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Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function

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Abstract

An explicit expression of the “Wigner operator” is derived, such that the Wigner function of a quantum state is equal to the expectation value of this operator with respect to the same state. This Wigner operator leads to a representation-independent procedure for establishing the correspondence between the inhomogeneous symplectic group applicable to linear canonical transformations in classical mechanics and the Weyl-metaplectic group governing the symmetry of unitary transformations in quantum mechanics.

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References

  1. R. G. Littlejohn,Phys. Rep. 138, 193 (1986).

    Google Scholar 

  2. G. S. Agarwal and E. Wolf,Phys. Rev. D 2, 2161, 2187, 2206 (1970), and references therein.

    Google Scholar 

  3. E. P. Wigner,Phys. Rev. 40, 749 (1932); T. F. Jordan and E. C. G. Sudarshan,J. Math. Phys. 2, 772 (1961).

    Google Scholar 

  4. R. F. O'Connell,Found. Phys. 13, 83 (1983).

    Google Scholar 

  5. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,Phys. Rep. 106, 121 (1984).

    Google Scholar 

  6. V. I. Tatarskii,Sov. Phys. Usp. 26, 311 (1983).

    Google Scholar 

  7. L. Wang and R. F. O'Connell,Found. Phys. 18, 1023 (1988).

    Google Scholar 

  8. Y. S. Kim and E. P. Wigner,Am. J. Phys. 58, 439 (1990).

    Google Scholar 

  9. Y. S. Kim and M. E. Noz,Phase Space Picture of Quantum Mechanics: Group Theoretical Approach (World Scientific, Singapore, 1991).

    Google Scholar 

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

    Google Scholar 

  11. H. Goldstein,Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, Massachusetts, 1980).

    Google Scholar 

  12. M. Moshinsky and T. H. Seligman,J. Math. Phys. 22, 1338 (1981).

    Google Scholar 

  13. N. N. Bogoliubov and D. V. Shirkov,Quantum Fields (Benjamin/Cummings, Reading, Massachusetts, 1983).

    Google Scholar 

  14. J. H. P. Colpa,Physica A 93, 327 (1978).

    Google Scholar 

  15. M. E. Taylor,Noncommutative Harmonic Analysis (American Mathematical Society, Providence, Rhode Island, 1986).

    Google Scholar 

  16. G. B. Folland,Harmonic Analysis in Phase Space (Princeton University Press, Princeton, New Jersey, 1989).

    Google Scholar 

  17. C. L. Mehta and E. C. G. Sudarshan,Phys. Rev. 138, B274 (1965). J. R. Klauder and B.-S. Skagerstam,Coherent States: Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).

    Google Scholar 

  18. D. Han, Y. S. Kim, and M. E. Noz,Phys. Rev. A 37, 807 (1988).

    Google Scholar 

  19. B. R. Mollow,Phys. Rev. 162, 1256 (1967).

    Google Scholar 

  20. K. E. Cahill and R. J. Glauber,Phys. Rev. 177, 1882 (1969).

    Google Scholar 

  21. A. Grossmann,Commun. Math. Phys. 48, 191 (1976).

    Google Scholar 

  22. A. Royer,Phys. Rev. A 15, 449 (1977).

    Google Scholar 

  23. H. Fan and H. R. Zaidi,Phys. Lett. A 124, 303 (1987).

    Google Scholar 

  24. R. Gilmore,Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, New York, 1974).

    Google Scholar 

  25. E. C. G. Sudarshan and N. Mukunda,Classical Dynamics: A Modern Perspective (Wiley, New York, New York, 1974).

    Google Scholar 

  26. R. London and P. L. Knight,J. Mod. Opt. 34, 709 (1987).

    Google Scholar 

  27. W. Zhang, D. H. Feng, and R. Gilmore,Rev. Mod. Phys. 62, 867 (1990).

    Google Scholar 

  28. X. Ma and W. Rhodes,Phys. Rev. A 41, 4625 (1990).

    Google Scholar 

  29. L. Yeh, “Decoherence of multimode thermal squeezed coherent states” (LBL-32101), inProceedings of the Harmonic Oscillator Workshop (NASA, 1992).

  30. S. Abe,J. Math. Phys. 33, 1690 (1992).

    Google Scholar 

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Yeh, L., Kim, Y.S. Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function. Found Phys 24, 873–884 (1994). https://doi.org/10.1007/BF02067652

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  • DOI: https://doi.org/10.1007/BF02067652

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