Coherent States and Applications in Mathematical Physics

Preface to the Second Edition
Preface to the First Edition
Contents
Part I Basic Euclidian Coherent States and Applications
1 Introduction to Coherent States
	1.1 A Very Brief Introduction to the Heisenberg Uncertainty Principle
	1.2 The Weyl–Heisenberg Group and the Canonical Coherent States
		1.2.1 The Weyl–Heisenberg Translation Operator
		1.2.2 The Coherent States of Arbitrary Profile
	1.3 The Coherent States of the Harmonic Oscillator
		1.3.1 Definition and Properties
		1.3.2 The Time Evolution of the Coherent State  for the Harmonic Oscillator Hamiltonian
		1.3.3 An Overcomplete System
	1.4 From Schrödinger to Bargmann–Fock Representation
2 Weyl Quantization and Coherent States
	2.1 Classical and Quantum Observables
		2.1.1 Group Invariance of Weyl Quantization
	2.2 Wigner Functions
	2.3 Coherent States and Operator Norms Estimates
	2.4 Product Rule and Applications
		2.4.1 The Moyal Product
		2.4.2 Functional Calculus
		2.4.3 Propagation of Observables
		2.4.4 Return to Symplectic Invariance of Weyl Quantization
	2.5 Husimi Functions, Frequency Sets and Propagation
		2.5.1 Frequency Sets
		2.5.2 About Frequency Sets of Eigenstates
	2.6 Wick Quantization
		2.6.1 General Properties
		2.6.2 Application to Semi-classical Measures
3 Quadratic Hamiltonians
	3.1 The Propagator of Quadratic Quantum Hamiltonians
	3.2 The Propagation of Coherent States
	3.3 The Metaplectic Transformations
	3.4 Representation of the Quantum Propagator in Terms  of the Generator of Squeezed States
	3.5 Representation of the Weyl Symbol of the Metaplectic Operators
	3.6 Traps
		3.6.1 The Classical Motion
		3.6.2 The Quantum Evolution
4 The Semiclassical Evolution of Gaussian Coherent States
	4.1 General Results and Assumptions
		4.1.1 Assumptions and Notations
		4.1.2 The Semiclassical Evolution of Generalized Coherent States
		4.1.3 Related Works and Other Results
	4.2 Application to the Spreading of Quantum Wave Packets
	4.3 Evolution of Coherent States and Bargmann Transform
		4.3.1 Formal Computations
		4.3.2 Weighted Estimates and Fourier–Bargmann Transform
		4.3.3 Large Time Estimates and Fourier–Bargmann Analysis
		4.3.4 Exponentially Small Estimates
	4.4 Application to the Scattering Theory
5 Trace Formulas and Coherent States
	5.1 Introduction
	5.2 The Semiclassical Gutzwiller Trace Formula
	5.3 Preparations for the Proof
	5.4 The Stationary Phase Computation
	5.5 A Pointwise Trace Formula and Quasi-Modes
		5.5.1 A Pointwise Trace Formula
		5.5.2 Quasi-Modes and Bohr–Sommerfeld Quantization Rules
Part II Coherent States in Non Euclidian Geometries
6 Quantization and Coherent States  on the 2-Torus
	6.1 Introduction
	6.2 The Automorphisms of the 2-Torus
	6.3 The Kinematics Framework and Quantization
	6.4 The Coherent States of the Torus
	6.5 The Weyl and Anti-Wick Quantizations on the 2-Torus
		6.5.1 The Weyl Quantization on the 2-Torus
		6.5.2 The Anti-Wick Quantization on the 2-Torus
	6.6 Quantum Dynamics and Exact Egorov's Theorem
		6.6.1 Quantization of SL(2, mathbbZ)
		6.6.2 The Egorov Theorem is Exact
		6.6.3 Propagation of Coherent States
	6.7 Equipartition of the Eigenfunctions of Quantized Ergodic Maps on the 2-Torus
	6.8 Spectral Analysis of Hamiltonian Perturbations
7 Spin Coherent States
	7.1 Introduction
	7.2 The Groups SO(3) and SU(2)
	7.3 The Irreducible Representations of SU(2)
		7.3.1 The Irreducible Representations of mathfraksu(2)
		7.3.2 The Irreducible Representations of SU(2)
		7.3.3 Irreducible Representations of SO(3) and Spherical Harmonics
	7.4 The Coherent States of SU(2)
		7.4.1 Definition and First Properties
		7.4.2 Some Explicit Formulas
	7.5 Coherent States on the Riemann Sphere
	7.6 Application to High Spin Inequalities
		7.6.1 Berezin-Lieb Inequalities
		7.6.2 High Spin Estimates
	7.7 More on High Spin Limit: From Spin Coherent States …
8 Pseudo-Spin Coherent States
	8.1 Introduction to the Geometry of the Pseudosphere, SO(2,1) and SU(1,1)
		8.1.1 Minkowski Model
		8.1.2 Lie Algebra
		8.1.3 The Disc and the Half-Plane Poincaré Representations of the Pseudosphere
	8.2 Unitary Representations of SU(1,1)
		8.2.1 Classification of the Possible Representations  of SU(1,1)
		8.2.2 Discrete Series Representations of SU(1,1)
		8.2.3 Irreducibility of Discrete Series
		8.2.4 Principal Series
		8.2.5 Complementary Series
		8.2.6 Realizations for Bosonic Systems
	8.3 Pseudo-Spin-Coherent States for Discrete Series
		8.3.1 Definition of Coherent States for Discrete Series
		8.3.2 Some Explicit Formulae
		8.3.3 Bargmann Transform and Large k Limit
	8.4 Coherent States for the Principal Series
	8.5 Generator of Squeezed States. Application
		8.5.1 The Generator of Squeezed States
		8.5.2 Application to Quantum Dynamics
	8.6 Wavelets and Pseudo-Spin Coherent States
9 The Coherent States of the Hydrogen Atom
	9.1 The mathbbS3 Sphere and the Group SO(4)
		9.1.1 Introduction
		9.1.2 Irreducible Representations of SO(4)
		9.1.3 Hyperspherical Harmonics and Spectral Decomposition of ΔmathbbS3
		9.1.4 The Coherent States for mathbbS3
	9.2 The Hydrogen Atom
		9.2.1 Generalities
		9.2.2 The Fock Transformation: A Map from L2(mathbbS3) Towards the Pure-Point Subspace of
	9.3 The Coherent States of the Hydrogen Atom
Part III More Advanced Results on Harmonic Coherent States and Applications
10 Characterizations of Harmonic Pseudodifferential Operators
	10.1 Operators Classes Associated with the Harmonic Oscillator
		10.1.1 The Harmonic Classes
		10.1.2 The Beals Commutator Classes
		10.1.3 Coherent States Classes
		10.1.4 The Matrix Class
		10.1.5 An Application to Time Dependent Perturbations  of Harmonic Oscillator
	10.2 The Semi-classical Setting
11 Herman-Kluk Approximation  for the Schrödinger Propagator
	11.1 Sub-quadratic Hamiltonians
	11.2 Semi-classical Fourier-Integral Operators
	11.3 The Herman-Kluk Semi-classical Approximation
		11.3.1 Proof of Theorem 57 by Deformation
		11.3.2 A Proof by Solving Transport Equations
		11.3.3 Error Estimates on Finite Time Intervals
	11.4 Large Time Estimates
	11.5 Application to the Aharonov-Bohm Effect
		11.5.1 The Van-Vleck Formula
		11.5.2 A Setting for the Time-Dependent Aharonov-Bohm Effect
	11.6 Application to the Spectrum of Periodic Hamiltonians Flow
12 Semi-classical Measures: Stationary  and Large Time Behavior
	12.1 Semi-classical Measures for Bound States
		12.1.1 More on the Weyl Law
		12.1.2 More About Semi-classical Measures
	12.2 Semi-Classical Quantum Ergodicity
	12.3 The Quantum Ergodic Birkhoff Theorem
	12.4 Time Dependent Semi-classical Measures
	12.5 Quantum Unique Ergodicity for a Random Hermite Basis
13 Open Quantum Systems and Coherent States
	13.1 Introduction to Open Quantum Systems
		13.1.1 A General Setting
		13.1.2 Coupled Quadratic Hamiltonians
		13.1.3 A Schrödinger Cat Model
	13.2 Computation of the Purity
		13.2.1 Assumption and Statements
		13.2.2 Proofs of the Statements of Sect.13.2.1
	13.3 Separability and Entanglement
		13.3.1 Statements
		13.3.2 Proofs of the Separability Results
		13.3.3 Proof of Theorem 72
	13.4 The Master Equation in the Quadratic Case
		13.4.1 General Quadratic Hamiltonians
		13.4.2 Time Evolution of Reduced Mixed States
		13.4.3 About the Schrödinger Cat
14 Adiabatic Decoupling and Time Evolution of Coherent States  for Multi-component Systems
	14.1 Semi-classical Analysis for Multi-components Systems
	14.2 Systems with a Scalar Leading Term
	14.3 Spin-Orbit Interaction
	14.4 Systems with Constant Multiplicities Eigenvalues
	14.5 Figure
Part IV Coherent States with Infinitely Many Degrees of Freedom
15 Bosonic Coherent States
	15.1 Introduction
	15.2 Fock Spaces
		15.2.1 Bosons and Fermions
		15.2.2 Bosons
	15.3 The Bosons Coherent States
	15.4 The Classical Limit for Large Systems of Bosons
		15.4.1 Introduction
		15.4.2 The Hepp Method
		15.4.3 Remainder Estimates in the Hepp Method
		15.4.4 Time Evolution of Coherent States
16 Fermionic Coherent States
	16.1 Introduction
	16.2 From Fermionic Fock Spaces to Grassmann Algebras
	16.3 Integration on Grassmann Algebra
		16.3.1 More Properties of Grassmann Algebras
		16.3.2 Calculus with Grassmann Numbers
		16.3.3 Gaussian Integrals
	16.4 Super-Hilbert Spaces and Operators
		16.4.1 A Space for Fermionic States
		16.4.2 Integral Kernels
		16.4.3 A Fourier Transform
	16.5 Coherent States for Fermions
		16.5.1 Weyl Translations
		16.5.2 Fermionic Coherent States
	16.6 Representations of Operators
		16.6.1 Trace
		16.6.2 Representation by Translations and Weyl Quantization
		16.6.3 Wigner–Weyl Functions
		16.6.4 The Moyal Product for Fermions
	16.7 Examples
		16.7.1 The Fermi Oscillator
		16.7.2 The Fermi-Dirac Statistics
		16.7.3 Quadratic Hamiltonians and Coherent States
		16.7.4 More on Quadratic Propagators
17 Supercoherent States—An Introduction
	17.1 Introduction
	17.2 Quantum Supersymmetry
	17.3 Classical Superspaces
		17.3.1 Morphisms and Spaces
		17.3.2 Super Algebra Notions
		17.3.3 Examples of Morphisms
	17.4 Super-Lie Algebras and Groups
		17.4.1 Super-Lie Algebras
		17.4.2 Supermanifolds, A Very Brief Presentation
		17.4.3 Super-Lie Groups
	17.5 Classical Supersymmetry
		17.5.1 A Short Overview of Classical Mechanics
		17.5.2 Supersymmetric Mechanics
		17.5.3 Supersymmetric Quantization
	17.6 Supercoherent States
	17.7 Phase Space Representations of Super Operators
	17.8 Application to the Dicke Model
Appendix A Tools for Integral Computations
A.1  Fourier Transform of Gaussian Functions
A.2  Sketch of Proof for Theorem 29摥映數爠eflinkstphthm295
A.3  A Determinant Computation
A.4  The Saddle Point Method
A.4.1  The One Real Variable Case
A.4.2  The Complex Variables Case
A.5  Kähler Geometry
Appendix B Remainder Estimate for the Moyal Product
Appendix C Semi-classical Functional Calculus
C.1  Introduction
C.2  Functional Calculus and Almost Analytic Extension
C.2.1  Weyl Calculus and Resolvent Estimates
C.2.2  End of the Proof of Theorem 10摥映數爠eflinkfuncal102
Appendix D Lie Groups and Coherent States
D.1  Lie Groups and Coherent States
D.2  On Lie Groups and Lie Algebras
D.2.1  Lie Algebras
D.2.2  Lie Groups
D.3  Representations of Lie Groups
D.3.1  General Properties of Representations
D.3.2  The Compact Case
D.3.3  The Non-compact Case
D.4  Coherent States According Gilmore-Perelomov
Appendix E Berezin Quantization and Coherent States
Appendix  References
Index of Concepts
Index of Symbols