The Wigner Transform 1st Edition by Maurice De Gosson ISBN B0728BMBY5 9781786343116

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ISBN 10: B0728BMBY5
ISBN 13: 9781786343116
Author: Maurice De Gosson

The Wigner Transform 1st Edition Table of contents:

Part I General Mathematical Framework

1 Phase Space Translations and Reflections

1.1 Some Notation

1.1.1 The spaces Rnx and Rnp

1.1.2 The symplectic structure of phase space

1.1.3 Some usual function spaces

1.1.4 Fourier transform

1.2 The Heisenberg–Weyl and Grossmann–Royer Operators

1.2.1 The displacement Hamiltonian

1.2.2 The Heisenberg–Weyl operators

1.2.3 The Grossmann–Royer parity operators

1.3 A Functional Relation Between T(z0) and R(z0)

1.4 Quantization of Exponentials

2 The Cross-Wigner Transform

2.1 Definitions of the Cross-Wigner Transform

2.1.1 First definition

2.1.2 Wigner’s definition

2.1.3 The Gabor transform and its variants

2.1.4 Extension to tempered distributions

2.2 Properties of the Cross-Wigner Transform

2.2.1 Elementary algebraic properties

2.2.2 Analytical properties and continuity

2.2.3 The marginal properties

2.2.4 Translating Wigner transforms

3 The Cross-Ambiguity Function

3.1 Definition of the Cross-Ambiguity Function

3.1.1 Definition using the Heisenberg–Weyl operator

3.1.2 Traditional definition

3.1.3 The Fourier–Wigner transform

3.2 Properties and Relation with the Wigner Transform

3.2.1 Properties of the cross-ambiguity function

3.2.2 Relation with the cross-Wigner transform

3.2.3 The maximum of the ambiguity function

4 Weyl Operators

4.1 The Notion of Weyl Operator

4.1.1 Weyl’s definition, and rigorous definitions

4.1.2 The distributional kernel of a Weyl operator

4.1.3 Relation with the cross-Wigner transform

4.2 Some Properties of the Weyl Correspondence

4.2.1 The adjoint of a Weyl operator

4.2.2 An L2 boundedness result

5 Symplectic Covariance

5.1 Symplectic Covariance Properties

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

5.1.2 Proof of the symplectic covariance property

5.1.3 Symplectic covariance of Weyl operators

5.2 Maximal Covariance

5.2.1 Antisymplectic matrices

5.2.2 The maximality property

5.2.3 The case of Weyl operators

6 The Moyal Identity

6.1 Precise Statement and Proof

6.1.1 The general Moyal identity

6.1.2 A continuity result

6.2 Reconstruction Formulas

6.2.1 Reconstruction using the cross-Wigner transform

6.2.2 Reconstruction using the cross-ambiguity function

6.3 The Wavepacket Transforms

6.3.1 Definition

6.3.2 Properties of the wavepacket transform

7 The Feichtinger Algebra

7.1 Definition and First Properties

7.1.1 Definition of S0(Rn)

7.1.2 Analytical properties of S0(Rn)

7.1.3 The algebra property of S0(Rn)

7.2 The Dual Space sd′(Rn)

7.2.1 Description of sd′(Rn)

7.2.2 The Gelfand triple (S0, L2, sd′)

8 The Cohen Class

8.1 Definition

8.1.1 The marginal conditions

8.1.2 Generalization of Moyal’s identity

8.1.3 The operator calculus associated with Q

8.2 Two Examples

8.2.1 The generalized Husimi distribution

8.2.2 The Born–Jordan transform

9 Gaussians and Hermite Functions

9.1 Wigner Transform of Generalized Gaussians

9.1.1 Generalized Gaussian functions

9.1.2 Explicit results

9.1.3 Cross-ambiguity function of a Gaussian

9.1.4 Hudson’s theorem

9.2 The Case of Hermite Functions

9.2.1 Short review of the Hermite and Laguerre functions

9.2.2 The Wigner transform of Hermite functions

9.2.3 The cross-Wigner transform of Hermite functions

9.2.4 Flandrin’s conjecture

10 Sub-Gaussian Estimates

10.1 Hardy’s Uncertainty Principle

10.1.1 The one-dimensional case

10.1.2 Two lemmas

10.1.3 The multidimensional Hardy uncertainty principle

10.2 Sub-Gaussian Estimates for the Wigner Transform

10.2.1 Statement of the result

10.2.2 First proof

10.2.3 Second proof

Part II Applications to Quantum Mechanics

11 Moyal Star Product and Twisted Convolution

11.1 The Moyal Product of Two Symbols

11.1.1 Definition of the Moyal product

11.1.2 Twisted convolution

11.2 Bopp Operators

11.2.1 Bopp shifts

11.2.2 Definition and justification of Bopp operators

11.2.3 The intertwining property

12 Probabilistic Interpretation of the Wigner Transform

12.1 Introduction

12.1.1 Back to Wigner

12.1.2 Averaging observables and symbols

12.2 The Strong Uncertainty Principle

12.2.1 Variances and covariances

12.2.2 The uncertainty principle

12.2.3 The quantum covariance matrix

12.3 The Notion of Weak Value

12.3.1 Definition of weak values

12.3.2 A complex phase space distribution

12.3.3 Reconstruction using weak values

13 Mixed Quantum States and the Density Operator

13.1 Trace Class Operators

13.1.1 Definition and general properties

13.1.2 The case of Weyl operators

13.2 The Density Operator

13.2.1 The Wigner transform of a mixed state

13.2.2 A characterization of density operators

13.2.3 Uncertainty principle for density operators

13.2.4 Covariance matrix

14 The KLM Conditions and the Narcowich–Wigner Spectrum

14.1 The Quantum Bochner Theorem

14.1.1 Bochner’s theorem

14.1.2 The quantum case: the KLM conditions

14.1.3 The quantum covariance matrix

14.2 The Narcowich–Wigner Spectrum

14.2.1 η-Positive functions

14.2.2 The Narcowich–Wigner spectrum of some states

15 Wigner Transform and Quantum Blobs

15.1 Quantum Blobs and Phase Space

15.1.1 Geometric definition of a quantum blob

15.1.2 Quantum phase space

15.2 Quantum Blobs and the Wigner Transform

15.2.1 The basic example

15.2.2 Covariance ellipsoid and quantum blobs

15.3 From One Quantum Blob to Another

15.3.1 The general case

15.3.2 Averaging over quantum blobs

Appendix A Sp(n) and Mp(n)

A.1 The Symplectic Group

A.2 The Metaplectic Group

A.3 The Inhomogeneous Metaplectic Group

Appendix B The Symplectic Fourier Transform

Appendix C Symplectic Diagonalization

C.1 Williamson’s Theorem

C.2 The Block-Diagonal Case

C.3 The Symplectic Case

C.4 The Symplectic Spectrum

Appendix D Symplectic Capacities

D.1 Gromov’s Non-squeezing Theorem

D.2 Symplectic Capacities

D.3 Properties

D.4 The Symplectic Capacity of an Ellipsoid

Bibliography

Index

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Tags: Maurice De Gosson, Wigner Transform