Foundations of Physics

, Volume 44, Issue 12, pp 1317–1335 | Cite as

On the Relation Between Gauge and Phase Symmetries

Article

Abstract

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from \(q\)and\(p\) to \(q\)or\(p\) in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s moment map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points.

Keywords

Quantum mechanics Gauge theories Moment map Symplectic reduction Symplectic “category” 

References

  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley Publishing Company, Reading (1978)MATHGoogle Scholar
  2. 2.
    Catren, G.: On classical and quantum objectivity. Found. Phys. 38, 470–487 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Catren, G.: Can classical description of physical reality be considered complete? In: Bitbol, M., Kerszberg, P., Petitot, J. (eds.) Constituting Objectivity: Transcendental Perspectives on Modern Physics, The Western Ontario Series in the Philosophy of Science, vol. 74, pp. 375–386. Springer-Verlag, Berlin (2009)Google Scholar
  4. 4.
    Catren, G.: Quantum ontology in the light of gauge theories. In: de Ronde, C., Aerts, S., Aerts, D. (eds.) Probing the Meaning of Quantum Mechanics: Physical, Philosophical, and Logical Perspectives. World Scientific Publishing, Singapore (2014)Google Scholar
  5. 5.
    Dirac, P.M.: Lectures on Quantum Mechanics. Dover Publications, New York (1964)Google Scholar
  6. 6.
    Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515–538 (1982)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  8. 8.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, New Jersey (1994)Google Scholar
  9. 9.
    Hochs, P., Landsman, N.P.: The Guillemin–Sternberg conjecture for noncompact groups and spaces. J. K-Theory 1, 473–533 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hörmander, L.: The analysis of Fourier integral operators, I. Acta Math. 127, 79–183 (1971)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Iglesias, P.: Symétries et moment. Editions Hermann, Paris (2000)Google Scholar
  12. 12.
    Isham, C.J.: Topological and global aspects of quantum theory. In: DeWitt, B.S., Stora, R. (eds.) Relativity, Groups and Topology II, Les Houches Session XL, pp. 1060–1290. North-Holland Publishing Company, Amsterdam (1983)Google Scholar
  13. 13.
    Kirillov, A.A.: Lectures on the Orbit Method. Graduate Studies in Mathematics, vol. 64. AMS, Providence (2004)Google Scholar
  14. 14.
    Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer Monographs in Mathematics. Springer-Verlag, New York (1998)CrossRefGoogle Scholar
  15. 15.
    Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Springer, Netherlands (1987)CrossRefMATHGoogle Scholar
  16. 16.
    Marsden, J.E., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer-Verlag, New York (1999)CrossRefMATHGoogle Scholar
  18. 18.
    Ortega, J.-P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction. Birkhäuser, Boston (2004)CrossRefMATHGoogle Scholar
  19. 19.
    Souriau, J.-M.: Structure of Dynamical Systems. A Symplectic View of Physics. Birkhäuser, Boston (1997)CrossRefMATHGoogle Scholar
  20. 20.
    Weinstein, A.: Symplectic geometry. Bull. Am. Math. Soc. 5, 1 (1981)CrossRefMATHGoogle Scholar
  21. 21.
    Xu, P.: Morita equivalence of Poisson manifolds. Commun. Math. Phys. 142, 493–509 (1991)ADSCrossRefMATHGoogle Scholar
  22. 22.
    Xu, P.: Classical intertwiner space and quantization. Commun. Math. Phys. 164, 473–488 (1994)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Laboratoire SPHERE (UMR 7219)Université Paris Diderot - CNRSParisFrance
  2. 2.Facultad de Filosofía y LetrasUniversidad de Buenos Aires - CONICETBuenos AiresArgentina

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