The Wigner Transform

by Maurice A De Gosson

Overview

This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.

Product Details

ISBN-13:	9781786343093
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	05/23/2017
Series:	Advanced Textbooks In Mathematics
Pages:	252
Product dimensions:	6.00(w) x 8.90(h) x 0.60(d)

Table of Contents

Preface vii

Introduction ix

Part I General Mathematical Framework 1

1 Phase Space Translations and Reflections 3

1.1 Some Notation 3

1.1.1 The spaces Rⁿ_x and Rⁿ_p 3

1.1.2 The symplectic structure of phase space 4

1.1.3 Some usual function spaces 5

1.1.4 Fourier transform 6

1.2 The Heisenberg-Weyl and Grossmann-Rover Operators 6

1.2.1 The displacement Hamiltonian 6

1.2.2 The Heisenberg Weyl operators 7

1.2.3 The Grossmann-Royer parity operators 9

1.3 A Functional Relation Between T(Z₀) and R(Z₀) 12

1.4 Quantization of Exponentials 14

2 The Cross-Wigner Transform 17

2.1 Definitions of the Cross-Wigner Transform 17

2.1.1 First definition 17

2.1.2 Wigner's definition 18

2.1.3 The Gabor transform and its variants 19

2.1.4 Extension to tempered distributions 20

2.2 Properties of the Cross-Wigner Transform 22

2.2.1 Elementary algebraic properties 22

2.2.2 Analytical properties and continuity 24

2.2.3 The marginal properties 26

2.2.4 Translating Wigner transforms 27

3 The Cross-Ambiguity Function 31

3.1 Definition of the Cross-Ambiguity Function 31

3.1.1 Definition using the Heisenberg Weyl operator 31

3.1.2 Traditional definition 32

3.1.3 The Fourier-Wigner transform 33

3.2 Properties and Relation with the Wigner Transform 34

3.2.1 Properties of the cross-ambiguity function 34

3.2.2 Relation with the cross-Wigner transform 35

3.2.3 The maximum of the ambiguity function 37

4 Weyl Operators 39

4.1 The Notion of Weyl Operator 39

4.1.1 Weyl's definition, and rigorous definitions 39

4.1.2 The distributional kernel of a Weyl operator 43

4.1.3 Relation with the cross-Wigner transform 47

4.2 Some Properties of the Weyl Correspondence 49

4.2.1 The adjoint of a Weyl operator 49

4.2.2 An L² boundedness result 50

5 Symplectic Covariance 53

5.1 Symplectic Covariance Properties 53

5.1.1 Review of some properties of Mp(n) and Sp(n) 53

5.1.2 Proof of the symplectic covariance property 55

5.1.3 Symplectic covariance of Weyl operators 58

5.2 Maximal Covariance 59

5.2.1 Antisymplectic matrices 60

5.2.2 The maximality property 62

5.2.3 The case of Weyl operators 64

6 The Moyal Identity 67

6.1 Precise Statement and Proof 67

6.1.1 The general Moyal identity 67

6.1.2 A continuity result 69

6.2 Reconstruction Formulas 70

6.2.1 Reconstruction using the cross-Wigner transform 70

6.2.2 Reconstruction using the cross-ambiguity function 71

6.3 The Wavepacket Transforms 72

6.3.1 Definition 72

6.3.2 Properties of the wavepacket transform 73

7 The Feichtinger Algebra 77

7.1 Definition and First Properties 77

7.1.1 Definition of S₀(Rⁿ) 77

7.1.2 Analytical properties of S₀(Rⁿ) 82

7.1.3 The algebra property of S₀(Rⁿ) 85

7.2 The Dual Space S'₀(Rⁿ) 86

7.2.1 Description of S'₀(Rⁿ) 86

7.2.2 The Gelfand triple (S₀, L², S'₀) 87

8 The Cohen Class 89

8.1 Definition 89

8.14 The marginal conditions 92

8.1.2 Generalization of Moyal's identity 94

8.1.3 The operator calculus associated with Q 95

8.2 Two Examples 97

8.2.1 The generalized Husimi distribution 97

8.2.2 The Born -Jordan transform 100

9 Gaussians and Hermite Functions 103

9.1 Wigner Transform of Generalized Gaussians 103

9.1.1 Generalized Gaussian functions 103

9.1.2 Explicit results 105

9.1.3 Cross-ambiguity function of a Gaussian 108

9.1.4 Hudson's theorem 109

9.2 The Case of Hermite Functions 109

9.2.1 Short review of the Hermite and Laguerre functions 109

9.2.2 The Wigner transform of Hermite functions 112

9.2.3 The cross-Wigner transform of Hermite functions 113

9.2.4 Flandrin's conjecture 116

10 Sub-Gaussian Estimates 119

10.1 Hardy's Uncertainty Principle 119

10.1.1 The one-dimensional case 119

10.1.2 Two lemmas 120

10.1.3 The multidimensional Hardy uncertainty principle 121

10.2 Sub-Gaussian Estimates for the Wigner Transform 123

10.2.1 Statement of the result 123

10.2.2 First proof 124

10.2.3 Second proof 126

Part II Applications to Quantum Mechanics 129

11 Moyal Star Product and Twisted Convolution 131

11.1 The Moyal Product of Two Symbols 131

11.1.1 Definition of the Moyal product 131

11.1.2 Twisted convolution 134

11.2 Bopp Operators 136

11.2.1 Bopp shifts 136

11.2.2 Definition and justification of Bopp operators 139

11.2.3 The intertwining property 141

12 Probabilistic Interpretation of the Wigner Transform 143

12.1 Introduction 143

12.1.1 Back to Wigner 143

12.1.2 Averaging observables and symbols 144

12.2 The Strong Uncertainty Principle 146

12.2.1 Variances and covarianccs 146

12.2.2 The uncertainty principle 147

12.2.3 The quantum covariance matrix 148

12.3 The Notion of Weak Value 150

12.3.1 Definition of weak values 150

12.3.2 A complex phase space distribution 152

12.3.3 Reconstruction using weak values 152

13 Mixed Quantum States and the Density Operator 155

13.1 Trace Class Operators 155

13.1.1 Definition and general properties 156

13.1.2 The case of Weyl operators 157

13.2 The Density Operator 159

13.2.1 The Wigner transform of a mixed state 159

13.2.2 A characterization of density operators 161

13.2.3 Uncertainty principle for density operators 162

13.2.4 Covariance matrix 165

14 The KLM Conditions and the Narcowich-Wigner Spectrum 169

14.1 The Quantum Bochner Theorem 169

11.1.1 Berliner's theorem 169

14.1.1 The quantum case: the KLM conditions 171

14.1.2 The quantum covariance matrix 175

14.2 The Nareowich-Wigner Spectrum 179

14.2.1 η-Positive functions 179

14.2.2 The Narcowich-Wigner spectrum of some states 180

15 Wigner Transform and Quantum Blobs 183

15.1 Quantum Blobs and Phase Space 183

15.1.1 Geometric definition of a quantum blob 184

15.1.2 Quantum phase space 185

15.2 Quantum Blobs and the Wigner Transform 186

15.2.1 The basic example 186

15.2.2 Covariance ellipsoid and quantum blobs 187

15.3 From One Quantum Blob to Another 189

15.3.1 The general case 190

15.3.2 Averaging over quantum blobs 192

Appendix A Sp(n) and Mp(n) 195

A.1 The Symplectic Group 195

A.2 The Metaplectic Group 198

A.3 The Inhomogeneous Metaplectic Group 200

Appendix B The Symplectic Fourier Transform 203

Appendix C Symplectic Diagonalization 207

C.1 Williamson's Theorem 207

C.2 The Block-Diagonal Case 209

C.3 The Symplectic Case 210

C.4 The Symplectic Spectrum 211

Appendix D Symplectic Capacities 215

D.1 Gromov's Non-squeezing Theorem 215

D.2 Symplectic Capacities 216

D.3 Properties 217

D.4 The Symplectic Capacity of an Ellipsoid 218

Bibliography 221

Index 227