The Wigner Transform

Preface
Introduction
Contents
Part I. General Mathematical Framework
	1. Phase Space Translations and Reflections
		1.1 Some Notation
			1.1.1 The spaces ℝⁿ_x and ℝⁿ_p
			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 \\hat{T}(z_0) and \\hat{R}(z_0)
		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
	Chapter 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 S\'₀(ℝⁿ)
			7.2.1. Description of S\'₀(ℝⁿ)
			7.2.2. The Gelfand triple (S₀, L², S\'₀)
	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
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[72]
Index
	A-H
	Hp-Tr
	Tw-W