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Geometric dequantization and the correspondence problem

  • Gérard G. Emch
Article

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.

Keywords

Manifold Field Theory Elementary Particle Quantum Field Theory Configuration Space 
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References

  1. Abraham, R., and Marsden, J. E. (1978).Foundations of Mechanics. Benjamin, Reading, Massachusetts.Google Scholar
  2. Aizenmann, M., Gallavotti, G., Goldstein, S., and Lebowitz, J. L. (1976).Communications in Mathematical Physics,48, 1–14.CrossRefGoogle Scholar
  3. Arnold, V. I. (1978).Mathematical Methods of Classical Mechanics. Springer, New York.Google Scholar
  4. Bargmann, V. (1954).Annals of Mathematics,59, 1–46.Google Scholar
  5. Chernoff, P. R. (1969). “Difficulties of Canonical Quantization,” unpublished lecture notes, Berkeley, California.Google Scholar
  6. Chernoff, P. R. (1981). “Mathematical Obstructions to Quantization.” Preprint, Berkeley, California.Google Scholar
  7. Dirac, P. A. M. (1930).The Principles of Quantum Mechanics. Clarendon Press, Oxford.Google Scholar
  8. Dixmier, J. (1969).Les C *-algèbres et leurs représentations. Gauthier-Villars, Paris.Google Scholar
  9. Emch, G. G. (1972).Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York.Google Scholar
  10. Emch, G. G. (1981). “Prequantization and KMS Structures.”International Journal of Theoretical Physics,20, 891–904.CrossRefGoogle Scholar
  11. Gallavotti, G., and Verboven, E. J. (1975).Nuovo Cimento,28B, 274–286.Google Scholar
  12. Godement, R. (1952).Transactions of the AMS,73, 496–556.Google Scholar
  13. Groenewold, H. J. (1946).Physica,12, 405–460.CrossRefGoogle Scholar
  14. Grossmann, A., Loupias, G., and Stein, E. M. (1968),Annales de l'Institut Fourier Grenoble,18, 343–368.Google Scholar
  15. Hepp, K. (1974).Communications in Mathematical Physics,35, 265–277.CrossRefGoogle Scholar
  16. Hormander, L. (1969).Linear Partial Differential Operators, 3rd. ed. Springer, New York.Google Scholar
  17. Hove, L. van (1951).Academie Royale de Belgique, Bulletin Classe des Sciences Memoires (5),37 610–620.Google Scholar
  18. Kastler, D. (1965).Communications in Mathematical Physics,1, 14–48.CrossRefGoogle Scholar
  19. Lavine, R. B. (1965). “The Weyl-Transform Fourier Analysis of Operators inL 2-Spaces,” Ph.D. thesis, MIT (unpublished).Google Scholar
  20. Mackey, G. W. (1963a).Mathematical Foundations of Quantum Mechanics. Benjamin, New York.Google Scholar
  21. Mackey, G. W. (1963b).Bulletin of the AMS,69, 628–686.Google Scholar
  22. Mackey, G. W. (1968).Induced Representations and Quantum Mechanics. Benjamin, New York.Google Scholar
  23. Mackey, G. W. (1975). InLie Groups and their Representations, I. M. Gelfand, ed. Hilgar, London, pp. 339–363.Google Scholar
  24. Mackey, G. W. (1976).The Theory of Unitary Group Representations. The University of Chicago Press, Chicago.Google Scholar
  25. Moyal, J. E. (1949).Proceedings of the Cambridge Philosophical Society,45, 99–124.Google Scholar
  26. Neumann, J. von (1931).Mathematische Annalen,104, 570–578.CrossRefMathSciNetGoogle Scholar
  27. Neumann, J. von (1932).Grundlagen der Quantenmechanik. Springer, Berlin.Google Scholar
  28. Perelomov, A. M. (1972).Communications in Mathematical Physics,26, 222–236.CrossRefGoogle Scholar
  29. Roepstorff, G. (1970).Communications in Mathematical Physics,19, 301–314.CrossRefGoogle Scholar
  30. Schroedinger, E. (1926).Naturwissenschaften,14, 664–666.CrossRefGoogle Scholar
  31. Segal, I. E. (1963).Math. Scand. 13, 31–43.Google Scholar
  32. Wigner, E. P. (1932).Physical Review,40, 749–759.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • Gérard G. Emch
    • 1
  1. 1.Departments of Mathematics and of PhysicsThe University of RochesterRochester

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