Skip to main content
SpringerLink
Log in

The transitions among classical mechanics, quantum mechanics, and stochastic quantum mechanics

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.

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

Access this article

Log in via an institution

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

  1. E. P. Wigner,Phys. Rev. 40, 749 (1932).

    Google Scholar 

  2. H. Weyl,The Theory of Groups and Quantum Mechanics (Dover, New York, 1950), pp. 274–275.

    Google Scholar 

  3. J. E. Moyal,Proc. Camb. Phil. Soc. 45, 99 (1959).

    Google Scholar 

  4. J. C. T. Pool,J. Math. Phys. 1, 66–76 (1966).

    Google Scholar 

  5. E. P. Wigner, “Quantum Mechanical Distribution Functions Revisited,” inPerspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds. (M.I.T. Press, Cambridge, Mass., 1971), pp. 25–36.

    Google Scholar 

  6. R. F. V. Anderson,J. Func. Anal. 4, 240–267 (1969);J. Func. Anal. 6, 110–115 (1970);J. Func. Anal. 9, 423–440 (1972).

    Google Scholar 

  7. R. L. Hudson,Rpt. Math. Phys. 6, 249–252 (1974).

    Google Scholar 

  8. H. Margenau and R. N. Hill,Prog. Theor. Phys. 26, 722–738 (1961).

    Google Scholar 

  9. R. Cressman,J. Func. Anal. 22, 405–419 (1976).

    Google Scholar 

  10. R. F. O'Connell and E. P. Wigner,Phys. Lett. 83A, 145–147 (1981).

    Google Scholar 

  11. C. Y. She and H. Heffner,Phys. Rev. 152, 1103–1110 (1966).

    Google Scholar 

  12. E. Prugovecki,Can. J. Phys. 45, 2173–2219 (1967).

    Google Scholar 

  13. J. M. Jauch and C. Piron,Helv. Phys. Acta. 40, 559–570 (1967).

    Google Scholar 

  14. J. L. Park and H. Margenau,Int. J. Theor. Phys. 1, 211–283 (1968).

    Google Scholar 

  15. E. B. Davies,J. Func. Anal. 6, 318–346 (1970).

    Google Scholar 

  16. E. B. Davies and J. T. Lewis,Comm. Math. Phys. 17, 239–260 (1970).

    Google Scholar 

  17. E. Prugovecki,Found. Phys. 4, 9–18 (1974).

    Google Scholar 

  18. E. Prugovecki,Found. Phys. 5, 557–571 (1975).

    Google Scholar 

  19. S. T. Ali and G. G. Emch,J. Math. Phys. 15, 176–182 (1974).

    Google Scholar 

  20. T. Louton,Pac. J. Math. 59, 147–159 (1975).

    Google Scholar 

  21. E. Prugovecki,J. Math. Phys. 17, 517–523 (1976).

    Google Scholar 

  22. E. Prugovecki,J. Math. Phys. 17, 1673–1681 (1976).

    Google Scholar 

  23. E. Prugovecki,J. Phys. A 9, 1851–1859 (1976).

    Google Scholar 

  24. E. Prugovecki,Int. J. Theor. Phys. 16, 321–331 (1977).

    Google Scholar 

  25. F. E. Schroeck, Jr., “Measures with Minimum Uncertainty on Non-Commutative Algebras with Application to Measurement Theory in Quantum Mechanics,” inMathematical Foundations of Quantum Theory, A. Marlow, ed. (Academic Press, Inc., New York, 1978), pp. 299–327.

    Google Scholar 

  26. S. T. Ali and E. Prugovecki,J. Math. Phys. 18, 219–228 (1977).

    Google Scholar 

  27. E. Prugovecki,Ann. Phys. 110, 102–121 (1978).

    Google Scholar 

  28. S. T. Ali and E. Prugovecki,Physica 89A, 501–521 (1977).

    Google Scholar 

  29. E. Prugovecki,Physica 91A, 202–228 (1978).

    Google Scholar 

  30. E. Prugovecki,Physica 91A, 229–248 (1978).

    Google Scholar 

  31. E. Prugovecki,Found. Phys. 9, 575–587 (1979).

    Google Scholar 

  32. F. E. Schroeck, Jr., “Measurement Theory with Instruments Treated Partially Quantum Mechanically,” inClassical, Semiclassical and Quantum Mechanical Problems in Mathematics, Chemistry and Physics, (Plenum Press, New York, 1981).

    Google Scholar 

  33. F. E. Schroeck, Jr.,J. Math. Phys. 22, 2562–2572 (1981).

    Google Scholar 

  34. E. Prugovecki,J. Math. Phys. 19, 2260–2270 (1978).

    Google Scholar 

  35. S. T. Ali,J. Math. Phys. 20, 1385–1391 (1979);21, 818–829 (1980).

    Google Scholar 

  36. S. T. Ali, R. Gagnon, and E. Prugoveki,Can. J. Phys. 59, 807–811 (1981).

    Google Scholar 

  37. E. Prugovecki,J. Math. Phys. 19, 2271–2277 (1978).

    Google Scholar 

  38. E. Prugovecki,Found. Phys. 11, 355–382 (1981) and references therein.

    Google Scholar 

  39. E. Prugovecki,Hadronic J. 4, 1118–1164 (1981).

    Google Scholar 

  40. B. O. Koopman,Proc. Natl. Acad. Sci. 17, 315–318 (1931).

    Google Scholar 

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

    Google Scholar 

  42. J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, 1955).

    Google Scholar 

  43. A. M. Gleason,J. Math. and Mech. 6, 885–893 (1957).

    Google Scholar 

  44. A. S. Wightman,Rev. Mod. Phys. 34, 845–872 (1962).

    Google Scholar 

  45. J. von Neumann,Math. Ann. 104, 570–578 (1931).

    Google Scholar 

  46. D. Bohm and B. J. Hiley,Found. Phys. 11, 179–203 (1981).

    Google Scholar 

  47. E. Prugovecki,Lett. Nuovo Cimento 32, 277 (1981).

    Google Scholar 

  48. S. T. Ali, to appear inLecture Notes in Mathematics.

  49. E. Prugoviecki,J. Phys. A 10, 543–549 (1977).

    Google Scholar 

  50. F. E. Schroeck, Jr.,Found. Phys. 12, 479–497 (1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Florida Atlantic University, Boca Raton, Florida

    Franklin E. Schroeck Jr.

Authors
  1. Franklin E. Schroeck Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schroeck, F.E. The transitions among classical mechanics, quantum mechanics, and stochastic quantum mechanics. Found Phys 12, 825–841 (1982). https://doi.org/10.1007/BF01884995

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01884995

Keywords

Search

Navigation