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Foundations of Physics

, Volume 40, Issue 5, pp 532–544 | Cite as

Classical and Non-relativistic Limits of a Lorentz-Invariant Bohmian Model for a System of Spinless Particles

  • Sergio Hernández-ZapataEmail author
  • Ernesto Hernández-Zapata
Article

Abstract

A completely Lorentz-invariant Bohmian model has been proposed recently for the case of a system of non-interacting spinless particles, obeying Klein-Gordon equations. It is based on a multi-temporal formalism and on the idea of treating the squared norm of the wave function as a space-time probability density. The particle’s configurations evolve in space-time in terms of a parameter σ with dimensions of time. In this work this model is further analyzed and extended to the case of an interaction with an external electromagnetic field. The physical meaning of σ is explored. Two special situations are studied in depth: (1) the classical limit, where the Einsteinian Mechanics of Special Relativity is recovered and the parameter σ is shown to tend to the particle’s proper time; and (2) the non-relativistic limit, where it is obtained a model very similar to the usual non-relativistic Bohmian Mechanics but with the time of the frame of reference replaced by σ as the dynamical temporal parameter.

Keywords

Bohmian mechanics Klein-Gordon equation Relativistic quantum mechanics Multi-temporal formalism Space-time probability density Conditional wave function 

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References

  1. 1.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden variables”, Parts 1 and 2. Phys. Rev. 89, 166–193 (1952) CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1993) Google Scholar
  3. 3.
    Durr, D., Teufel, S.: Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, Berlin (2009) Google Scholar
  4. 4.
    Berndl, K., Daumer, M., Durr, D., Goldstein, S., Zanghi, N.: A survey of Bohmian mechanics. Nuovo Cimento 110B, 737–750 (1995) ADSGoogle Scholar
  5. 5.
    Durr, D., Goldstein, S., Zanghi, N.: Quantum mechanics, randomness, and deterministic reality. Phys. Lett. A 172, 6–12 (1992) CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Durr, D., Goldstein, S., Zanghi, N.: A global equilibrium as the foundation for quantum randomness. Found. Phys. 23, 721–738 (1993) CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Durr, D., Goldstein, S., Zanghi, N.: Bohmian mechanics and quantum equilibrium. In: Albeverio, S., Cattaneo, U., Merlini, D. (eds.): Stochastic Processes, Physics and Geometry, vol. II, pp. 221–232. World Scientific, Singapore (1995) Google Scholar
  8. 8.
    Durr, D., Goldstein, S., Zanghi, N.: Quantum equilibrium and the role of operators as observables in Quantum Theory. J. Stat. Phys. 116, 959–1055 (2004) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Tumulka, R.: Understanding Bohmian Mechanics: a dialogue. Am. J. Phys. 72(9), 1220–1226 (2004) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Bricmont, J.: Http://www.fyma.ucl.ac.be/files/meaningWF.pdf. Cited 8 Oct 2009 (2009)
  11. 11.
    Berndl, K., Durr, D., Goldstein, S., Zanghi, N.: Nonlocality, Lorentz invariance, and Bohmian quantum theory. Phys. Rev. A 53(4), 2062–2073 (1996) CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Nikolic, H.: Time in Relativistic and Nonrelativistic Quantum Mechanics. Int. J. Quantum Inf. 7(3), 595–602 (2009) zbMATHCrossRefGoogle Scholar
  13. 13.
    Nikolic, H.: Relativistic quantum mechanics and the Bohmian interpretation. Found. Phys. Lett. 18(6), 549–561 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nikolic, H.: Covariant many-fingered time Bohmian interpretation of quantum field theory. Phys. Lett. A 348(3–6), 166–171 (2006) CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Nikolic, H.: Relativistic Bohmian interpretation of quantum mechanics. Conference on the present status of Quantum Mechanics. AIP Conf. Proc. 844, 272–280 (2006) CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Nikolic, H.: Quantum mechanics: myths and facts. Found. Phys. 37(11), 1563–1611 (2007) zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Nikolic, H.: Probability in relativistic quantum mechanics and foliation of spacetime. Int. J. Mod. Phys. A 22(32), 6243–6251 (2007) zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Nikolic, H.: Probability in relativistic Bohmian mechanics of particles and strings. Found. Phys. 38(9), 869–881 (2008) zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Allori, V., Durr, D., Goldstein, S., Zanghi, N.: Seven steps towards the classical world. J. Opt. B 4, 482–488 (2002) ADSGoogle Scholar
  20. 20.
    Einstein, A.: Grundgedancen und Methoden der Relativitätstheorie, in ihrer Entwivehung darqestellt. In: The Collected Papers of Albert Einstein. Princeton University Press, Princeton (2002) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Sergio Hernández-Zapata
    • 1
    Email author
  • Ernesto Hernández-Zapata
    • 2
  1. 1.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoDistrito FederalMéxico
  2. 2.Departamento de Física, Matemáticas e IngenieríaUniversidad de SonoraH. CaborcaMéxico

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