Deterministic Chaos in Infinite Quantum Systems

1. Front Matter

   Pages I-VI

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2. Introduction

   • Fabio Benatti

   Pages 1-2

3. Classical Ergodic Theory

   • Fabio Benatti

   Pages 3-52

4. Algebraic Approach to Classical Ergodic Theory

   • Fabio Benatti

   Pages 53-68

5. Infinite Quantum Systems

   • Fabio Benatti

   Pages 69-140

6. Connes-Narnhofer-Thirring Entropy

   • Fabio Benatti

   Pages 141-207

7. Appendix

   • Fabio Benatti

   Pages 209-212

8. Back Matter

   Pages 213-225

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The purpose of this volume is to give a detailed account of a series of re­ sults concerning some ergodic questions of quantum mechanics which have the past six years following the formulation of a generalized been addressed in Kolmogorov-Sinai entropy by A.Connes, H.Narnhofer and W.Thirring. Classical ergodicity and mixing are fully developed topics of mathematical physics dealing with the lowest levels in a hierarchy of increasingly random behaviours with the so-called Bernoulli systems at its apex showing a structure that characterizes them as Kolmogorov (K-) systems. It seems not only reasonable, but also inevitable to use classical ergodic theory as a guide in the study of ergodic behaviours of quantum systems. The question is which kind of random behaviours quantum systems can exhibit and whether there is any way of classifying them. Asymptotic statistical independence and, correspondingly, complete lack of control over the distant future are typical features of classical K-systems. These properties are fully characterized by the dynamical entropy of Kolmogorov and Sinai, so that the introduction of a similar concept for quantum systems has provided the opportunity of raising meaningful questions and of proposing some non-trivial answers to them. Since in the following we shall be mainly concerned with infinite quantum systems, the algebraic approach to quantum theory will provide us with the necessary analytical tools which can be used in the commutative context, too.

• Entropie
• Ergodentheorie
• Infinite Quantum System
• chaos
• deterministic chaos
• entropy
• ergodic theory
• quantum chaos

• Dipartimento di Fisica Teorica, Università di Trieste, Trieste, Italy

  Fabio Benatti

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