Quantum dynamical systems / by Robert Alicki and Mark Fannes.

Format:

Book

Author/Creator:

Alicki, R. (Robert), 1951-

Contributor:

Fannes, M.

Language:

English

Subjects (All):

Quantum theory.

Nonlinear theories.

Differentiable dynamical systems.

Physical Description:

xiv, 278 pages : illustrations ; 24 cm

Place of Publication:

Oxford ; New York : Oxford University Press, [2001]

Summary:

The present book provides a general framework for studying quantum and classical dynamical systems, both finite and infinite, conservative and dissipative. Special attention is paid to the use of statistical and geometrical techniques, such as multitime correlation functions, quantum dynamical entropy, and non-commutative Lyapunov exponents, for systems with a complex evolution. The material is presented in a concise but self-contained and mathematically friendly way with main ideas introduced and illustrated by numerous examples which are directly connected to the relevant physics.

Contents:

2 Basic tools for quantum mechanics 4

2.1 Hilbert spaces and operators 5

2.1.1 Vector spaces 5

2.1.2 Banach and Hilbert spaces 7

2.1.3 Geometrical properties of Hilbert spaces 9

2.1.4 Orthonormal bases 10

2.1.5 Subspaces and projectors 11

2.1.6 Linear maps between Banach spaces 13

2.1.7 Linear functionals and Dirac notation 14

2.1.8 Adjoints of bounded operators 18

2.1.9 Hermitian, unitary and normal operators 19

2.1.10 Partial isometries and polar decomposition 21

2.1.11 Spectra of operators 22

2.1.12 Unbounded operators 24

2.2 Measures 25

2.2.1 Measures and integration 25

2.2.2 Distributions 29

2.2.3 Hilbert spaces of functions 30

2.2.4 Spectral measures 31

2.3 Probability in quantum mechanics 34

2.3.1 Pure states 35

2.3.2 Mixed states, density matrices 37

2.4 Observables in quantum mechanics 38

2.4.1 Compact operators 38

2.4.2 Weyl quantization 40

2.5 Composed systems 44

2.5.1 Direct sums 45

2.5.2 Tensor products 46

2.5.3 Observables and states of composite systems 48

3 Deterministic dynamics 54

3.1 Deterministic quantum dynamics 55

3.1.1 Time-independent Hamiltonians 55

3.1.2 Perturbations of Hamiltonians 57

3.1.3 Time-dependent Hamiltonians 58

3.1.4 Periodic perturbations and Floquet operators 59

3.1.5 Kicked dynamics 61

3.2 Classical limits 62

3.3 Classical differentiable dynamics 66

3.4 Self-adjoint Laplacians on compact manifolds 70

4 Spin chains 73

4.1 Local observables 73

4.2 States of a spin system 75

4.3 Symmetries and dynamics 77

5 Algebraic tools 82

5.1 C*-algebras 82

5.3 States and representations 90

5.4 Dynamical systems and von Neumann algebras 96

6 Fermionic dynamical systems 103

6.1 Fermions in Fock space 103

6.1.1 Fock space 103

6.1.2 Creation and annihilation 105

6.1.3 Second quantization 108

6.2 The CAR-algebra 113

6.2.1 Canonical anticommutation relations 113

6.2.2 Quasi-free automorphisms 114

6.2.3 Quasi-free states 117

7 Ergodic theory 122

7.1 Ergodicity in classical systems 122

7.2 Ergodicity in quantum systems 125

7.2.1 Asymptotic Abelianness 125

7.2.2 Multitime correlations 132

7.2.3 Fluctuations around ergodic means 135

7.3 Lyapunov exponents 139

7.3.1 Classical dynamics 140

7.3.2 Quantum dynamics 143

8 Quantum irreversibility 146

8.1 Measurement theory 146

8.2 Open quantum systems 151

8.3 Complete positivity 153

8.4 Quantum dynamical semigroups 160

8.5 Quasi-free completely positive maps 164

9 Entropy 169

9.1 von Neumann entropy 171

9.1.1 Technical preliminaries 171

9.1.2 Properties of von Neumann's entropy 174

9.1.3 Mean entropy 176

9.1.4 Entropy of quasi-free states 178

9.2 Relative entropy 181

9.2.1 Finite-dimensional case 181

9.2.2 Maximum entropy principle 182

9.2.3 Algebraic setting 184

10 Dynamical entropy 186

10.1 Operational partitions 188

10.2 Dynamical entropy 195

10.2.1 Symbolic dynamics 195

10.2.2 The entropy 197

10.3 Some technical results 200

10.4.1 The quantum shift 205

10.4.2 The free shift 207

10.4.3 Infinite entropy 208

10.4.4 Powers-Price shifts 209

11 Classical dynamical entropy 213

11.1 The Kolmogorov-Sinai invariant 213

11.2 H-density 218

12 Finite quantum systems 225

12.1 Quantum chaos 225

12.1.1 Time scales 226

12.1.2 Spectral statistics 226

12.1.3 Semi-classical limits 227

12.2 The kicked top 227

12.2.1 The model 227

12.2.2 The classical limit 228

12.2.3 Kicked mean-field Heisenberg model 228

12.2.4 Chaotic properties 229

12.3 Gram matrices 230

12.4 Entropy production 235

13 Model systems 240

13.1 Entropy of the quantum cat map 240

13.2 Ruelle's inequality 243

13.2.1 Non-commutative Riemannian structures 244

13.2.2 Non-commutative Lyapunov exponents 246

13.2.3 Ruelle's inequality 247

13.3 Quasi-free Fermion dynamics 249

13.3.1 Description of the model 250

13.3.2 Main result 251

13.3.3 Sketch of the proof 251.

Notes:

Includes bibliographical references (pages 259-268) and index.

ISBN:

0198504004

OCLC:

47282772