Advertisement

Foundations of Physics

, Volume 14, Issue 9, pp 821–848 | Cite as

Stochastic phase spaces, fuzzy sets, and statistical metric spaces

Article

Abstract

This paper is devoted to the study of the notion of the phase-space representation of quantum theory in both the nonrelativisitic and the relativisitic cases. Then, as a derived concept, the stochastic phase space is introduced and its connections with fuzzy set theory and probabilistic topological (in particular, metric) spaces are discussed.

Keywords

Phase Space Quantum Theory Stochastic Phase Probabilistic Topological Stochastic Phase Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. T. Ali and E. Prugovečki,J. Math. Phys. 18, 219 (1977).Google Scholar
  2. 2.
    S. T. Ali and E. Prugovečki,Physica A 89, 501 (1977).Google Scholar
  3. 3.
    E. Prugovečki,Phys. Rev. D 18, 3655 (1978).Google Scholar
  4. 4.
    E. Prugovečki,Hadronic J. 4, 1118 (1981).Google Scholar
  5. 5.
    E. Prugovečki,Found. Phys. 11, 355, 501 (1981).Google Scholar
  6. 6.
    E. Prugovečki,Stochastic Quantum Mechanics and Quantum Space time (D. Reidel, Dordrecht, 1984).Google Scholar
  7. 7.
    S. T. Ali, R. Gagnon, and E. Prugovečki,Can. J. Phys. 59, 807 (1981).Google Scholar
  8. 8.
    E. Prugovečki,Lett. Nuovo Cimento 32, 481 (1981).Google Scholar
  9. 9.
    J. A. Brooke and W. Guz,Lett. Nuovo Cimento 35, 265 (1982).Google Scholar
  10. 10.
    J. A. Brooke and W. Guz,Nuovo Cimento A78, 221 (1983).Google Scholar
  11. 11.
    K. Menger,Proc. Natl. Acad. Sci. U.S.A. 28, 535 (1942).Google Scholar
  12. 12.
    K. Menger,Proc. Natl. Acad. Sci. U.S.A. 37, 226 (1951).Google Scholar
  13. 13.
    W. Guz,Inter. J. Theor. Phys. 23, 157 (1984).Google Scholar
  14. 14.
    N. Dunford and J. T. Schwartz,Linear Operators, Vol. 1 (Wiley-Interscience, New York, 1958).Google Scholar
  15. 15.
    R. Schatten,Norm Ideals of Completely Continuous Operators (Springer, Berlin, 1960).Google Scholar
  16. 16.
    J. R. Ringrose,Compact Non-self-adjoint Operators (Van Nostrand, New York, 1971).Google Scholar
  17. 17.
    K. Yosida,Functional Analysis (Springer, Berlin, 1968), 2nd edn.Google Scholar
  18. 18.
    S. T. Ali and E. Prugovečki, “Extended Harmonic Analysis of Phase-Space Representations for the Galilei Group, University of Toronto preprint (1983).Google Scholar
  19. 19.
    J.-M. Lévy-Leblond, inGroup Theory and Its Applications, Vol. 2, E. M. Loebl, ed. (Academic Press, New York, 1971).Google Scholar
  20. 20.
    F. E. Schroeck, Jr.,J. Math. Phys. 22, 2562 (1981).Google Scholar
  21. 21.
    F. E. Schroeck, Jr.,Found. Phys. 12, 825 (1982).Google Scholar
  22. 22.
    L. A. Zadeh,Inf. Control 8, 338 (1965).Google Scholar
  23. 23.
    V. S. Varadarajan,Geometry of Quantum Theory, Vol. 1 (Van Nostrand, Princeton, 1968).Google Scholar
  24. 24.
    C. Piron,Foundations of Quantum Mechanics (Benjamin, New York, 1976).Google Scholar
  25. 25.
    M. J. Frank,J. Math. Anal. Appl. 34, 67 (1971).Google Scholar
  26. 26.
    A. Wald,Proc. Natl. Acad. Sci. U.S.A. 29, 196 (1943).Google Scholar
  27. 27.
    A. Špaček,Czech. Math. J. 6, 72 (1956).Google Scholar
  28. 28.
    K. Menger, B. Schweizer, and A. Sklar,Czech. Math. J. 9, 459 (1959).Google Scholar
  29. 29.
    B. Schweizer and A. Sklar,Pacific J. Math. 10, 313 (1960).Google Scholar
  30. 30.
    B. Schweizer and A. Sklar,Theory Probab. Appl. 7, 447 (1962).Google Scholar
  31. 31.
    H. Sherwood,J. London Math. Soc. 44, 441 (1969).Google Scholar
  32. 32.
    R. R. Stevens,Fundam. Math. 61, 259 (1968).Google Scholar
  33. 33.
    J. B. Brown,Z. Wahrscheinlichkeitstheor. Verw. Geb. 6, 62 (1966).Google Scholar
  34. 34.
    A. Wintner,Asymptotic Distributions and Infinite Convolutions (University of Michigan Press, Ann Arbor, 1938).Google Scholar
  35. 35.
    J. Ehlers, inGeneral Relativity and Cosmology, R. K. Sachs, ed. (Academic Press, New York, 1971).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • W. Guz
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations