Skip to main content
SpringerLink
Log in

Logical reformulation of quantum mechanics. IV. Projectors in semiclassical physics

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

This is a technical paper providing the proofs of three useful theorems playing a central role in two kinds of physical applications: an explicit logical and mathematical formulation of the interpretation of quantum mechanics and the corresponding description of irreversibility. The Appendix contains a brief mathematical introduction to microlocal analysis. Three theorems are derived in the text: (A) Associating a projector in Hilbert space with a macroscopic regular cell in classical phase space. (B) Specifying the algebra of the projectors associated with different cells. (C) Showing the connection between the classical motion of cells and the Schrödinger evolution of projectors for a class of regular Hamiltonians corresponding approximately to deterministic systems as described within the framework of quantum mechanics. Applications to the interpretation of quantum mechanics are given and the consequences for irreversibility will be given later.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Omnès, Logical reformulation of quantum mechanics I, II, III,J. Stat. Phys., to appear.

  2. E. Schrödinger,Ann. Phys. (Leipzig)79:489 (1926).

    Google Scholar 

  3. W. Heisenberg,Die Physikalischen Prinzipien der Quantum Theorie (Hirzel, Leipzig, 1930).

    Google Scholar 

  4. P. Ehrenfest,Z. Physik 45:455 (1927).

    Google Scholar 

  5. K. Hepp,Commun. Math. Phys. 35:265 (1974).

    Google Scholar 

  6. R. J. Glauber,Phys. Rev. 131:2766 (1963).

    Google Scholar 

  7. J. Ginibre and G. Velo,Commun. Math. Phys. 66:37 (1979);Ann. Phys. (NY)128:243 (1980).

    Google Scholar 

  8. G. Hagedorn,Commun. Math. Phys. 71:77 (1980).

    Google Scholar 

  9. G. Hagedorn,Ann. Phys. (NY)135:58 (1981).

    Google Scholar 

  10. J. Von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1955).

    Google Scholar 

  11. G. W. Mackey,Mathematical Foundations of Quantum Mechanics (Benjamin, New York, 1963).

    Google Scholar 

  12. L. Hörmander,The Analysis of Linear Partial Differential Operators (Springer, Berlin, 1985).

    Google Scholar 

  13. Yu. V. Egorov,Uspekhi Mat. Nauk 24:235 (1969).

    Google Scholar 

  14. L. Hörmander,Ark. Mathematik 17:297 (1979).

    Google Scholar 

  15. I. Daubechies, A. Grossmann, and Y. Meyer,J. Math. Phys. 27:1271 (1986).

    Google Scholar 

  16. V. P. Maslov,Théorie des perturbations et méthodes asymptotiques (Dunod, Paris, 1972).

    Google Scholar 

  17. R. Griffiths,J. Stat. Phys. 36:219 (1984).

    Google Scholar 

  18. H. Weyl,Bull. Am. Math. Soc. 9:56, 115 (1950).

    Google Scholar 

  19. E. P. Wigner,Phys. Rev. 40:749 (1932).

    Google Scholar 

  20. C. L. Fefferman,Bull. Am. Math. Soc. 9:129 (1983).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Energies (Laboratoire associé au Centre National de la Recherche Scientifique), Université de Paris-Sud, 91405, Orsay, France

    Roland Omnès

Authors
  1. Roland Omnès
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Omnès, R. Logical reformulation of quantum mechanics. IV. Projectors in semiclassical physics. J Stat Phys 57, 357–382 (1989). https://doi.org/10.1007/BF01023649

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01023649

Key words

Search

Navigation