Deterministic Chaos in Infinite Quantum Systems

1 Introduction.- 2 Classical Ergodic Theory.- 2.1 Irreversibility.- 2.1.1 Coarse-Graining.- 2.1.2 Correlations.- 2.1.3 Abstract Dynamical Systems.- 2.1.4 Spectral Theory.- 2.2 Entropy.- 2.2.1 Randomness and Entropy.- 2.2.2 The Entropy of Kolmogorov and Sinai.- 2.2.3 Kolmogorov Systems.- 2.3 Topological Properties of Dynamical Systems.- 2.3.1 Topological Dynamics.- 2.3.2 Topological Entropy.- 3 Algebraic Approach to Classical Ergodic Theory.- 3.1 Abelian C* Dynamical Systems.- 3.2 Abelian W* Dynamical Systems.- 3.3 W* Algebras: KS-Entropy and K-Systems.- 3.4 C* Algebras: Classical Topological Entropy.- 4 Infinite Quantum Systems.- 4.1 Useful Tools from Finite Quantum Systems.- 4.1.1 Density Matrices and von Neumann Entropy.- 4.1.2 Relative Entropy and Completely Positive Maps.- 4.2 GNS-Construction.- 4.2.1 Fermions, Bosons and Toy Models.- 4.3 Ergodic Properties in Quantum Systems.- 4.3.1 Galilei-Invariant Two-Body Interactions.- 4.4 Algebraic Quantum Kolmogorov Systems.- 5 Connes-Narnhofer-Thirring Entropy.- 5.1 Basic Ideas and Construction 1.- 5.2 Basic Ideas and Construction 2.- 5.3 CNT-Entropy: Applications.- 5.3.1 Dynamical Entropy of Quasi-Free Automorphisms.- 5.3.2 CNT-Entropy and Thermodynamics.- 5.4 Short History of the Topic and Latest Developments.- 5.5 Entropic Quantum Kolmogorov Systems.- 5.6 Ideas for a Non-commutative Topological Entropy.- 6 Appendix.- References.- Index of Symbols.