Skip to main content
Log in

Geometric dequantization and the correspondence problem

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

On the way to settle a conjecture proposed by Mackey, we first present in detail a complete solution to the correspondence problem for systems whose configuration space isR n. We then indicate how this can be considered as a first step in the elaboration of a geometric dequantization program which would extend the results to more general manifolds.

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

Access this article

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

  • Abraham, R., and Marsden, J. E. (1978).Foundations of Mechanics. Benjamin, Reading, Massachusetts.

    Google Scholar 

  • Aizenmann, M., Gallavotti, G., Goldstein, S., and Lebowitz, J. L. (1976).Communications in Mathematical Physics,48, 1–14.

    Article  Google Scholar 

  • Arnold, V. I. (1978).Mathematical Methods of Classical Mechanics. Springer, New York.

    Google Scholar 

  • Bargmann, V. (1954).Annals of Mathematics,59, 1–46.

    Google Scholar 

  • Chernoff, P. R. (1969). “Difficulties of Canonical Quantization,” unpublished lecture notes, Berkeley, California.

  • Chernoff, P. R. (1981). “Mathematical Obstructions to Quantization.” Preprint, Berkeley, California.

  • Dirac, P. A. M. (1930).The Principles of Quantum Mechanics. Clarendon Press, Oxford.

    Google Scholar 

  • Dixmier, J. (1969).Les C *-algèbres et leurs représentations. Gauthier-Villars, Paris.

    Google Scholar 

  • Emch, G. G. (1972).Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York.

    Google Scholar 

  • Emch, G. G. (1981). “Prequantization and KMS Structures.”International Journal of Theoretical Physics,20, 891–904.

    Article  Google Scholar 

  • Gallavotti, G., and Verboven, E. J. (1975).Nuovo Cimento,28B, 274–286.

    Google Scholar 

  • Godement, R. (1952).Transactions of the AMS,73, 496–556.

    Google Scholar 

  • Groenewold, H. J. (1946).Physica,12, 405–460.

    Article  Google Scholar 

  • Grossmann, A., Loupias, G., and Stein, E. M. (1968),Annales de l'Institut Fourier Grenoble,18, 343–368.

    Google Scholar 

  • Hepp, K. (1974).Communications in Mathematical Physics,35, 265–277.

    Article  Google Scholar 

  • Hormander, L. (1969).Linear Partial Differential Operators, 3rd. ed. Springer, New York.

    Google Scholar 

  • Hove, L. van (1951).Academie Royale de Belgique, Bulletin Classe des Sciences Memoires (5),37 610–620.

    Google Scholar 

  • Kastler, D. (1965).Communications in Mathematical Physics,1, 14–48.

    Article  Google Scholar 

  • Lavine, R. B. (1965). “The Weyl-Transform Fourier Analysis of Operators inL 2-Spaces,” Ph.D. thesis, MIT (unpublished).

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

    Google Scholar 

  • Mackey, G. W. (1963b).Bulletin of the AMS,69, 628–686.

    Google Scholar 

  • Mackey, G. W. (1968).Induced Representations and Quantum Mechanics. Benjamin, New York.

    Google Scholar 

  • Mackey, G. W. (1975). InLie Groups and their Representations, I. M. Gelfand, ed. Hilgar, London, pp. 339–363.

    Google Scholar 

  • Mackey, G. W. (1976).The Theory of Unitary Group Representations. The University of Chicago Press, Chicago.

    Google Scholar 

  • Moyal, J. E. (1949).Proceedings of the Cambridge Philosophical Society,45, 99–124.

    Google Scholar 

  • Neumann, J. von (1931).Mathematische Annalen,104, 570–578.

    Article  MathSciNet  Google Scholar 

  • Neumann, J. von (1932).Grundlagen der Quantenmechanik. Springer, Berlin.

    Google Scholar 

  • Perelomov, A. M. (1972).Communications in Mathematical Physics,26, 222–236.

    Article  Google Scholar 

  • Roepstorff, G. (1970).Communications in Mathematical Physics,19, 301–314.

    Article  Google Scholar 

  • Schroedinger, E. (1926).Naturwissenschaften,14, 664–666.

    Article  Google Scholar 

  • Segal, I. E. (1963).Math. Scand. 13, 31–43.

    Google Scholar 

  • Wigner, E. P. (1932).Physical Review,40, 749–759.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Emch, G.G. Geometric dequantization and the correspondence problem. Int J Theor Phys 22, 397–420 (1983). https://doi.org/10.1007/BF02083286

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02083286

Keywords

Navigation