Advertisement

Does Quantum Chaos Exist?

  • Andreas Knauf
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 574)

Abstract

The usual operational definition of the term ‘Quantum Chaos’, meaning a quantum system whose classical counterpart is non-integrable, is not self-contained. However it is argued that there cannot be any intrinsic definition of chaoticity of a finite quantum system which is not based on some kind of semiclassical limit.

Unlike for finite systems, the quantum dynamical entropy of infinite systems may be strictly positive. However, an example shows that this quantity may be lowered by interactions which lead to an increase of classical dynamical entropy.

Keywords

Lyapunov Exponent Semiclassical Limit Classical Counterpart Quantum Entropy Qualitative Distinction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asch, J., Knauf, A.: Motion in Periodic Potentials. Nonlinearity 11, 175–200 (1998)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Asch, J., Knauf, A.: Quantum Transport on KAM Tori. Commun. Math. Phys. 205, 113–128 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Benatti, F.: Deterministic Chaos in Infinite Quantum Systems. Trieste Notes in Physics. Berlin: Springer 1993Google Scholar
  4. 4.
    Benatti F., Hudetz, T., Knauf, A.: Quantum Chaos and Dynamical Entropy. Commun. Math. Phys. 198, 607–688 (1998)zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Connes, A., Narnhofer, H., Thirring, W.: Dynamical Entropy for C* algebras and von Neumann Algebras. Commun. Math. Phys. 112, 691 (1987).zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Gutzwiller, M.: Chaos in Classical and Quantum Mechanics. Berlin, Heidelberg, New York: Springer; 1990zbMATHGoogle Scholar
  7. 7.
    Helffer, B.: h-pseudodifferential operators and applications: an introduction. Rauch, Jeffrey (ed.) et al., Quasiclassical methods. Proceedings based on talks given at the IMA workshop, Minneapolis, MN, USA, May 22–26, 1995. New York, NY: Springer. IMA Vol. Math. Appl. 95, 1–49 (1997)Google Scholar
  8. 8.
    Knauf, A.: Ergodic and Topological Properties of Coulombic Periodic Potentials. Commun. Math. Phys. 110, 89–112 (1987)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Knauf, A.: Coulombic Periodic Potentials: The Quantum Case. Annals of Physics 191, 205–240 (1989)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Knauf, A., Sinai, Ya.: Classical Nonintegrability, Quantum Chaos. DMV-Seminar Band 27. Basel: Birkhäuser 1997Google Scholar
  11. 11.
    Lazutkin, V. F.: KAM theory and semiclassical approximations to eigenfunctions. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 24. Berlin; New York: Springer 1993.Google Scholar
  12. 12.
    Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer 1993.zbMATHGoogle Scholar
  13. 13.
    Sarnak, P.: Arithmetic Quantum Chaos. Israel Math. Conf. Proc. 8, 183–236 (1995)Google Scholar
  14. 14.
    Schnirelman, A.I.: Ergodic Properties of Eigenfunctions. Usp. Math. Nauk. 29, 181–182 (1974)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Andreas Knauf
    • 1
  1. 1.Mathematisches Institut der Universität Erlangen-NürnbergErlangen

Personalised recommendations