International Journal of Theoretical Physics

, Volume 50, Issue 3, pp 774–791 | Cite as

Modal-Hamiltonian Interpretation of Quantum Mechanics and Casimir Operators: The Road Toward Quantum Field Theory

  • Juan Sebastián Ardenghi
  • Mario Castagnino
  • Olimpia Lombardi
Article

Abstract

The general aim of this paper is to extend the Modal-Hamiltonian interpretation of quantum mechanics to the case of relativistic quantum mechanics with gauge U(1) fields. In this case we propose that the actual-valued observables are the Casimir operators of the Poincaré group and of the group U(1) of the internal symmetry of the theory. Moreover, we also show that the magnitudes that acquire actual values in the relativistic and in the non-relativistic cases are correctly related through the adequate limit.

Keywords

Hamiltonian Casimir Interpretation Actualization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lombardi, O., Castagnino, M.: Stud. Hist. Philos. Mod. Phys. 39, 380 (2008) CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Castagnino, M., Lombardi, O.: J. Phys. Conf. Ser. 128, 012014 (2008) CrossRefADSGoogle Scholar
  3. 3.
    Ardenghi, J.S., Castagnino, M., Lombardi, O.: Found. Phys. 39, 1023 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Lombardi, O., Castagnino, M., Ardenghi, J.S.: Theoria 24, 5 (2009) MATHGoogle Scholar
  5. 5.
    Lombardi, O., Castagnino, M., Ardenghi, J.S.: Stud. Hist. Philos. Mod. Phys. 41, 93 (2010) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Dieks, D., Vermaas, P.E.: The Modal Interpretation of Quantum Mechanics. Kluwer Academic, Dordrecht (1998) MATHCrossRefGoogle Scholar
  7. 7.
    Harshman, N.L., Wickramasekara, S.: Phys. Rev. Lett. 98, 080406 (2007) CrossRefADSGoogle Scholar
  8. 8.
    Harshman, N.L., Wickramasekara, S.: Open Syst. Inf. Dyn. 14, 341 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kochen, S., Specker, E.: J. Math. Mech. 17, 59 (1967) MATHMathSciNetGoogle Scholar
  10. 10.
    Vermaas, P.E.: Stud. Hist. Philos. Mod. Phys. 27, 133 (1996) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Weinberg, S.: The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, Cambridge (1995) Google Scholar
  12. 12.
    Bose, S.K.: Commun. Math. Phys. 169, 385 (1995) MATHCrossRefADSGoogle Scholar
  13. 13.
    Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover, New York (1931) MATHGoogle Scholar
  14. 14.
    Ballentine, L.: Quantum Mechanics, a Modern Development. World Scientific, Singapore (1998) MATHGoogle Scholar
  15. 15.
    Wigner, E.P.: Ann. Math. 40, 149 (1939) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Bargmamm, V.: Ann. Math. 59, 1 (1954) CrossRefGoogle Scholar
  17. 17.
    Levy-Leblond, J.M.: J. Math. Phys. 4, 776 (1963) MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Brading, K., Castellani, E.: Symmetries and invariances in classical physics. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics. North-Holland, Amsterdam (2007) Google Scholar
  19. 19.
    Minkowski, H.: Space and time. In: Perrett, W., Jeffrey, G.B. (eds.) The Principle of Relativity. A Collection of Original Memoirs on the Special and General Theory of Relativity. Dover, New York (1923) Google Scholar
  20. 20.
    Weyl, H.: Symmetry. Princeton University Press, Princeton (1952) MATHGoogle Scholar
  21. 21.
    Auyang, S.Y.: How is Quantum Field Theory Possible? Oxford University Press, Oxford (1995) Google Scholar
  22. 22.
    Nozick, R.: Invariances: The Structure of the Objective World. Harvard University Press, Harvard (2001) Google Scholar
  23. 23.
    Brading, K., Castellani, E. (eds.): Symmetries in Physics: Philosophical Reflections. Cambridge University Press, Cambridge (2003) Google Scholar
  24. 24.
    Earman, J.: Philos. Sci. 69, S209 (2002) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Earman, J.: Philos. Sci. 71, 1227 (2004) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Bacry, H., Levy Léblond, J.-M.: J. Math. Phys. 9, 1605 (1968) MATHCrossRefADSGoogle Scholar
  27. 27.
    Haag, R.: Questions in quantum physics: a personal view. arXiv:hep-th/0001006 (2006)
  28. 28.
    Cariñena, J.F., del Olmo, M.A., Santander, M.: J. Phys. A, Math. Gen. 14, 1 (1981) CrossRefADSGoogle Scholar
  29. 29.
    Ardenghi, J.S., Campoamor-Stursberg, R., Castagnino, M.: J. Math. Phys. 50, 103526 (2009) CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Greiner, W.: Relativistic Quantum Mechanics. Springer, Berlin (2000) MATHGoogle Scholar
  31. 31.
    Colussi, V., Wickramasekara, S.: Ann. Phys. 323, 3020 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Le Bellac, M., Levy-Léblond, J.M.: Nuovo Cimento 14B, 217 (1973) Google Scholar
  33. 33.
    Coleman, S., Mandula, J.: Phys. Rev. 159, 1251 (1967) MATHCrossRefADSGoogle Scholar
  34. 34.
    Kaku, M.: Quantum Field Theory, A Modern Introduction. Oxford University Press, New York (1993) Google Scholar
  35. 35.
    Weinberg, S.: The Quantum Theory of Fields. Vol. III: Supersymmetry. Cambridge University Press, New York (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Juan Sebastián Ardenghi
    • 1
  • Mario Castagnino
    • 2
  • Olimpia Lombardi
    • 3
  1. 1.CONICET-IAFEUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.CONICET-IAFE-IFIRUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.CONICETUniversidad de Buenos AiresBuenos AiresArgentina

Personalised recommendations