Advertisement

Foundations of Physics

, Volume 40, Issue 12, pp 1885–1901 | Cite as

Geometrizing Relativistic Quantum Mechanics

  • F. T. FalcianoEmail author
  • M. Novello
  • J. M. Salim
Article

Abstract

We propose a new approach to describe quantum mechanics as a manifestation of non-Euclidean geometry. In particular, we construct a new geometrical space that we shall call Qwist. A Qwist space has a extra scalar degree of freedom that ultimately will be identified with quantum effects. The geometrical properties of Qwist allow us to formulate a geometrical version of the uncertainty principle. This relativistic uncertainty relation unifies the position-momentum and time-energy uncertainty principles in a unique relation that recover both of them in the non-relativistic limit.

Keywords

Foundations of quantum mechanics Bohm-de Broglie interpretation Weyl integrable space Non-Euclidean geometry 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Weyl, H.: Sitz. Preuss. Akad. Wiss. 26, 465 (1918) Google Scholar
  2. 2.
    London, F.: Z. Phys. 42, 375 (1927) CrossRefADSGoogle Scholar
  3. 3.
    Riemann: In: Smith, D.E. (ed.) A Source Book in Mathematics, vol. 2. Dover, New York (1959) Google Scholar
  4. 4.
    Israelit, M.: Found. Phys. 28, 205 (1998) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Israelit, M.: Found. Phys. 29, 1303 (1999) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Israelit, M.: Found. Phys. 32, 295 (2002) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Israelit, M.: Found. Phys. 32, 945 (2002) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Canuto, V., Adams, P.J., Hsieh, S.H., Tsiang, E.: Phys. Rev. D 16, 1643 (1977) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Rosen, N.: Found. Phys. 13, 363 (1983) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Koch, B.: arxiv:0810.2786 [hep-th] (2008)
  11. 11.
    Koch, B.: arxiv:0901.4106 [gr-qc] (2009)
  12. 12.
    Santamato, E.: Phys. Rev. D 29, 216–222 (1984) CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Holland, P., Kyprianidis, A., Vigier, J.P.: Phys. Lett. A 107, 376 (1985) CrossRefADSGoogle Scholar
  14. 14.
    Gueret, Ph., Holland, P., Kyprianidis, A., Vigier, J.P.: Phys. Lett. A 107, 379 (1985) CrossRefADSGoogle Scholar
  15. 15.
    Dirac, P.M.: Proc. R. Soc. Lond. A 333, 403 (1973) CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Anderson, J.L.: Principles of Relativity Physics. Academic Press, London (1967) Google Scholar
  17. 17.
    Novello, M., Oliveira, L.A.R., Salim, J.M., Elbaz, E.: Int. J. Mod. Phys. D 1(4), 641–677 (1992) zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Perlick, V.: Class. Quantum Gravity 8, 1369 (1991) zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Bohm, D.: Phys. Rev. 85, 166 (1952) CrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Bohm, D.: Phys. Rev. 85, 180 (1952) CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Bohm, D., Hiley, B.J.: The Undivided Universe. Routledge, London (1993) Google Scholar
  22. 22.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993) CrossRefGoogle Scholar
  23. 23.
    de Broglie, L.: Non-Linear Wave Mechanics: A Causal Interpretation. Elsevier, Amsterdam (1960) zbMATHGoogle Scholar
  24. 24.
    Halbwachs, F.: Théorie Relativiste des Fluides a Spin. Gauthier-Villars, Paris (1960) Google Scholar
  25. 25.
    Holland, P.R.: Found. Phys. 17, 345–363 (1987) CrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Cufaro-Petroni, N., Dewdney, C., Holland, P., Kyprianidis, T., Vigier, J.P.: Phys. Lett. A 106, 368–370 (1984) CrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Kyprianidis, A.: Phys. Lett. A 111, 111–116 (1985) CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Santamato, E.: J. Math. Phys. 25, 2477–2480 (1984) CrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Novello, M., Salim, J.M., Falciano, F.T.: arxiv:0901.3741 [gr-qc] (2009)
  30. 30.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Elsevier, Oxford (1975) Google Scholar
  31. 31.
    Synge, J.L.: Relativity: The Special Theory. North-Holland, Amsterdam (1958) Google Scholar
  32. 32.
    Tolman, R.C.: Relativity Thermodynamics and Cosmology. Oxford University Press, Oxford (1962) Google Scholar
  33. 33.
    Feshbach, H., Villars, F.: Rev. Mod. Phys. 30, 24 (1958) zbMATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Rosen, N.: Found. Phys. 12, 213 (1982) CrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Wheeler, J.T.: Phys. Rev. D 41, 431 (1990) CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Shojai, F., Shojai, A.: Gravit. Cosmol. 9, 163 (2003) zbMATHMathSciNetADSGoogle Scholar
  37. 37.
    Shojai, F., Shojai, A.: arxiv:gr-qc/0404102 (2004)
  38. 38.
    Carroll, R.: arxiv:0705.3921 [gr-qc] (2008)

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Instituto de Cosmologia Relatividade Astrofisica ICRA—CBPFRio de JaneiroBrazil

Personalised recommendations