دانلود کتاب Coherent States and Applications in Mathematical Physics (Theoretical and Mathematical Physics)

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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