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 C 1=−(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.
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REFERENCES
I. M. Gelfand and M. A. Neumark, Izv. Akad. Nauk SSSR 11, 411 (1947).
V. Bargmann, Ann. of Math. 48, 568 (1947).
Harish-Chandra, Proc. Roy. Soc. London Ser. A 189, 372 (1947).
I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5 (Gos. Izd. Fiz. Mat. Lit., Moscow, 1962) [in Russian].
N. Mukunda and R. Simon, J. Math. Phys. 36, 5170 (1995).
L. S. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, 1992, p. 530).
J. Schwinger, Particles, Sources and Fields, Vol. 1 (Addison–Wesley, Reading, MA, 1970, p. 274).
A. Staruszkiewicz, Ann. Phys. (N.Y.) 190, 354 (1989).
A. Staruszkiewicz, Erratum, Acta Phys. Pol. B 23, 959 (1992).
G. Wróbel, “Investigations in the quantum theory of the electric charge,” Ph.D. thesis, Cracow 1998, unpublished.
N. Mukunda, J. Math. Phys. 9, 417 (1968).
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Staruszkiewicz, A. Quantum Mechanics of the Supplementary Series. Foundations of Physics 32, 1863–1876 (2002). https://doi.org/10.1023/A:1022366616717
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DOI: https://doi.org/10.1023/A:1022366616717