T (g1 g2) = T (g1)T (g2)and T (e) = Id , the identity operator in V , where e is the identity element of G. It fol-lows that T (g–1) = (T (g ))–1. That is, T (g ) is a homomorphism of G into the groupof nonsingular continuous linear transformations of V .

A representation is unitary if the linear operators T (g ) are unitary with respectto the inner product 〈·|·〉 on V . That is, 〈T (g ) v1|T (g ) v2〉 = 〈v1|v2〉 for all vectorsv1, v2 in V . A representation is irreducible if there is no nontrivial subspace V0 ⊂ Vsuch that for all vectors vo ∈ V0, T (g ) vo is in V0 for all g ∈ G. That is, there is nonontrivial subspace V0 of V which is invariant under the operators T (g ).

Let G be a transformation group of a set S. Let V be a linear space of functionsf (s) for s ∈ S . For each invariant subspace V0 ⊂ V we have a representation of thegroup G by shift operators: (T (g ) f )(s) = f (g–1 · s). A multiplier representation isof the form (T (g ) f )(s) = A(g–1, s) f (g–1 · s), where A(g , s) is an automorphic factorsatisfying A(g1 g2, s) = A(g1, g2 · s)A(g2, s) and A(e, s) = 1.


304 Appendix B The Basics of Lie Algebra, Lie Groups, and Their Representations

B.2Lie Algebras

Let g be a complex Lie algebra, that is, a complex vector space with an antisymmet-ric bracket [·, ·] that satisfies the Jacobi identity

[[X , Y ], Z ] + [[Y , Z ], X ] + [[Z , X ], Y ] = 0 , ∀X , Y , Z ∈ g . (B1)

For X , Y ∈ g, the relation (adX )(Y ) = [X , Y ] gives a linear map ad: g → End g

(endomorphisms of g), called the adjoint representation of g. Next, if dim g < ∞, itmakes sense to define

B (X , Y ) = Tr[(adX )(adY )] , X , Y ∈ g . (B2)

B is a symmetric bilinear form on g, called the Killing form of g. Alternatively, onemay choose a basis {X j , j = 1, . . . , n} in g, in terms of which the commutationrelations read

[X i , X j ] =n∑

k=1

cki j X k , i , j = 1, . . . , n , (B3)

where cki j are called the structure constants and n = dim g. Then it is easy to see

that g i j =∑n

k,m=1 cmik ck

jm = B (X i , X j ) defines a metric on g, called the Cartan–Killing metric.

The Lie algebra g is said to be simple, or semisimple, if it contains no nontrivialideal, or Abelian ideal. A semisimple Lie algebra may be decomposed into a directsum of simple ones. Furthermore, g is semisimple if and only if the Killing formis nondegenerate (Cartan’s criterion).

Let g be semisimple. Choose in g a Cartan subalgebra h, that is, a maximal nilpo-tent subalgebra (it is in fact maximal Abelian and unique up to conjugation). Thedimension � of h is called the rank of g. A root of g with respect to h is a linear formon h, α ∈ h∗, for which there exists X =/ 0 in g such that (adH)X = α(H)X ,∀H ∈ h.

Then one can find (Cartan, Chevalley) a basis {Hi , E α} of g, with the followingproperties. {H j , j = 1, . . . , �} is a basis of h and each generator E α is indexed bya nonzero root α, in such a way that the commutation relations (B3) may be writtenin the following form:

[Hi , H j ] = 0 , j = 1, . . . , � (B4)

[Hi , E α] = α(Hi )E α , i = 1, . . . , � (B5)

[E α, E –α] = Hα ==�∑

i=1

αi Hi ∈ h , (B6)

[E α, E �] = N α� E α+� , (B7)

where N α� = 0 if α + � is not a root. Let Δ denote the set of roots of g. Note thatthe nonzero roots come in pairs, α ∈ Δ⇔ –α ∈ Δ, and no other nonzero multiple

B.2 Lie Algebras 305

of a root is a root. Accordingly, the set of nonzero roots may be split into a subsetΔ+ of positive roots and the corresponding subset of negative roots Δ– = {–α, α ∈Δ+}. The set Δ+ is contained in a simplex (convex pyramid) in h∗, the edges ofwhich are the so-called simple positive roots, that is, positive roots which cannotbe decomposed as the sum of two other positive roots. Of course, the same holdsfor Δ–. The consideration of root systems is the basis of the Cartan classificationof simple Lie algebras into four infinite series A�, B �, C �, D � and five exceptionalalgebras G2, F 4, E 6, E 7, E 8 [225].

In addition to the Lie algebra g, one may also consider the universal envelopingalgebra U(g), which consists of all polynomials in the elements of g, taking into ac-count the commutation relations (B3). Of special interest are the so-called Casimirelements, that generate the center of U(g), and in particular the quadratic Casimirelement C 2 =

∑ni ,k=1 g ikX i X k , where {X i} is the basis of g dual to {X i}, that is,

B (X i , X j ) = δij and g ik is the Cartan–Killing metric. The element C 2 does not de-

pend on the choice of the basis {X i}. In a Cartan–Chevalley basis {H j , E α}, onegets

C 2 =�∑

j=1

(H j )2 +∑α∈Δ

E αE –α . (B8)

In the same way, one defines a real Lie algebra as a real vector space with an anti-symmetric bracket [·, ·] that satisfies the Jacobi identity. The two concepts are closelyrelated. If g is a real Lie algebra, one can define its complexification gc by complex-ifying it as a vector space and extending the Lie bracket by linearity. If dim g = n,the complex dimension of gc is still n, but its real dimension is 2n. Conversely, if g

is a complex Lie algebra, and one restricts its parameter space to real numbers, oneobtains a real form gr , that is, a real Lie algebra whose complexification is again g.A given complex Lie algebra has in general several nonisomorphic real forms (alsoclassified by Cartan), among them a unique compact one, characterized by the factthat the Cartan–Killing metric g i j is negative-definite. For instance, the complexLie algebra A2 yields two real forms, su(3), the compact one, and su(2, 1).

For physical applications, it is not so much the Lie algebras themselves that mat-ter, but their representations in Hilbert spaces. The key ingredient for building thelatter, also due to Cartan, is the notion of a weight vector. Let T be a representation ofthe Lie algebra g in the Hilbert space h, that is, a linear map of g into the operatorson h such that

T ([X , Y ]) = [T (X ), T (Y )] , ∀X , Y ∈ g . (B9)

The representation T is called Hermitian if T (X ∗) = T (X )∗, for every X ∈ g.By Schur’s lemma, it follows that the Casimir elements of U(g) are simultane-

ously diagonalizable in any irreducible Hermitian representation of g, and in facttheir eigenvalues characterize the representation uniquely.

Let {H j , E α} be a Cartan–Chevalley basis of the complexified Lie algebra gc of g.Then a weight for T is a linear form α ∈ h∗, for which there exists a nonzero vector


|λ〉 ∈ h such that [T (H) – α(H)]n |λ〉 = 0, for all H ∈ h and some n. Then |λ〉 iscalled a weight vector if n = 1, that is, T (H)|λ〉 = α(H)|λ〉, ∀H ∈ h. In particular,|λ〉 is called a highest (or lowest) weight vector if

T (E α+ )|λ〉 = 0 , ∀α+ ∈ Δ+ , (or T (E α– )|λ〉 = 0 , ∀α– ∈ Δ–) , (B10)

where Δ+, or Δ–, is the set of the positive, or negative, roots. The interest of thisnotion is the fundamental result obtained by Cartan which says that the irreducible,finite-dimensional, Hermitian representations of simple Lie algebras are in one-to-one correspondence with highest-weight vectors for the geometrical constructionof representations in those terms).

We illustrate this with the simplest example, namely, su(2), with the familiarangular momentum basis { Jo , J+, J–} and Δ± = {±α}. In the unitary spin-j ir-reducible representation D j of SU (2), of dimension 2 j + 1, with standard basis{| j , m〉 |m = – j , . . . , j}, the highest (or lowest) weight vector | j , j〉 (or | j , – j〉) sat-isfies the relation J+| j , j〉 = 0 (or J–| j , – j〉 = 0) and has an isotropy subalgebrau(1) (by this, we mean the subalgebra that annihilates the given vector). Notice thathere C 2 = J2

o + J+ J– + J– J+ is indeed diagonal in D j , with eigenvalue j ( j + 1).

B.3Lie Groups

A Lie group G may be defined in several ways, for instance, as a smooth manifoldwith a group structure such that the group operations (g1, g2) �→ g1 g2, g �→ g–1

are C k , for some k v 2. This actually implies that all group operations are in fact(real) analytical. A Lie group is said to be simple, or semisimple, if it has no nontrivialinvariant subgroup, or Abelian invariant subgroup.

Let G be a Lie group. For g ∈ G consider the map Lg : G → G with Lg (g ′) = g g ′.The derivative of this map at g ′ ∈ G, denoted T g ′ (Lg ), sets up an isomorphismT g ′ (Lg ) : T g ′G → T g g ′G between the tangent spaces at g and g g ′. For any vectorX ∈ T eG (the tangent space at the identity), let us define a vector field X on G byX g = T e(Lg )X . Such a vector field is said to be left-invariant. It can be demonstratedthat the usual Lie bracket [X , Y ] of two left-invariant vector fields is again a left-invariant vector field. Using this fact, we see that if Y g = T e(Lg )Y , Y ∈ T eG,the relation [X , Y ] = [X , Y ] defines a Lie bracket on the tangent space, T eG, atthe identity of the Lie group G. Thus, equipped with this bracket relation, T eGbecomes a Lie algebra which we denote by g.

Next, it can be shown that for any X ∈ g, there exists a unique analytical homo-morphism, t �→ θX (t) of R into G such that

θX (0) =ddt

θX (t)|t=0 = X ,

and for each X ∈ g we then write exp X = θX (1). The map X �→ exp X is called theexponential map and it defines a homeomorphism between an open neighborhood

B.3 Lie Groups 307

N 0 of the origin 0 ∈ g and an open neighborhood N e of the identity element e of G.Each θX (t) defines a one-parameter subgroup of G, with infinitesimal generator X,

θX (t) = exp(tX ) ,

and every one-parameter subgroup is obtained in this way. If G is a matrix group,the elements X of the Lie algebra are also matrices and the exponential map comesout in terms of matrix exponentials.

Using this tool, we may sketch the fundamental theorems of Lie as follows:

• Every Lie group G has a unique Lie algebra g, obtained as the vector spaceof infinitesimal generators of all one-parameter subgroups, in other words,the tangent space T eG, at the identity element of G.

• Given a Lie algebra g, one may associate with it, by the exponential mapX �→ exp X , a unique connected and simply connected Lie group G, withLie algebra g (G is called the universal covering of G). Any other connectedLie group G with the same Lie algebra g is of the form G = G/D , where Dis an invariant discrete subgroup of G.

Furthermore, a Lie group G is simple, or semisimple, if and only if its Lie algebra g

is simple, or semisimple.Here again, one may start with a real Lie group G, build its complexification Gc , andfind all real forms of Gc . One, and only one, of them is compact (it corresponds, ofcourse, to the compact real form of the Lie algebra). For instance, the complex Liegroup SL(2, C) has two real forms, SU (2) and SU (1, 1), the former being compact.

A Lie group has natural actions on its Lie algebra and its dual. These are theadjoint and coadjoint actions, respectively, and may be understood in terms of theexponential map. For g ∈ G, g ′ �→ g g ′ g–1 defines a differentiable map from G toitself. The derivative of this map at g ′ = e is an invertible linear transformation,Adg , of T eG (or equivalently, of g) onto itself, giving the adjoint action. Thus, fort ∈ (–ε, ε), for some ε > 0, such that exp(tX ) ∈ G and X ∈ g,

Y =ddt

[g exp(tX ) g–1]|t=0 := Adg (X ) (B11)

is a tangent vector in T eG = g. If G is a matrix group, the adjoint action is simply

Adg (X ) = gX g–1 . (B12)

Now considering g �→ Adg as a function on G with values in End g, its derivativeat the identity, g = e, defines a linear map ad : g → End g. Thus, if g = exp X , thenAdg = exp(adX ), and it can be verified that (adX )(Y ) = [X , Y ].

The corresponding coadjoint actions are now obtained by dualization: the coad-joint action Ad#

g of g ∈ G on the dual g∗, of the Lie algebra, is given by

〈Ad#g (X ∗); X 〉 = 〈X ∗; Adg–1 (X )〉 , X ∗ ∈ g∗ , X ∈ g , (B13)

where 〈·; ·〉 == 〈·; ·〉g∗,g denotes the dual pairing between g∗ and g. Once again, the(negative of the) derivative of the map g �→ Ad#

g at g = e is a linear transformation


ad# : g→ End g∗, such that for any X ∈ g, (ad#)(X ) is the map

〈(ad#)(X )(X ∗); Y 〉 = 〈X ∗; (ad)(X )(Y )〉 , X ∗ ∈ g∗ , Y ∈ g. (B14)

Clearly, Ad#g = exp(–ad#X ). If we introduce a basis in g and represent Adg by a ma-

trix in this basis, then, in terms of the dual basis in g∗, Ad#g is represented by the

transposed inverse of this matrix.Under the coadjoint action, the vector space g∗ splits into a union of disjoint

coadjoint orbits

OX ∗ = {Ad#g (X ∗)|g ∈ G} . (B15)

According to the Kirillov–Souriau–Kostant theory (see [226]), each coadjoint orbitcarries a natural symplectic structure. In addition, Ω is G-invariant. This impliesin particular that the orbit is of even dimension and carries a natural G-invariant(Liouville) measure. Therefore, a coadjoint orbit is a natural candidate for realizingthe phase space of a classical system and hence a starting point for a quantizationprocedure.

Semisimple Lie groups have several interesting decompositions. In the sequel,we present three of them, the Cartan, the Iwasawa, and the Gauss decompositions.

(1) Cartan decomposition

This is the simplest case. Given a semisimple Lie group G, its real Lie algebra g

always possesses a Cartan involution, that is, an automorphism θ : g → g, withsquare equal to the identity,

θ[X , Y ] = [θ(X ), θ(Y )] , ∀X , Y ∈ g , θ2 = Id , (B16)

and such that the symmetric bilinear form Bθ(X , Y ) = –B (X , θY ) is positive-def-inite, where B is the Cartan–Killing form. Then the Cartan involution θ yields aneigenspace decomposition

g = k ⊕ p (B17)

of g into +1 and –1 eigenspaces. It follows that

[k, k] ⊆ k , [k, p] ⊆ p , [p, p] ⊆ k . (B18)

Assume for simplicity that the center of G is finite, and let K denote the analyticalsubgroup of G with Lie algebra k. Then:

• K is closed and maximal-compact,

• there exists a Lie group automorphism Θ of G, with differential θ, such thatΘ2 = Id and the subgroup fixed by Θ is K,

• the mapping K ~ p→ G given by (k, X ) �→ k exp X is a diffeomorphism.

One may also write the diffeomorphism as (X , k) �→ exp X k == p k.

B.3 Lie Groups 309

(2) Iwasawa decomposition

Any connected semisimple Lie group G has an Iwasawa decomposition into threeclosed subgroups, namely, G = K AN , where K is a maximal compact subgroup,A is Abelian, and N nilpotent, and the last two are simply connected. This meansthat every element g ∈ G admits a unique factorization g = kan, k ∈ K , a ∈A, n ∈ N , and the multiplication map K ~ A ~ N → G given by (k, a, n) �→ kan isa diffeomorphism.

Assume that G has a finite center. Let M be the centralizer of A in K, that is,M = {k ∈ K : ka = ak, ∀a ∈ A} (if the center of G is not finite, the definitionof M is slightly more involved). Then P = MAN is a closed subgroup of G, calledthe minimal parabolic subgroup. The interest in this subgroup is that the quo-tient manifold X = G/P ~ K /M carries the unitary irreducible representationsof the principal series of G (which are induced representations), in the sense thatthese representations are realized in the Hilbert space L2(X , ν), with ν the naturalG-invariant measure. To give a concrete example, take G = SOo (3, 1), the Lorentzgroup. Then the Iwasawa decomposition reads

SOo (3, 1) = SO (3) · A ·N , (B19)

where A ~ SOo (1, 1) ~ R is the subgroup of Lorentz boosts in the z-direction andN ~ C is two-dimensional and Abelian. Then M = SO (2), the subgroup of rotationsaround the z-axis, so that X = G/P ~ SO (3)/SO (2) ~ S2, the 2-sphere.

A closely related decomposition is the so-called K AK decomposition. Let G againbe a semisimple Lie group with a finite center, K a maximal compact subgroup,and G = K AN the corresponding Iwasawa decomposition. Then every element inG has a decomposition as k1ak2 with k1, k2 ∈ K and a ∈ A. This decompositionis in general not unique, but a is unique up to conjugation. A familiar example ofa K AK decomposition is the expression of a general rotation γ ∈ SO (3) as theproduct of three rotations, parameterized by the Euler angles:

γ = m(ψ) u(θ) m(ϕ) , (B20)

where m and u denote rotations around the z-axis and the y-axis, respectively.

(3) Gauss decomposition

Let G again be a semisimple Lie group and Gc = exp gc the corresponding com-plexified group. If b is a subalgebra of gc , we call it maximal (in the sense ofPerelomov [10]) if b ⊕ b = gc , where b is the conjugate of b in gc . Let {H j , E α} bea Cartan–Chevalley basis of the complexified Lie algebra gc .

Then Gc possesses remarkable subgroups:

• Hc , the Cartan subgroup generated by {H j}.• B±, the Borel subgroups, which are maximal connected solvable subgroups,

corresponding to the subalgebras b±, generated by {H j , E α |α ∈ Δ±}; if b

is maximal, then b+ = b and b– = b generate Borel subgroups.


• Z±, the connected nilpotent subgroups generated by {E α± |α± ∈ Δ±}.

The interest in these subgroups is that almost all elements of Gc admit a Gaussdecomposition:

g = z+hz– = b+z– = z+b– , z± ∈ Z± , h ∈ Hc , b± ∈ B± . (B21)

It follows that the quotients X + = Gc/B– and X – = B+\Gc are compact complexhomogeneous manifolds, on which Gc acts by holomorphic transformations.

B.3.1Extensions of Lie algebras and Lie groups

It is useful to have a method for constructing a group G from two smaller ones,one of them at least becoming a closed subgroup of G. Several possibilities areavailable. Here, we describe the two simplest ones.

(1) Direct product

This is the most trivial solution, which consists in glueing the two groups togeth-er, without interaction. Given two (topological or Lie) groups G1, G2, their directproduct G = G1 ~ G2 is simply their Cartesian product, endowed with the grouplaw:

(g1, g2)(g ′1, g ′2) = (g1 g ′1, g2 g ′2) , g1, g ′1 ∈ G1, g2, g ′2 ∈ G2 . (B22)

With the obvious identifications g1 ~ (g1, e2), g2 ~ (e1, g2), where e j denotes theneutral element of G j , j = 1, 2, it is clear that both G1 and G2 are invariant sub-groups of G1 ~ G2. In the case of Lie groups, the notion of direct product corre-sponds to that of direct sum of the corresponding Lie algebras, g = g1 ⊕ g2, andagain both g1 and g2 are ideals of g.

(2) Semidirect product

A more interesting construction arises when one of the groups, say, G2, acts onthe other one, G1, by automorphisms. More precisely, there is given a homomor-phism α from G2 into the group Aut G1 of automorphisms of G1. Although thegeneral definition may be given as in the first case, we consider only the case whereG1 == V is Abelian, in fact a vector space (hence group operations are noted addi-tively), and G2 == S is a subgroup of Aut V . Then we define the semidirect productG = V � S as the Cartesian product, endowed with the group law:

(v , s)(v ′, s ′) = (v + αs (v ′), ss ′) , v , v ′ ∈ V , s, s ′ ∈ S . (B23)

The law (B23) entails that the neutral element of G is (0, eS ) and the inverse of (v , s)is (v , s)–1 = (–α–1

s (v ), s–1) = (–αs–1 (v ), s–1). It is easy to check that V is an invariantsubgroup of G, while S is not in general. As a matter of fact, S is invariant if and

B.3 Lie Groups 311

only if the automorphism α is trivial, that is, the product is direct. Indeed one hasreadily

(v , s)(v ′, eS )(v , s)–1 = (αs (v ′) , eS ) ∈ V ,

and

(v , s)(0, s ′)(v , s)–1 = (v , ss ′)(–α–1s (v ), s–1) = (v – αss ′s–1 (v ), ss ′s–1) .

In addition to the Weyl–Heisenberg group GWH = R � R2 that we discussed inChapter 3, the following groups are examples of semidirect products of this typeare:

• The Euclidean group E (n) = Rn � SO (n);

• The Poincaré group P↑+ (1, 3) = Rn � SOo (1, 3), where the second factor isthe Lorentz group;

• The similitude group SIM(n) = Rn � (R+∗ ~ SO (n)), where R+

∗ is the groupof dilations, whereas SO (n) denotes the rotations, as in the first example(since these two operations commute, one gets here a direct product).

313

Appendix CSU(2) Material

In this appendix, we just give a list of formulas concerning the group SU (2) andits representations, material necessary for the construction of sigma-spin coher-ent states and the resulting fuzzy sphere. They are essentially extracted from Tal-man [74] and Edmonds [75], and also from [212].

C.1SU(2) Parameterization

SU (2) � � =

(�0 + i�3 –�2 + i�1

�2 + i�1 �0 – i�3

). (C1)

In bicomplex angular coordinates,

�0 + i�3 = cos ωeiψ1 , �1 + i�2 = sin ωeiψ2 (C2)

0 � ω � π2

, 0 � ψ1, ψ2 < 2π (C3)

and so

SU (2) � � =

(cos ωeiψ1 i sin ωeiψ2

i sin ωe–iψ2 cos ωe–iψ1

), (C4)

in agreement with Talman [74].

C.2Matrix Elements of SU(2) Unitary Irreducible Representation

D jm1m2 (�) = (–1)m1–m2

[( j + m1)!( j – m1)!( j + m2)!( j – m2)!

]1/2

~∑

t

(�0 + i�3) j–m2–t

( j – m2 – t)!(�0 – i�3) j+m1–t

( j + m1 – t)!

~(–�2 + i�1)t+m2–m1

(t + m2 – m1)!(�2 + i�1)t

t !, (C5)


314 Appendix C SU (2) Material

in agreement with Talman. With angular parameters the matrix elements of theunitary irreducible representation of SU (2) are given in terms of Jacobi polynomi-als [18] by

D jm1m2 (�) =e–im1(ψ1+ψ2)e–im2(ψ1–ψ2)im2–m1

√( j – m1)!( j + m1)!( j – m2)!( j + m2)!

~1

2m1(1 + cos 2ω)

m1 +m22 (1 – cos 2ω)

m1 –m22 ~

~ P (m1–m2,m1+m2)j–m1

(cos 2ω) , (C6)

in agreement with Edmonds [75] (up to an irrelevant phase factor).

C.3Orthogonality Relations and 3j Symbols

Let us equip the SU (2) group with its Haar measure:

μ(d�) = sin 2ω dω dψ1 dψ2 , (C7)

in terms of the bicomplex angular parameterization. Note that the volume of SU (2)with this choice of normalization is 8π2. The orthogonality relations satisfied by thematrix elements D j

m1m2 (�) read as∫SU (2)

D jm1m2 (�) D j ′

m′1m′2(�) μ(d�) =

8π2

2 j + 1δ j j ′δm1m′1

δm2m′2. (C8)

In connection with the reduction of the tensor product of two unitary irreduciblerepresentations of SU (2), we have the following equivalent formula involving theso-called 3j symbols (proportional to Clebsch–Gordan coefficients), in the Talmannotation:

D jm1m2 (�) D j ′

m′1m′2(�) =

∑j ′′m′′1 m′′2

(2 j ′′ + 1)

(j j ′ j ′′

m1 m′1 m′′1

)

~

(j j ′ j ′′

m2 m′2 m′′2

)D j ′′

m′′1 m′′2(�) , (C9)

∫SU (2)

D jm1m2 (�) D j ′

m′1m′2(�) D j ′′

m′′1 m′′2(�) μ(d�) =

8π2

(j j ′ j ′′

m1 m′1 m′′1

)(j j ′ j ′′

m2 m′2 m′′2

). (C10)

C.4 Spin Spherical Harmonics 315

One of the multiple expressions of the 3 j symbols (in the convention that they areall real) is given by(

j j ′ j ′′

m m′ m′′

)= (–1) j– j ′–m′′

[( j + j ′ – j ′′)!( j – j ′ + j ′′)!(– j + j ′ + j ′′)!

( j + j ′ + j ′′ + 1)!

]1/2

~[( j + m)!( j – m)!( j ′ + m′)!( j ′ – m′)!( j ′′ + m′′)!( j ′′ – m′′)!

]1/2

~∑

s

(–1)s 1s!( j ′ + m′ – s)!( j – m – s)!( j ′′ – j ′ + m + s)!

~1

( j ′′ – j – m′ + s)!( j + j ′ – j ′′ – s)!. (C11)

C.4Spin Spherical Harmonics

The spin spherical harmonics, as functions on the 2-sphere S2, are defined as fol-lows:

σY jμ(r) =

√2 j + 1

4π

[D j

μσ (� (Rr ))]∗

= (–1)μ–σ

√2 j + 1

4πD j

–μ–σ (� (Rr ))

=

√2 j + 1

4πD j

σμ(�† (Rr )

), (C12)

where � (Rr ) is a (nonunique) element of SU (2) which corresponds to the spacerotation Rr which brings the unit vector e3 to the unit vector r with polar coordi-nates

r =

⎧⎪⎨⎪⎩x1 = sin θ cos φ ,

x2 = sin θ sin φ ,

x3 = cos θ .

(C13)

We immediately infer from the definition (6.55) the following properties:(σY jμ(r)

)�= (–1)σ–μ

–σY j–μ(r) , (C14)

μ= j∑μ=– j

∣∣σY jμ(r)∣∣2

=2 j + 1

4π. (C15)

Let us recall here the correspondence (homomorphism) � = �(R) ∈ SU (2)↔R ∈SO (3) � SU (2)/Z2:

r ′ = (x ′1, x ′2, x ′3) = R · r ←→ (C16)

(ix ′3 –x ′2 + ix ′1

x ′2 + ix ′1 –ix ′3

)= �

(ix3 –x2 + ix1

x2 + ix1 –ix3

)�† . (C17)


In the particular case (C12), the angular coordinates ω, ψ1, ψ2 of the SU (2)-element� (Rr ) are constrained by

cos 2ω = cos θ , sin 2ω = sin θ , so 2ω = θ, (C18)

ei(ψ1+ψ2) =ieiφ so ψ1 + ψ2 = φ +π2

. (C19)

Here we should pay special attention to the range of values for the angle φ, de-pending on whether j and consequently σ and m are half-integer or not. If j ishalf-integer, then angle φ should be defined mod (4π), whereas if j is integer, itshould be defined mod (2π).

We still have one degree of freedom concerning the pair of angles ψ1, ψ2. Weleave open the option concerning the σ-dependent phase factor by putting

i–σeiσ(ψ1–ψ2) def= eiσψ , (C20)

where ψ is arbitrary. With this choice and considering (C5), we get the expressionsfor the spin spherical harmonics in terms of φ, θ/2, and ψ:

σY jμ(r) = (–1)σeiσψeiμφ

√2 j + 1

4π

√( j + μ)!( j – μ)!( j + σ)!( j – σ)!

~

(cos

θ2

)2 j ∑t

(–1)t

(j – σ

t

)(j + σ

t + σ – μ

)(tan

θ2

)2t+σ–μ

,

(C21)

= (–1)σeiσψeiμφ

√2 j + 1

4π

√( j + μ)!( j – μ)!( j + σ)!( j – σ)!

~

(sin

θ2

)2 j ∑t

(–1) j–t+μ–σ

(j – σt – μ

)(j + σt + σ

)(cot

θ2

)2t+σ–μ

, (C22)

which are not in agreement with the definitions of Newman and Penrose [77],Campbell [76] (note that there is a mistake in the expression given by Campbell,in which a cos θ

2 should read cot θ2 ), and Hu and White [78]. Besides the presence

of different phase factors, the disagreement is certainly due to a different relationbetween the polar angle θ and the Euler angle.

Now, considering (C6), we get the expression for the spin spherical harmonics interms of the Jacobi polynomials, valid in the case in which μ± σ > –1:

σY jμ(r) = (–1)μeiσψ

√2 j + 1

4π

√( j – μ)!( j + μ)!( j – σ)!( j + σ)!

~12μ (1 + cos θ)

μ+σ2 (1 – cos θ)

μ–σ2 P (μ–σ,μ+σ)

j–μ (cos θ) eiμφ . (C23)

C.5 Transformation Laws 317

For other cases, it is necessary to use alternative expressions based on the rela-tions [18]

P (–l,�)n (x) =

(n+�l

)(nl

) (x – 1

2

)l

P (l,�)n–l (x) , P (α,�)

0 (x) = 1 . (C24)

Note that with σ = 0 we recover the expression of the normalized spherical har-monics

0Y jm (r) = Y jm (r )

= (–1)m

√2 j + 1

4π

√( j – m)!( j + m)!

1j ! 2m (sin θ)mP (m,m)

j–m (cos θ) eimφ

=

√2 j + 1

4π

√( j – m)!( j + m)!

P mj (cos θ)eimφ ,

(C25)

since we have the following relation between associated Legendre polynomials andJacobi polynomials:

P (m,m)j–m (z) = (–1)m2m (1 – z2)– m

2j !

( j + m)!P m

j (z) , (C26)

for m > 0. We recall also the symmetry formula

P –mj (z) = (–1)m ( j – m)!

( j + m)!P m

j (z) . (C27)

Our expression for spherical harmonics is rather standard, in agreement with Ar-fken [227, 228].19)

C.5Transformation Laws

We consider here the transformation law of the spin spherical harmonics underthe rotation group. From the relation

RRtRr = Rr (C28)

for any R ∈ SO (3), and from the homomorphism �(RR′) = �(R)�(R′) betweenSO (3) and SU (2), we deduce from the definition (6.55) of the spin spherical har-

19) Sometimes (e.g., Arfken [227]), theCondon–Shortley phase (–1)m is prependedto the definition of the spherical harmonics.Talman adopted this convention.


monics the transformation law

σY jμ(tR · r ) =

√2 j + 1

4πD j

σμ(

�† (RtR·r ))

=

√2 j + 1

4πD j

σμ(�†

(tRRr

))=

√2 j + 1

4πD j

σμ(

�† (Rr ) � (R))

=

√2 j + 1

4π

∑ν

D jσν

(�† (Rr )

)D j

νμ (� (R))

=∑

ν

σY jν(r)D jνμ (� (R)) ,

(C29)

as expected if we think of the special case (σ = 0) of the spherical harmonics.Given a function f (r) on the sphere S2 belonging to the (2 j + 1)-dimensional

Hilbert spaceHσ j and a rotationR ∈ SO (3), we define the representation operatorDσ j (R) by(

Dσ j (R) f)

(r) = f (R–1 · r) = f (tR · r) . (C30)

Thus, in particular,(Dσ j (R)σY jμ

)(r) =σ Y jμ(tR · r) . (C31)

The generators of the three rotationsR(a) , a = 1, 2, 3, around the three usual axes,are the angular momentum operator in the representation. When σ = 0, we recoverthe spherical harmonics, and these generators are the usual angular momentumoperators J ( j)

a (or simply Ja), a = 1, 2, 3, for that representation. In the general caseσ =/ 0, we call them Λ(σ j )

a . We study their properties below.

C.6Infinitesimal Transformation Laws

Recalling that the components Ja = –iεabc xb∂c of the ordinary angular momentumoperator are given in spherical coordinates by

J3 = –i∂φ ,

J+ = J1 + i J2 = eiφ(∂θ + i cot θ∂φ

),

J– = J1 – i J2 = –e–iφ(∂θ – i cot θ∂φ

),

(C32)

we have for the “sigma-spin” angular momentum operators:

Λσ j3 = J3 = –i∂φ , (C33)

Λσ j+ = Λσ j

1 + iΛσ j2 = J+ + σ csc θeiφ , (C34)

C.7 Integrals and 3 j Symbols 319

Λσ j– = Λσ j

1 – iΛσ j2 = J– + σ csc θe–iφ . (C35)

They obey the expected commutation rules,[Λσ j

3 , Λσ j±

]= ±Λσ j

± ,[

Λσ j+ , Λσ j

–

]= 2Λσ j

3 . (C36)

These operators are the infinitesimal generators of the action of SU (2) on the spinspherical harmonics:

Λσ j3 σY jμ = μ σY jμ (C37)

Λσ j+ σY jμ =

√( j – μ)( j + μ + 1) σY jμ+1 (C38)

Λσ j– σY jμ =

√( j + μ)( j – μ + 1) σY jμ–1 . (C39)

C.7Integrals and 3j Symbols

Specifying (C8) for the spin spherical harmonics leads to the following orthogonal-ity relations which are valid for integer j (and consequently integer σ):∫

S2σY jμ(r) σY j ′ν(r) μ(dr ) = δ j j ′δμν . (C40)

We recall that in the integer case, the range of values assumed by the angle φ is0 � φ < 2π. Now, if we consider half-integer j (and consequently σ), the range ofvalues assumed by the angle φ becomes 0 � φ < 4π. The integration above has tobe carried out on the “doubled” sphere S2 and an extra normalization factor equalto 1√

2is needed in the expression of the spin spherical harmonics.

For a given integer σ the set{

σY jμ, –∞ � μ �∞, j � max (0, σ, m)}

forms anorthonormal basis of the Hilbert space L2(S2). Indeed, at μ fixed so that μ± σ � 0,the set {√

2 j + 14π

√( j – μ)!( j + μ)!( j – σ)!( j + σ)!

12μ (1 + cos θ)

μ+σ2 (1 – cos θ)

μ–σ2

~ P (μ–σ,μ+σ)j–μ (cos θ) , j � μ

}(C41)

is an orthonormal basis of the Hilbert space L2([–π, π], sin θ dθ). The same holdsfor other ranges of values of μ by using alternative expressions such as (C24)for Jacobi polynomials. Then it suffices to view L2(S2) as the tensor productL2([–π, π], sin θ dθ)

⊗L2(S1). Similar reasoning is valid for half-integer σ. Then,

the Hilbert space to be considered is the space of “fermionic” functions on thedoubled sphere S2, that is, such that f (θ, φ + 2π) = – f (θ, φ).


Specifying (C9) for the spin spherical harmonics leads to

σY jμ(r)σ′Y j ′μ′ (r) =∑j ′′μ′′σ′′

√(2 j + 1)(2 j ′ + 1)(2 j ′′ + 1)

4π

~

(j j ′ j ′′

μ μ′ μ′′

)(j j ′ j ′′

σ σ′ σ′′

)σ′′Y j ′′μ′′ (r) . (C42)

We easily deduce from (C42) and with the constraint that σ+σ′+σ′′ = 0 the followingintegral involving the product of three spherical spin harmonics (in the integercase, but analogous formula exists in the half-integer case):∫

S2σY jμ(r)σ′Y j ′μ′ (r)σ′′Y j ′′μ′′ (r) μ(dr ) =

√(2 j + 1)(2 j ′ + 1)(2 j ′′ + 1)

4π

~

(j j ′ j ′′

μ μ′ μ′′

)(j j ′ j ′′

σ σ′ σ′′

). (C43)

Note that this formula is independent of the presence of a constant phase factorof the type eiσψ in the definition of the spin spherical harmonics because of thea priori constraint σ +σ′ +σ′′ = 0. On the other hand, we have to be careful in apply-ing (C43) because of this constraint, that is, since it has been derived from (C42)on the ground that σ′′ was already fixed at the value σ′′ = –σ – σ′. Therefore, thecomputation of∫

S2σY jμ(r)σ′Y j ′μ′ (r)σ′′Y j ′′μ′′ (r) μ(dr ) (C44)

for an arbitrary triplet (σ, σ′, σ′′) should be carried out independently.

C.8Important Particular Case: j = 1

In the particular case j = 1, we get the following expressions for the spin sphericalharmonics:

σY 10(r) = eiσψ

√3

4π1√

(1 + σ)!(1 – σ)!

(cot

θ2

)σ

cos θ , (C45)

σY 11(r) = –eiσψ

√3

4π1√

2(1 + σ)!(1 – σ)!

(cot

θ2

)σ

sin θ eiφ , (C46)

σY 1–1(r) = (–1)σe–iσψ

√3

4π1√

2(1 + σ)!(1 – σ)!

(tan

θ2

)σ

sin θ e–iφ .

(C47)

For σ = 0, we recover familiar formulas connecting spherical harmonics to compo-nents of the vector on the unit sphere:

Y 10(r) =

√3

4πcos θ =

√3

4πx3 , (C48)

C.9 Another Important Case: σ = j 321

Y 11(r) = –

√3

4π1√2

sin θeiφ = –

√3

4πx1 + ix2√

2, (C49)

Y 1–1(r) =

√3

4π1√2

sin θe–iφ =

√3

4πx1 – ix2√

2. (C50)

C.9Another Important Case: σ = j

For σ = j , owing to the relations (6.63), the spin spherical harmonics reduce totheir simplest expressions:

j Y jμ(r ) = (–1) j ei jψ

√2 j + 1

4π

√(2 j

j + μ

)(cos

θ2

) j+μ (sin

θ2

) j–μ

eiμφ .

(C51)

They are precisely the states which appear in the construction of the Gilmore–Radcliffe coherent states. Otherwise said, the latter and related quantization arejust particular cases of our approach.

323

Appendix DWigner–Eckart Theorem for CoherentState Quantized Spin Harmonics

We give more precision on the rotational covariance properties of some operatorsobtained in Chapter 17 from the coherent state quantization of spin spherical har-monics.

By construction, the operators AνY kn acting on Hσ j are tensorial-irreducible. In-deed, under the action of the representation operator Dσ j (R) inHσ j , due to (6.83),the rotational invariance of the measure andN (r ), and (C29), they transform as

Dσ j (R)AνY kn Dσ j (R–1) =∫

S2νY kn(r) |R · r ; σ〉〈R · r ; σ| N (r) μ(dr)

=∫

XνY kn(R–1 · r) |r ; σ〉〈r ; σ| N (r ) μ(dr)

=∑

n′

D kn′n (� (R))

∫S2

νY kn′ (r) |r ; σ〉〈r ; σ| N (r) μ(dr)

=∑

n′

AνY kn′Dkn′n (� (R)) .

(D1)

Therefore, the Wigner–Eckart theorem [75] tells us that the matrix elements ofthe operator Aν Y kn with respect to the spin spherical harmonic basis

{σY jm

}are

given by[AνY kn

]mm′

= (–1) j–m

(j j k

–m m′ n

)K(ν, σ, j , k) . (D2)

Note that the presence of the 3 j symbol in (D2) implies the selection rules n + m′ =m and the triangular rule 0 � k � 2 j . The proportionality coefficient K can becomputed directly from (17.6) by choosing therein suitable values of m, m′.

On the other hand, we have by definition (C29,C31)∑n′

νY kn′D kn′n (� (R)) = Dνk (R)νY kn .

Thus, from the formula above,

Dσ j (R)AνY kn D j (R–1) = ADνk (R)νY kn.


324 Appendix D Wigner–Eckart Theorem for Coherent State Quantized Spin Harmonics

In the special case ν = 0,

Dσ j (R)AY kn D j (R–1) = AD0k (R)Y kn. (D3)

Its infinitesimal version for each of the three rotationsRi reads as[Λ(σ j )

i , AY kn

]= A J (k)

i Y kn. (D4)

325

Appendix ESymmetrization of the Commutator

We want to prove that

S([

J3, Jα11 Jα2

2 Jα33

])=

[J3, S

(Jα1

1 Jα22 Jα3

3

)],

where Ji is a representation of so(3).Let us first comment on the symmetrization:

S(

Jα11 Jα2

2 Jα33

)=

1l !

∑σ∈Sl

J iσ(1) . . . Jiσ(l) ,

where l = α1 + α2 + α3. The terms of the sum are not all distinct, since the exchangeof, say, two J1 gives the same term: each term appears in fact α1!α2!α3! times, sothere are l !/(α1!α2!α3!) distinct terms. This is the number of sequences of length l,with values in {1, 2, 3}, where there are αi occurrences of the value i (for i = 1, 2, 3).One denotes this set as U α1,α2,α3 . After grouping of identical terms, one obtains

S(

Jα11 Jα2

2 Jα33

)=

α1!α2!α3!l !

∑u∈U α1,α2,α3

Ju1 . . . Jul ,

where all the terms of the summation are now different.Let us now calculate S

([J3, Jα1

1 Jα22 Jα3

3

]). First, we write[

J3, Jα11 Jα2

2 Jα33

]=

[J3, Jα1

1

]Jα2

2 Jα33︸︷︷︸

A

+ Jα11

[J3, Jα2

2

]Jα3

3︸︷︷︸B

,

with

A =α1∑

k=1

J1 . . . J1︸︷︷︸k–1 terms

J2 J1 . . . J1︸︷︷︸α1–k terms

Jα22 Jα3

3 .

The different terms in A give the same symmetrized operator. Thus,

S(A) = α1S(

Jα1–11 Jα2+1

2 Jα33

)(E1)


326 Appendix E Symmetrization of the Commutator

= α1(α1 – 1)!(α2 + 1)!α3!

l !

∑u∈U α1–1,α2+1,α3

Ju1 . . . Jul . (E2)

Similarly, for B,

S(B ) = –α2(α1 + 1)!(α2 – 1)!α3!

l !

∑u∈U α1+1,α2–1,α3

Ju1 . . . Jul .

Now we calculate

I =[

J3, S( Jα11 Jα2

2 Jα33 )

]=

α1!α2!α3!l !

∑u∈U α1,α2,α3

l∑k=1

Ju1 . . . Juk–1 [ J3, Juk ] Juk+1 . . . Jul .

The sum splits into two parts, according to the value of uk = 1 or 2:

I = A′ + B ′ ,

with

A′ =α1!α2!α3!

l !

∑u∈U α1,α2 ,α3

∑k|uk=1

Ju1 . . . Juk–1 J2 Juk+1 . . . Jul

and

B ′ = –α1!α2!α3!

l !

∑u∈U α1,α2,α3

∑k|uk=2

Ju1 . . . Juk–1 J1 Juk+1 . . . Jul .

Let us examine the constituents of A′. There are of the form Ju1 . . . Jul , withu ∈ U α1–1,α2+1,α3 . Their number is l !/(α1!α2!α3!) ~ α1, but they are not all different.Each monomial emerges from a term where a J1 has been transformed into a J2.Since there are α2 +1 occurrences of J2 in each term, each monomial appears α2 +1times. We now group these identical terms:

A′ =α1!α2!α3!

l !(α2 + 1)

∑?

Ju1 . . . Jul .

It remains to determine the definition set of the summation. Let us first estimatethe number of its terms, namely,

N =l !

α1!α2!α3!α1

α2 + 1=

l !(α1 – 1)!(α2 + 1)!α3!

.

This is the number of elements in U α1–1,α2+1,α3 . On the other hand, all the elementsof U α1–1,α2+1,α3 appear. In the contrary case, the retransformation of a J2 into aJ1 would provide some elements not appearing in I, which cannot be true. It re-

Appendix E Symmetrization of the Commutator 327

sults that the sum comprises exactly all symmetrized expressions of Jα1–11 Jα2+1

2 Jα33 .

Thus,

A′ =α1!α2!α3!

l !(α2 + 1)

∑u∈U α1–1,α2+1,α3

Ju1 . . . Jul

= α1(α1 – 1)!(α2 + 1)!α3!

l !

∑u∈U α1–1,α2+1,α3

Ju1 . . . Jul

= S(A) .

The application of the same method to B ′ leads to the proof.

329

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227 Arfken, G. (1985) Spherical Harmonics andIntegrals of the Products of Three SphericalHarmonics. 12.6 and 12.9 in MathematicalMethods for Physicists, p. 680 and p. 698,3rd ed., Academic Press, Orlando, FL.

228 Weisstein, E.W. Spherical Harmon-ic, MathWorld – A Wolfram Web Re-source. http://mathworld.wolfram.com/SphericalHarmonic.html.

339

Index

aAction–angle 140, 150Affine group Aff(R) 130Aguiar 48Ali 207Angle–action 32Annihilation operator 15Anosov diffeomorphism 260Anticommutation rules 179Arfken 317Arnold cat 260Ashtekar 256Autocorrelation function 155Avalanche photodiodes 63

bBaker–Campbell–Hausdorff 84, 184Barnett 232Bayes 58Bayesian 147

– duality 71, 76– statistical inference 70, 295

Bayes’s theorem 292Berezin 112, 194, 212Berezin integral

– Majid–Rodríguez-Plaza 219Bernoulli process 82Bessel function 150Bloch theorem 264Borel function 290Borel set 216Bouzouina 259Brownian regularization 47Bundle section 124Busch 234

cCampbell 316Canonical commutation rule 16, 196

Cartan– classification 305– decomposition 123, 126– involution 124, 308– subalgebra 185, 188, 304

Cartan–Killing metric 304Casimir operator 146Catlike superpositions 49Characteristic

– length 13– momentum 15

Classical observable 211Classical spin 79Clebsch–Gordan 314Closed-loop measurement 63Coherent radiation 22Coherent state family 73Coherent states (CS) 3, 148, 211

– atomic 79– Barut–Girardello 117, 153– Bloch 79– canonical 4– fermionic 179– finite set 224– k-fermionic 219– motion on discrete sets 257– on the circle 77– para-Grassmann 219– phase 235– quantization 110– Schrödinger 19– sigma-spin 93, 274– SO(2r) 186– spin 79, 180– spin or SU(2) or Bloch or atomic 81– squeezed 167– standard 20– SU(1,1) 117, 119– SU(2) 79


340 Index

– torus 265– two-photon 165– U(r) 182– vector 248

Commutation rule 251Continuous-time regularization 45Continuous wavelet analysis (CWT) 131Continuum limit 45Contraction 81Cook 61Correlation function 22Coset (group) 183Cosmological constant 283Counting interval 61Covariance matrix 168Creation operator 15

dDaubechies 46De Bièvre 259de Sitter hyperboloid 283de Sitter space-time 244Decomposition

– Cartan 308– Gauss 309– Iwasawa 309

Dense (family) 29Density

– matrix 22, 40– operator 22

Dequantization– CS 195

Detection efficiency 66Dicke model 103Dirac 199, 230Dirac comb 239Disentanglement 84, 129, 174Displacement amplitude 57Distribution

– Bernoulli 55, 296– beta 120, 299– binomial 111, 297– Cauchy 300– conditional posterior 76– conditional probability 70– degenerate 296– gamma 69–70, 299– Gaussian 155– hypergeometrical 297– inferred 71– negative binomial 120, 297– normal 300– Poisson 69–70, 102, 155, 298

– posterior 82– retrodictive 71– sub-Poissonian 155– super-Poissonian 155– uniform 296– Wigner semicircle 203, 299

Distribution “P” 42Distribution “Q” 41Distribution “R” 41Dolinar receiver 57dos Santos 48Dynamical algebra 143Dynamical system 260

– ergodic 260

eEdmonds 88, 314Egorov theorem 271Energy

– free 107, 116Englis 207Expected value 294Experimental protocol 75Extension

– central 16External-cavity diode laser 63Extremal states 80, 182, 186

fFeedback amplitude 57–58Feynman 44–45Fiber-optic intensity modulator 63Fock–Bargmann 185

– basis 127– Hilbert space 17, 193– spin CS 86– SU(1,1) CS 119

Fock space 21Form

– 2-form 32Frame 207, 212, 220Fundamental weights 181Fuzzy

– hyperboloid 284– manifold 208– sphere 278

Fuzzy sphere 273

gGabor transform 39Gaboret 39Gamma transform 196Gaussian

– convolution 195

Index 341

– distribution 29, 43– measure 17

Generating function 16, 88Geometrical probability 217Geometry

– hyperbolic 118– Lobatcheskian 118

Geremia 50, 61Glauber 20, 170Glogower 231Green function 99Grosse 274Group

– de Sitter SO0(1,2) 283– de Sitter SO0(1,4) 283– direct product 310– discrete Weyl–Heisenberg 263– P+

↑ 283– semidirect product 310– SL(2,Z) 260– SL(2,R) 123– SO(2r + 1) 187– SO(2r) 185– Sp(2,R) 165– Spin(2r) 186– stability 186– SU(1,1) 123– SU(2) 87, 179, 226, 306, 313– U(r), SU(r) 181– Weyl–Heisenberg 262

hHaar basis 227Hamilton–Jacobi–Bellman equation 59Heisenberg inequality 25, 31, 169Helstrom bound 50, 52Hepp 103, 109Hermite polynomial 14, 203Hermitian space 74Hioe 109Hirota 56Hollenhorst 165Homodyne detection 167Hopf

– fibration 85– map 85

Husimi function 41Hyperbolic automorphism 260Hypergeometrical polynomial 125

iInequalities

– Berezin–Lieb 115

– Cauchy–Schwarz 30– Peierls–Bogoliubov 114– spin CS 85

Inequality– Cauchy–Schwarz 170

Infinite square well 137, 245

j3j symbols 276, 314Jacobi polynomials 88, 125Jordan algebra 214, 229

kKennedy receiver 54Killing vectors 283Kirillov–Souriau–Kostant theory 308Klauder 20, 45–46, 135, 194Kolmogorov 291Kähler

– manifold 118, 184, 187– potential 118, 123, 184

lLadder operator 30Lahti 234Landsman 207Lattice regularization 45Lie algebra 304

– root 304– semisimple 304– simple 304– weight 305

Lie group 306– (co)adjoint action 307– coadjoint orbit 308– exponential map 306

Lie theorem 307Lieb 103, 109, 111–112, 212Light

– amplitude-squeezed 170– phase-squeezed 170

Likelihood function 71, 76, 82Likelihood ratio 59Limit

– classical 112– thermodynamical 112

Louisell 231Lower symbol 253Lowest weight 126

mMach–Zehnder interferometer 63Madore 208, 273, 278Mandel parameter 155

342 Index

Martin 61Mean-field approximation 182, 189Measurable function 290Measure 290

– Bohr 150– Haar 89– Liouville 32– pinned Wiener measure 48– probability 291– rotationally invariant 80

Mixed state 40Mixing map 261Mode 21Moment problem

– Stieltjes 147Momentum

– representation 14, 264Möbius transformation 118

nNewman 316Nieto 165Noncommutative geometry 208Noncommutativity 38Normal law 77Number

– basis 15– operator 15

oObservable

– CS quantizable 194, 200– quantizable 212

Observation set 72Operator

– angle 230, 243– Casimir 127, 245, 284– compact 115– cosine, sine 231– displacement 37, 100, 167, 183– ladder for spin 80– number 229– Pegg–Barnett phase 232– phase 230, 235– position 228, 250– spin angular momentum 93– squeezing 167– symmetrization 263

Optimal control 60Optimality principle 59Order

– antinormal 42– normal 43

Orthogonality relations 91Oscillator

– driven 98Overcomplete (family) 29Overcompleteness 28Overlap of coherent states 26Overlap of CS 85, 151, 185

pPara-Grassmann algebra 218Partition function 105, 114Path integral 44Pauli matrices 83, 104, 225Pegg 232Penrose 316Perelomov 87, 117, 127Phase quadrature 167Phase space

– classical 72Phase trajectory 137, 140Phase transition 108Poincaré

– halfplane 118– metric 118

Poisson 55– process 298– summation 77–78, 199

Poisson algebra 208Polarization 21Polychronakos model 205Popov 231Position

– representation 14, 264Positive-operator-valued measure (POVM)

28, 50Posterior probability distribution 71, 120Potential

– chemical 190– deformation 190– Hartree–Fock 190– pairing 190

POVM 52, 75, 217Prešnajder 274Prior measure 71, 76, 82, 120Probability

– conditional 291– density 294– distribution 293

Probability space 291Probe 35, 73Propagator 45, 47Punctured disk 118Pöschl–Teller potential 139, 142

Index 343

qq-deformed polynomial 219Quantization

– anti-Wick 194– Berezin–Toeplitz 194– Bohr–Sommerfeld 142– canonical 196, 208– coherent states 110, 208– deformation 208– fuzzy 273– geometrical 208– hyperboloid 284– motion on circle 241– motion on discrete sets 257– motion on torus 269– 2-sphere 275– Weyl 267

Quantum error probability 52Quantum information processing 49Quantum key distribution 49Quantum limit 50Quantum measurement 49Quantum processing 73, 207Quaternion 83

rRadiation

– thermal 43Random variable 293Reciprocal torus 262Representation 303

– discrete series 125, 145– irreducible 303– irreducible unitary 37– multiplier 303– projective 38– SO0(1,2) principal series 245– spinorial 188– square integrable 73– unitary 36, 303

Reproducing– kernel 29– kernel space 29

Reproducing Hilbert space 75, 86Resolution of the identity or unity 4Resolution of the unity 73, 152, 194Revival

– fractional 153– quantum 153– time 141, 153

Riemann sphere 81, 86Riesz theorem 210Rotating-wave approximation 103–104

sS matrix 101, 177Saddle point approximation 195Sasaki 56Schrödinger

– equation 14, 44, 46, 109Schur lemma 39, 73, 128, 266Schwartz space 200, 263Semidirect product 131, 172Separable Hilbert space 74shell structure 189Shot noise 61Sigma-algebra 213, 289

– Borel 290Signal analysis 39Sphere S2 80Spherical harmonics 86Spin-flip 111Spin spherical harmonics

79, 89, 111, 274, 315Squeezed states 165Stone–von Neumann theorem 37Sudarshan 20, 195Superradiance 103Susskind 231Symbol 30

– contravariant 30, 112, 195, 212– covariant 112, 195, 212– lower 25, 30, 112, 195, 212– upper 112, 195, 212

Symmetrization– commutator 325

Symplectic– area 85– correction 31– Fourier transform 42– group 165– Lie algebra 16– phase 20

tTalman 88, 314Temperature

– critical 108Tempered distributions 200, 261Temporal evolution 158Temporal stability 32, 149Temporally stable 25Theta function 77–78, 151Time–frequency plane 72Time–frequency representation 39Time-scale half-plane 72Time-scale transform 122

344 Index

Torus 118– two-dimensional 259

Total (family) 29Transformation

– Hartree–Fock–Bogoliubov 189– homographic 122– Möbius 122

Transmission coefficient 54Two-photon

– Lie algebra 172

uUIR

– discrete Weyl–Heisenberg 263– SO0(1,2) principal series 285– SU(1,1) 125, 145– SU(2) 88, 277– SU(2) matrix elements 314

Uncertainty relation 31Unit disk 117Unity

– resolution 26

vVacuum energy 31

wWang 109Wavelet 39, 131Wavelet family 73wavelet(s) 6

– 1-D Mexican hat, Marr 132– 1-D Morlet 133

Weak sense 27Weyl formula 36Weyl–Heisenberg

– group 38Weyl-Heisenberg

– algebra 16Wigner distribution 43Wigner–Eckart theorem 281, 323Wigner–Ville transform 44Windowed Fourier transform 39

yYarunin 232

zZassenhaus 84

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