Positive and Negative Joint Quantum Distributions

  • Leon Cohen
Part of the NATO ASI Series book series (NSSB, volume 135)


There have been many attempts to write a proper joint quantum probability distribution that is positive and gives the correct individual quantum distributions of position and momentum (marginals). The general impression that has prevailed is that there is something in quantum mechanics which prevents the writing of such a distribution. The two most common reasons given why proper distributions cannot exist is the uncertainty principle and the fact that position and momentum do not commute. None the less we shall show that positive distributions do exist, are easy to obtain, and an infinite number of them can be generated readily.


Wave Function Joint Distribution Classical Function Uncertainty Principle Quantum Operator 
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  1. 1.
    L. Cohen and Y. I. Zaparovanny, J. Math. Phys. 21, 794, (1980).MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    L. Cohen, J. Chem. Phys. 80 4277 (1984).ADSCrossRefGoogle Scholar
  3. 3.
    L. Cohen, ICASSP 84, 41B. 1. 1 (1984).Google Scholar
  4. 4.
    L. Cohen, J. Math. Phys., 25 2402 (1984).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    L. Cohen and T. Posch, Positive Time-Frequency Distribution Functions, IEEE Trans, on Acoustics, Speech and Signal Processing, (to appear).Google Scholar
  6. 6.
    P. D. Finch and R. Groblicki, Found, of Physics, 14 549 (1984).MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    L. Cohen, J. Math. Phys. 7, 781 (1966).ADSCrossRefGoogle Scholar
  8. 8.
    L. Cohen, Phil, of Science, 33, 317 (1966).CrossRefGoogle Scholar
  9. 9.
    H. Margenau and L. Cohen, in: Quantum Theory and Reality, ed. Mario Bunge p. 71, Springer-Verlag, New York (1967).Google Scholar
  10. 10.
    L. Cohen, in: Foundations and Philosophy of Quantum Mechanics, ed. A. Hooker, p. 69 D. Reidel, New York (1973).Google Scholar
  11. 11.
    L. Cohen, J. Math. Phys. 17, 1863 (1976).ADSCrossRefGoogle Scholar
  12. 12.
    L. Cohen, J. Chem. Phys. 70, 788 (1978).ADSCrossRefGoogle Scholar
  13. 13.
    E. Wigner, Phys. Rev. 40 749 (1932).ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Wigner, in: Perspectives in Quantum Theory, eds. W. Yourgrau and A. van der Merwe, p. 25 M.I.T., Cambridge, (1971).Google Scholar
  15. 15.
    J. E. Moyal, >Proc. Cambridge Phil. Soc. 45 99 (1949).ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    For a general review of the Wigner distribution function see the forthcoming article: M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner, Distribution Functions in Physics: Fundamentals, Physics Reports (to appear).Google Scholar
  17. 17.
    T. Takabayasi, Progr. Theoret. Phys. 11, 341 (1954).MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    H. Margenau and R. N. Hill, Prog. Theoret. Phys. 26, 722 (1961).Google Scholar
  19. 19.
    C. L. Mehta, J. Math. Phys. 5, 677 (1964).ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    O. von Roos, Phys. Rev. 119, 1174 (1960).MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    G. J. Ruggeri, Prog. Theoret. Phys. 46 1703 (1971).ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    J. R. Shewell, Am. J. Phys. 27., (1959).Google Scholar
  23. 23.
    H. J. Groenewold, Physica 12, 405 (1946).MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    H. Weyl, The Theory of Groups and Quantum Mechanics, E. P. Dutton and Co., New York, p. 275 (1931).Google Scholar
  25. 25.
    N. H. McCoy, Proc. Natl. Acad. Sei. U.S.A. 18 674 (1932).ADSzbMATHCrossRefGoogle Scholar
  26. 26.
    M. Born and P. Jordan, Z. Phys. 34 873 (1925).Google Scholar
  27. 27.
    N. Cartwright, Found, of Phys. 4, 127 (1974).MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    L. Cohen, Found, of Phys. 6, 739 (1976).ADSCrossRefGoogle Scholar
  29. 29.
    See, for example, the following papers and the references therein: Morgenstern, Mitt. Math. Stat. 8, 234 (1956); E.J. Gumbel, Rev. Fac. Ci. Univ. Lisboa Ser. 2A CI. Mat.7., 179 (1959); D.J.G. Farlie, Biometrica 50, 499 (1963).Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • Leon Cohen
    • 1
  1. 1.Hunter College of The City UniversityNew YorkUSA

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