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Foundations of Physics

, Volume 32, Issue 12, pp 1863–1876 | Cite as

Quantum Mechanics of the Supplementary Series

  • Andrzej Staruszkiewicz
Article

Abstract

Unitary, irreducible representations of the proper, orthochronous Lorentz group comprise the main series and the supplementary series. The main series is spanned by the complete set of eigenstates of the self-adjoint Casimir operator C1=−(1/2)M μν M μν , where M μν are generators of Lorentz transformations. The supplementary series has no such interpretation; moreover it is spurious from the point of view of functional analysis as it does not enter into the integral representation of an arbitrary test function. The author describes the physical context within which the supplementary series arises, nevertheless, in a natural way: it arises if we exponentiate a massless scalar quantum field living in three-dimensional de Sitter space-time and if the appropriate coupling constant is small enough.

Lorentz group electric charge fine structure constant 

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REFERENCES

  1. 1.
    I. M. Gelfand and M. A. Neumark, Izv. Akad. Nauk SSSR 11, 411 (1947).Google Scholar
  2. 2.
    V. Bargmann, Ann. of Math. 48, 568 (1947).Google Scholar
  3. 3.
    Harish-Chandra, Proc. Roy. Soc. London Ser. A 189, 372 (1947).Google Scholar
  4. 4.
    I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5 (Gos. Izd. Fiz. Mat. Lit., Moscow, 1962) [in Russian].Google Scholar
  5. 5.
    N. Mukunda and R. Simon, J. Math. Phys. 36, 5170 (1995).Google Scholar
  6. 6.
    L. S. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, 1992, p. 530).Google Scholar
  7. 7.
    J. Schwinger, Particles, Sources and Fields, Vol. 1 (Addison–Wesley, Reading, MA, 1970, p. 274).Google Scholar
  8. 8.
    A. Staruszkiewicz, Ann. Phys. (N.Y.) 190, 354 (1989).Google Scholar
  9. 9.
    A. Staruszkiewicz, Erratum, Acta Phys. Pol. B 23, 959 (1992).Google Scholar
  10. 10.
    G. Wróbel, “Investigations in the quantum theory of the electric charge,” Ph.D. thesis, Cracow 1998, unpublished.Google Scholar
  11. 11.
    N. Mukunda, J. Math. Phys. 9, 417 (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Andrzej Staruszkiewicz
    • 1
  1. 1.Institute of PhysicsJagellonian UniversityKrakówPoland

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