Skip to main content
SpringerLink
Log in

Quantum mechanics as a deformation of classical mechanics

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.

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

    Article  Google Scholar 

  2. Weyl, H., The Theory of Groups and Quantum Mechanics, Dover, 1931.

  3. Agarwal E.S. and Wolf E., Phys. Rev. D2, 2161 (1970).

    Google Scholar 

  4. Kuang Chi Liu, J. Math. Phys. 17, 859 (1976).

    Google Scholar 

  5. Moyal J.E., Proc. Cambridge Phil. Soc. 45, 99 (1949).

    Google Scholar 

  6. Flato M., Lichnerowicz A., and Sternheimer D., C.R. Acad. Sci. Paris A279, 877 (1974); Compositio Mathematica 31, 47 (1975); J. Math. Phys. 17, 1745 (1976).

    Google Scholar 

  7. Vey J., Comment. Math. Helvet. 50, 421 (1975).

    Google Scholar 

  8. Flato M., Lichnerowicz A., and Sternheimer D., C.R. Acad. Sci. Paris A283, 19 (1976).

    Google Scholar 

  9. Gerstenhaber M., Ann. Math. 79, 59 (1964).

    Google Scholar 

  10. Souriau J.M., Structure des Systèmes Dynamiques, Dunod, Paris, 1970.

    Google Scholar 

  11. Kostant, B., in Lecture Notes in Mathematics, No. 170, Springer, 1970, pp. 87–208; M. Vergne, Bull. Math. Soc. Fr. 100, 301 (1972).

  12. Dirac P.A.M., Lectures on Quantum Mechanics, Belfer Graduate School of Sciences Monograph Series No. 2 (Yeshiva University, New York, 1964).

    Google Scholar 

  13. Lichnerowicz A., C.R. Acad. Sci. Paris A280, 523 (1974).

    Google Scholar 

  14. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., to be published.

  15. Avez A. and Lichnerowicz A., C.R. Acad. Sci. Paris A275, 113 (1972).

    Google Scholar 

Download references

Author information

Author notes

  1. M. Flato

    Present address: Physique Mathématique, Collège de France, France

Authors and Affiliations

  1. Département de Mathématiques, Université de Paris 6, 75230, Paris Cedex 05, France

    F. Bayen

  2. University of California, 90024, Los Angeles, Calif., USA

    M. Flato & C. Fronsdal

  3. Université de Dijon, 21000, Dijon, France

    A. Lichnerowicz

  4. Collège de France, 75231, Paris Cedex 05, France

    D. Sternheimer

Authors
  1. F. Bayen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Flato
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Fronsdal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Lichnerowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. D. Sternheimer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported in part by the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bayen, F., Flato, M., Fronsdal, C. et al. Quantum mechanics as a deformation of classical mechanics. Lett Math Phys 1, 521–530 (1977). https://doi.org/10.1007/BF00399745

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00399745

Keywords

Search

Navigation