Advertisement

Foundations of Physics

, Volume 37, Issue 11, pp 1540–1562 | Cite as

Typicality vs. Probability in Trajectory-Based Formulations of Quantum Mechanics

  • Bruno GalvanEmail author
Article

Abstract

Bohmian mechanics represents the universe as a set of paths with a probability measure defined on it. The way in which a mathematical model of this kind can explain the observed phenomena of the universe is examined in general. It is shown that the explanation does not make use of the full probability measure, but rather of a suitable set function deriving from it, which defines relative typicality between single-time cylinder sets. Such a set function can also be derived directly from the standard quantum formalism, without the need of an underlying probability measure. The key concept for this derivation is the quantum typicality rule, which can be considered as a generalization of the Born rule. The result is a new formulation of quantum mechanics, in which particles follow definite trajectories, but which is based only on the standard formalism of quantum mechanics.

Keywords

Typicality Trajectory-based formulations of quantum mechanics Bohmian mechanics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allori, V., Zanghì, N.: What is Bohmian mechanics. Int. J. Theor. Phys. 43, 1743 (2004), quant-ph/0112008 CrossRefGoogle Scholar
  2. 2.
    Bell, J.S.: The measurement theory of Everett and de Broglie’s pilot wave. In: Bell, J.S. (ed.) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987) Google Scholar
  3. 3.
    Bell, J.S.: Quantum mechanics for cosmologists. In: Bell, J.S. (ed.) Speakable and Unspeakable in Quantum Mechanics, op. cit Google Scholar
  4. 4.
    Bohm, D.: A suggested interpretation in terms of “Hidden Variables”: Part I. Phys. Rev. 85, 166 (1952) CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bohm, D.: A suggested interpretation in terms of “Hidden Variables”: Part II. Phys. Rev. 85, 180 (1952) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bohm, D., Hiley, B.J., Kaloyerou, P.N.: An ontological basis for the quantum theory. Phys. Rep. 6, 321 (1987) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Deotto, E., Ghirardi, G.C.: Bohmian mechanics revisited. Found. Phys. 28, 1 (1998), quant-ph/9704021 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dereziński, J., Gerard, C.: Scattering Theory of Classical and Quantum N-Particle Systems. Springer, New York (1997). Also available at the url http://www.fuw.edu.pl/~derezins/ Google Scholar
  9. 9.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 67, 843 (1992), quant-ph/0308039 zbMATHCrossRefGoogle Scholar
  10. 10.
    Dürr, D., Goldstein, S., Zanghì, 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. Kluwer Academic, Dordrecht (1996), quant-ph/9511016 Google Scholar
  11. 11.
    DeWitt, B., Graham, N. (eds.): The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, Cambridge (1973) Google Scholar
  12. 12.
    Everett, H.: Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 454 (1957) CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Goldstein, S.: Boltzmann’s approach to statistical mechanics. In: Bricmont, J., Dürr, D., Galavotti, M.C., Ghirardi, G., Petruccione, F., Zanghì, N. (eds.) Chance in Physics: Foundations and Perspectives. Lecture Notes in Physics, vol. 574. Springer, Berlin (2001), cond-mat/0105242 CrossRefGoogle Scholar
  14. 14.
    Griffiths, R.B.: Consistent Quantum Mechanics. Cambridge University Press, Cambridge (2002) Google Scholar
  15. 15.
    Mott, N.: The wave mechanics of alpha-ray tracks. Proc. R. Soc. A 126, 79 (1929). Reprinted as Sec.I-6 of Quantum Theory and Measurement, Wheeler, J.A., Zurek, W.H. Princeton (1983) CrossRefADSGoogle Scholar
  16. 16.
    Peruzzi, G., Rimini, A.: Quantum measurements in a family of hidden-variables theories. Found. Phys. Lett. 9, 505 (1996), quant-ph/9607004 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Reed, M., Simon, B.: Functional Analysis, p. 290. Academic Press, New York (1972) Google Scholar
  18. 18.
    Struyve, W., Westman, H.: A new pilot-wave model for quantum field theory. In: Bassi, A., Dürr, D., Weber, T., Zanghì, N. (eds.) Quantum Mechanics: Are there Quantum Jumps? On the Present Status of Quantum Mechanics. AIP Conference Proceedings, vol. 844. American Institute of Physics (2006), quant-ph/0602229 Google Scholar
  19. 19.
    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003), quant-ph/0105127 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.TrentoItaly

Personalised recommendations