Foundations of Physics

, Volume 40, Issue 4, pp 356–367 | Cite as

On the Relationship Between the Wigner-Moyal and Bohm Approaches to Quantum Mechanics: A Step to a More General Theory?

  • B. J. Hiley


In this paper we show that the three main equations used by Bohm in his approach to quantum mechanics are already contained in the earlier paper by Moyal which forms the basis for what is known as the Wigner-Moyal approach. This shows, contrary to the usual perception, that there is a deep relation between the two approaches. We suggest the relevance of this result to the more general problem of constructing a quantum geometry.


Quantum mechanics Bohm approach Wigner-Moyal approach Quantum geometries 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bohm, D., Bub, J.: A proposed solution to the measurement problem in quantum mechanics by a hidden variable theory. Rev. Mod. Phys. 38, 453–469 (1966) zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Bohm, D., Bub, J.: A refutation of a proof by Jauch and Piron that hidden variables can be excluded in quantum mechanics. Rev. Mod. Phys. 38, 470–475 (1966) zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Penrose, R.: Twistor algebra. J. Math. Phys. 8, 345–366 (1967) zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Wheeler, J.A.: Superspace and the nature of quantum geometrodynamics. In: DeWitt, C.M., Wheeler, J.A. (eds.) Battelle Rencontres, pp. 242–307. Benjamin, New York (1968) Google Scholar
  5. 5.
    Penrose, P.: On the nature of quantum geometry. In: Klauder, J.R. (ed.) Magic without Magic. Freeman, San Francisco (1972) Google Scholar
  6. 6.
    Rodrigues, Jr. W.A.: Algebraic and Dirac-Hestenes spinors and spinor fields. J. Math. Phys. 45, 2908–2944 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Francis, M.R., Kosowsky, A.: The construction of spinors in geometric algebra. Ann. Phys. 317, 383–409 (2005) zbMATHMathSciNetADSGoogle Scholar
  8. 8.
    Penrose, R.: The Road to Reality. Vintage Books, London (2004) Google Scholar
  9. 9.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables, I. Phys. Rev. 85, 166–179 (1952) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables, II. Phys. Rev. 85, 180–193 (1952) CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Dürr, D., Goldstein, S., Zanghi, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843–907 (1992) zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Dürr, D., Goldstein, S., Zanghi, N.: Bohmian mechanics as the foundation of quantum mechanics. In: Cushing, J.T., Fine, A., Goldstein, S. (eds.) Bohmian Mechanics and Quantum Theory: An Appraisal. Boston Studies in the Philosophy of Science, vol. 184, p. 26. Kluwer, Dordrecht (1996) Google Scholar
  13. 13.
    Bohm, D.: Wholeness and the Implicate Order. Routledge, London (1980) Google Scholar
  14. 14.
    Bohm, D., Hiley, B.J.: Twistors. Rev. Briasileira de Fisica, Volume Especial, Os 70 anos de Mario Schönberg, 1–26 (1984) Google Scholar
  15. 15.
    Bohm, D.: Time, the implicate order and pre-space. In: Griffen, D.R. (ed.) Physics and the Ultimate Significance of Time, pp. 172–176. SUNY Press, New York (1986) and pp. 177–208 Google Scholar
  16. 16.
    Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, London (1993) Google Scholar
  17. 17.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932) zbMATHCrossRefADSGoogle Scholar
  18. 18.
    Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 45, 99–123 (1949) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Bohm, D.: Chance and Causality in Modern Physics. Routledge & Kegan Paul, London (1957) CrossRefGoogle Scholar
  20. 20.
    Feynman, R.P.: Negative probability. In: Hiley, B.J., Peat, F.D. (eds.) Quantum Implications, pp. 235–248. Routledge & Kegan Paul, London (1987) Google Scholar
  21. 21.
    Bohm, D.: Comments on a letter concerning the causal interpretation of quantum theory. Phys. Rev. 89, 319–320 (1953) zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    de Gosson, M.: The quantum motion of half-densities and the derivation of Schrödinger’s equation. J. Phys. A: Math. Gen. 31, 4239–4247 (1998) zbMATHCrossRefADSGoogle Scholar
  23. 23.
    Zeh, H.D.: Why Bohm’s quantum theory? Found. Phys. Lett. 12, 197–200 (1999) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Weyl, H.: In: The Theory of Groups and Quantum Mechanics, p. 274. Dover, London (1931) Google Scholar
  25. 25.
    Takabayasi, T.: The formulation of quantum mechanics in terms of ensemble in phase space. Prog. Theor. Phys. 11, 341–374 (1954) zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Carruthers, P., Zachariasen, F.: Quantum collision theory with phase-space distributions. Rev. Mod. Phys. 55, 245–285 (1983) CrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Schempp, W.J.: Harmonic Analysis on the Heisenberg Nilpotent Lie Group with Applications to Signal Theory. Pitman Research Notes in Mathematics Series, vol. 147. Longman Scientific & Technical, New York (1986) zbMATHGoogle Scholar
  28. 28.
    Moran, W., Manton, J.H.: In: Byrne, J.S. (ed.) Computational Noncommutative Algebra and Applications. NATO Science Series, pp. 339–362. Kluwer Academic, Amsterdam (2004) Google Scholar
  29. 29.
    Hiley, B.J.: Towards a dynamics of moments: the role of algebraic deformation and inequivalent vacuum states. In: Bowden K.G. (ed.) Correlations Proc. ANPA, vol. 23, pp. 104–134 (2001) Google Scholar
  30. 30.
    de Gosson, M.: Phys. Lett. A 330, 161–167 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    van Oystaeyen, F.: Virtual Topology and Functor Geometry. Lecture Notes in Pure and Applied Mathematics, vol. 256. Chapman and Hall, New York (2007) Google Scholar
  32. 32.
    Hiley, B.J.: Phase space descriptions of quantum phenomena. In: Khrennikov, A. (ed.) Proc. Int. Conf. Quantum Theory: Reconsideration of Foundations, vol. 2, pp. 267–286. Växjö University Press, Växjö (2003) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.TPRU, BirkbeckUniversity of LondonLondonUK

Personalised recommendations