Foundations of Physics

, Volume 39, Issue 9, pp 1023–1045 | Cite as

Quantum Mechanics: Modal Interpretation and Galilean Transformations

  • Juan Sebastian Ardenghi
  • Mario Castagnino
  • Olimpia Lombardi
Article

Abstract

The aim of this paper is to consider in what sense the modal-Hamiltonian interpretation of quantum mechanics satisfies the physical constraints imposed by the Galilean group. In particular, we show that the only apparent conflict, which follows from boost-transformations, can be overcome when the definition of quantum systems and subsystems is taken into account. On this basis, we apply the interpretation to different well-known models, in order to obtain concrete examples of the previous conceptual conclusions. Finally, we consider the role played by the Casimir operators of the Galilean group in the interpretation.

Keywords

Modal-Hamiltonian interpretation Galilean group Casimir operators 

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References

  1. 1.
    Lombardi, O., Castagnino, M.: A modal-Hamiltonian interpretation of quantum mechanics. Stud. Hist. Philos. Mod. Phys. 39, 380–443 (2008) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Castagnino, M., Lombardi, O.: The role of the Hamiltonian in the interpretation of quantum mechanics. J. Phys. Conf. Ser. 28, 012014 (2008) CrossRefADSGoogle Scholar
  3. 3.
    van Fraassen, B.C.: A formal approach to the philosophy of science. In: Colodny, R. (ed.) Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, pp. 303–366. University of Pittsburgh Press, Pittsburgh (1972) Google Scholar
  4. 4.
    van Fraassen, B.C.: Semantic analysis of quantum logic. In: Hooker, C.A. (ed.) Contemporary Research in the Foundations and Philosophy of Quantum Theory, pp. 80–113. Reidel, Dordrecht (1973) Google Scholar
  5. 5.
    van Fraassen, B.C.: The Einstein-Podolsky-Rosen paradox. Synthese 29, 291–309 (1974) CrossRefGoogle Scholar
  6. 6.
    Dieks, D., Vermaas, P.E.: The Modal Interpretation of Quantum Mechanics. Kluwer, Dordrecht (1998) MATHGoogle Scholar
  7. 7.
    Harshman, N.L., Wickramasekara, S.: Galilean and dynamical invariance of entanglement in particle scattering. Phys. Rev. Lett. 98, 080406 (2007) CrossRefADSGoogle Scholar
  8. 8.
    Harshman, N.L., Wickramasekara, S.: Tensor product structures, entanglement, and particle scattering. Open Sys. Inf. Dyn. 14, 341–351 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967) MATHMathSciNetGoogle Scholar
  10. 10.
    Ballentine, L.: Quantum Mechanics: A Modern Development. World Scientific, Singapore (1998) MATHGoogle Scholar
  11. 11.
    Tinkham, M.: Group Theory and Quantum Mechanics. Dover, New York (1992) Google Scholar
  12. 12.
    Meijer, P., Bauer, E.: Group Theory. The Application to Quantum Mechanics. Dover, New York (2004) Google Scholar
  13. 13.
    Weinberg, S.: The Quantum Theory of Fields, vol. I: Foundations. Cambridge University Press, Cambridge (1995) Google Scholar
  14. 14.
    Bose, S.K.: The Galilean group in 2+1 space-times and its central extension. Commun. Math. Phys. 169, 385–395 (1995) MATHCrossRefADSGoogle Scholar
  15. 15.
    Brading, K., Castellani, E.: Symmetries and invariances in classical physics. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics, pp. 1331–1367. North-Holland, Amsterdam (2007) CrossRefGoogle Scholar
  16. 16.
    Cohen-Tannoudji, C., Diu, B., Lalöe, F.: Quantum Mechanics. Wiley, New York (1977) Google Scholar
  17. 17.
    Laue, H.: Space and time translations commute, don’t they? Am. J. Phys. 64, 1203–1205 (1996) CrossRefADSGoogle Scholar
  18. 18.
    Peleg, Y., Pnini, R., Zaarur, E.: Theory and Problems of Quantum Mechanics. McGraw-Hill, New York (1998) Google Scholar
  19. 19.
    Tung, W.K.: Group Theory in Physics. World Scientific, Singapore (1985) Google Scholar
  20. 20.
    Wigner, E.P.: On the unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Bargmann, V.: On unitary ray representations of continuous groups. Ann. Math. 59, 1–46 (1954) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Lévy-Leblond, J.M.: Galilei group and nonrelativistic quantum mechanics. J. Math. Phys. 4, 776–788 (1963) MATHCrossRefADSGoogle Scholar
  23. 23.
    Dynkin, E.B.: Calculation of the coefficients in the Campbell-Hausdorff formula. Dokl. Akad. Nauk 57, 323–326 (1947) MATHMathSciNetGoogle Scholar
  24. 24.
    Greiner, W., Reinhardt, J.: Field Quantization. Springer, Berlin (1995) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Juan Sebastian Ardenghi
    • 1
  • Mario Castagnino
    • 2
  • Olimpia Lombardi
    • 3
  1. 1.CONICET - IAFEBuenos AiresArgentina
  2. 2.CONICET - IAFE - IFIRBuenos AiresArgentina
  3. 3.CONICET - UBABuenos AiresArgentina

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