Quantum Signatures of Chaos

Foreword to the First Edition
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
	References
2 Time Reversal and Unitary Symmetries
	2.1 Autonomous Classical Flows
	2.2 Spinless Quanta
	2.3 Spin-1/2 Quanta
	2.4 Hamiltonians Without T Invariance
	2.5 T Invariant Hamiltonians, T2 = 1
	2.6 Unitary Symmetries
	2.7 Kramers\' Degeneracy for T Invariant Hamiltonians, T2=-1
	2.8 T Invariant Hamiltonians with T2=-1 and Additional Unitary Symmetries
	2.9 T Invariance with T2=-1 and no Unitary Symmetries
	2.10 Nonconventional Time Reversal
	2.11 Stroboscopic Maps for Periodically Driven Systems
	2.12 Time Reversal for Maps
	2.13 Canonical Transformations for Floquet Operators
	2.14 Universality
	2.15 Universality for the Kicked Top: The Level Spacing Distribution
	2.16 Beyond Dyson\'s Threefold Way
		2.16.1 Spectral Mirror Symmetries
		2.16.2 Universality in Non-standard Symmetry Classes
		2.16.3 Hamiltonian Matrix Structures and Canonical Transformations for Non-standard Symmetries
		2.16.4 Physical Realizations of Non-standard Symmetry Classes in Fermionic Systems
		2.16.5 Quantum Mechanical Realisation of Non-standard Symmetry Classes in Finite Dimensional Hilbert Spaces: Two Coupled Tops
		2.16.6 Non-standard Universality for Two Coupled Tops
	2.17 Problems
	References
3 Level Repulsion
	3.1 Preliminaries
	3.2 Symmetric Versus Nonsymmetric H or F
	3.3 Kramers\' Degeneracy
	3.4 Universality Classes of Level Repulsion
	3.5 Nonstandard Symmetry Classes
		3.5.1 The Chiral Symmetry Classes
		3.5.2 Classes CI and BDIIIν
		3.5.3 Classes BDν and C
	3.6 Experimental Observation of Level Repulsion
	3.7 Problems
	References
4 Level Clustering
	4.1 Preliminaries
	4.2 Invariant Tori of Classically Integrable Systems
	4.3 Einstein–Brillouin–Keller Approximation
	4.4 Level Crossings for Integrable Systems
	4.5 Poissonian Level Sequences
	4.6 Superposition of Independent Spectra
	4.7 Periodic Orbits and the Semiclassical Density of Levels
	4.8 Level Density Fluctuations for Integrable Systems
	4.9 Exponential Spacing Distribution for Integrable Systems
	4.10 Equivalence of Different Unfoldings
	4.11 Problems
	References
5 Random-Matrix Theory
	5.1 Preliminaries
	5.2 Gaussian Ensembles of Hermitian Matrices
		5.2.1 Gaussian Orthogonal Ensemble
		5.2.2 Gaussian Unitary Ensemble
		5.2.3 Gaussian Symplectic Ensemble
		5.2.4 Arbitrary Matrix Dimension
	5.3 Riemann Geometry for Matrix Ensembles: Invariance of the Volume Element
		5.3.1 Change of Variables on a Manifold
		5.3.2 Geometry for Matrix Ensembles: Invariance of the Volume Element dN H
	5.4 More of Riemann Geometry for Matrix Ensembles: Eigenvalue Distributions
	5.5 Eigenvalue Distributions for the Non-standard Symmetry Classes
	5.6 Level Spacing Distributions (Wigner Surmizes)
	5.7 Average Level Density (GUE)
	5.8 Dyson\'s Circular Ensembles
		5.8.1 Circular Unitary Ensemble
		5.8.2 COE and CSE
		5.8.3 Poissonian Ensemble
	5.9 Same Spectral Correlations in Circular and Gaussian Ensembles for N→∞
	5.10 Eigenvector Distributions
		5.10.1 Single-Vector Density
		5.10.2 Joint Density of Eigenvectors
	5.11 Ergodicity of the Level Density
	5.12 Asymptotic Level Spacing Distributions
	5.13 Determinants as Gaussian Grassmann Integrals
	5.14 Two-Point Correlations of the Level Density
		5.14.1 Two-Point Correlator and Form Factor
		5.14.2 Form Factor for the Poissonian Ensemble
		5.14.3 Form Factor for the CUE
		5.14.4 Form Factor for the COE
		5.14.5 Form Factor for the CSE
	5.15 Newton\'s Relations
		5.15.1 Traces Versus Secular Coefficients
		5.15.2 Solving Newton\'s Relations
	5.16 Selfinversiveness and Riemann–Siegel Lookalike
	5.17 Higher Correlations of the Level Density
		5.17.1 Correlation and Cumulant Functions
		5.17.2 Ergodicity of the CUE Form Factor
		5.17.3 Ergodicity of the CUE Two-Point Correlator
		5.17.4 Joint Density of Traces of Large CUE Matrices
	5.18 Correlations of Secular Coefficients
	5.19 Unfolding Spectra
	5.20 Fidelity of Kicked Tops to Random-Matrix Theory
	5.21 Problems
	References
6 Supersymmetry and Sigma Model for Random Matrices
	6.1 Preliminaries
	6.2 Semicircle Law for the Gaussian Unitary Ensemble
		6.2.1 The Green Function and Its Average
		6.2.2 An Aside: Complex Conjugation of Grassmann Variables
		6.2.3 The GUE Average
		6.2.4 Doing the Superintegral
		6.2.5 Two Remaining Saddle-Point Integrals
	6.3 Superalgebra
		6.3.1 Motivation and Generators of Grassmann Algebras
		6.3.2 Supervectors, Supermatrices
		6.3.3 Superdeterminants
		6.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices
		6.3.5 Diagonalizing Supermatrices
	6.4 Superintegrals
		6.4.1 Some Bookkeeping for Ordinary Gaussian Integrals
		6.4.2 Recalling Grassmann Integrals
		6.4.3 Gaussian Superintegrals
		6.4.4 Some Properties of General Superintegrals
		6.4.5 Integrals over Supermatrices, Parisi–Sourlas–Efetov–Wegner Theorem
		6.4.6 Asymptotic Analysis of SuSy Integrals: Massive and Zero Modes
	6.5 The Semicircle Law Revisited
	6.6 The Two-Point Function of the Gaussian Unitary Ensemble
		6.6.1 Generating Function
		6.6.2 Unitarity vs Pseudo-Unitarity and Superanalytic Hubbard-Stratonovich Transformation
		6.6.3 Efetov\'s Sigma Model
		6.6.4 Rational Parametrization
	6.7 Two-Point Functions of the Circular Ensembles
		6.7.1 Generating Function
	6.8 Zero Dimensional Sigma Model for the CUE
		6.8.1 Color-Flavor Transformation
		6.8.2 The Q Manifold
		6.8.3 Riemann Geometry for Supermatrix Ensembles: Flat Measure d(Q)=d(Z,Z̃)
		6.8.4 Proof of the Color-Flavor Transformation (6.8.2)
		6.8.5 Evaluation of the Generating Function and the Two-Point Correlator for the CUE
		6.8.6 Evaluation of the Grassmann Integral G
	6.9 The Zero Dimensional Sigma Model for COE and CSE
	6.10 Universality of Spectral Fluctuations: Non-GaussianEnsembles
		6.10.1 Delta Functions of Grassmann Variables
		6.10.2 Generating Function
	6.11 Problems
	References
7 Ballistic Sigma Model for Individual Unitary Maps and Graphs
	7.1 Preliminaries
	7.2 Generation Function for the Two-Point Correlator
	7.3 Reduction to the Zero Dimensional Sigma Model and Condition of Validity
	7.4 Perturbative Account of Fluctuations
		7.4.1 Fluctuations on the Q Manifold
		7.4.2 Small Parameters
		7.4.3 Semiclassical Limit
		7.4.4 Conditions for Universal Behavior
	7.5 Quantum Graphs
		7.5.1 Directed Graphs and Their Spectra
		7.5.2 The Sigma Model Approach to Spectral Statistics
	References
8 Quantum Localization
	8.1 Preliminaries
	8.2 Localization in Anderson\'s Hopping Model
	8.3 The Kicked Rotator as a Variant of Anderson\'s Model
	8.4 Lloyd\'s Model
	8.5 The Classical Diffusion Constant as the Quantum Localization Length
	8.6 Absence of Localization for the Kicked Top
	8.7 The Rotator as a Limiting Case of the Top
	8.8 Banded Random Matrices
		8.8.1 Banded Matrices Modelling Thick Wires
		8.8.2 Inverse Participation Ratio and Localization Length
		8.8.3 Sigma Model
		8.8.4 Implementing the One Dimensional Sigma Model
	8.9 Sigma Model for the Kicked Rotor
		8.9.1 A Rotor Without Time Reversal Invariance
		8.9.2 Inverse Participation Ratio
		8.9.3 Sigma Model
		8.9.4 Slow Modes
	8.10 Problems
	References
9 Classical Hamiltonian Chaos
	9.1 Preliminaries
	9.2 Phase Space, Hamilton\'s Equations and All That
	9.3 Action as a Generating Function
	9.4 Linearized Flow and Its Jacobian Matrix
	9.5 Liouville Picture
	9.6 Symplectic Structure
	9.7 Lyapunov Exponents
	9.8 Stretching Factors and Local Stretching Rates
	9.9 Poincaré Map
	9.10 Stroboscopic Maps of Periodically Driven Systems
	9.11 Varieties of Chaos; Mixing and Effective Equilibration
	9.12 The Sum Rule of Hannay and Ozorio de Almeida
		9.12.1 Maps
		9.12.2 Flows
	9.13 Propagator and Zeta Function
	9.14 Exponential Stability of the Boundary Value Problem
	9.15 Sieber-Richter Self-Encounter and Partner Orbit
		9.15.1 Non-technical Discussion
		9.15.2 Quantitative Discussion of 2-Encounters
	9.16 l-Encounters and Orbit Bunches
	9.17 Densities of Arbitrary Encounter Sets
	9.18 Concluding Remarks
	9.19 Problems
	References
10 Semiclassical Roles for Classical Orbits
	10.1 Preliminaries
	10.2 Van Vleck Propagator
		10.2.1 Maps
		10.2.2 Flows
	10.3 Gutzwiller\'s Trace Formula
		10.3.1 Maps
		10.3.2 Flows
		10.3.3 Weyl\'s Law
		10.3.4 Limits of Validity and Outlook
	10.4 Lagrangian Manifolds and Maslov Theory
		10.4.1 Lagrangian Manifolds
		10.4.2 Elements of Maslov Theory
		10.4.3 Maslov Indices as Winding Numbers
	10.5 Riemann-Siegel Look-Alike
	10.6 Spectral Two-Point Correlator
		10.6.1 Real and Complex Correlator
		10.6.2 Local Energy Average
		10.6.3 Generating Function
		10.6.4 Periodic-Orbit Representation
	10.7 Diagonal Approximation
		10.7.1 Unitary Class
		10.7.2 Orthogonal Class
	10.8 Off-Diagonal Contributions, Unitary Symmetry Class
		10.8.1 Structures of Pseudo-Orbit Quadruplets
		10.8.2 Diagrammatic Rules
		10.8.3 Example of Structure Contributions: A Single 2-Encounter
		10.8.4 Cancellation of All Encounter Contributionsfor the Unitary Class
	10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class
		10.9.1 Matrix Elements for Ports and Contraction Lines for Links
		10.9.2 Wick\'s Theorem and Link Summation
		10.9.3 Signs
		10.9.4 Proof of Contraction Rules, Unitary Case
		10.9.5 Emergence of a Sigma Model
	10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class
		10.10.1 Structures
		10.10.2 Leading-Order Contributions
		10.10.3 Symbols for Ports and Contraction Lines for Links
		10.10.4 Gauss and Wick
		10.10.5 Signs
		10.10.6 Proof of Contraction Rules, Orthogonal Case
		10.10.7 Sigma Model
	10.11 Outlook
	10.12 Mixed Phase Space
	10.13 Problems
	References
11 Level Dynamics
	11.1 Preliminaries
	11.2 Fictitious Particles (Pechukas-Yukawa Gas)
	11.3 Conservation Laws
	11.4 Intermultiplet Crossings
	11.5 Level Dynamics for Classically Integrable Dynamics
	11.6 Two-Body Collisions
	11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations
		11.7.1 Ergodicity
		11.7.2 Collision Time
		11.7.3 Universality
	11.8 Equilibrium Statistics
	11.9 Random-Matrix Theory as Equilibrium Statistical Mechanics
		11.9.1 General Strategy
		11.9.2 A Typical Coordinate Integral
		11.9.3 Influence of a Typical Constant of the Motion
		11.9.4 The General Coordinate Integral
		11.9.5 Concluding Remarks
	11.10 Dynamics of Rescaled Energy Levels
	11.11 Level Curvature Statistics
	11.12 Level Velocity Statistics
	11.13 Dyson\'s Brownian-Motion Model
	11.14 Local and Global Equilibrium in Spectra
	11.15 Problems
	References
12 Dissipative Systems
	12.1 Preliminaries
	12.2 Hamiltonian Embeddings
	12.3 Time-Scale Separation for Probabilities and Coherences
	12.4 Dissipative Death of Quantum Recurrences
	12.5 Complex Energies and Quasi-Energies
	12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion
	12.7 Poissonian Random Process in the Plane
	12.8 Ginibre\'s Ensemble of Random Matrices
		12.8.1 Normalizing the Joint Density
		12.8.2 The Density of Eigenvalues
		12.8.3 The Reduced Joint Densities
		12.8.4 The Spacing Distribution
	12.9 General Properties of Generators
	12.10 Universality of Cubic Level Repulsion
		12.10.1 Antiunitary Symmetries
		12.10.2 Microreversibility
	12.11 Dissipation of Quantum Localization
		12.11.1 Zaslavsky\'s Map
		12.11.2 Damped Rotator
		12.11.3 Destruction of Localization
	12.12 Problems
	References
Index