Abstract
The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.
References
Gel'fand, I. M., andM. L. Tseitlin: Dokl. Akad. Nauk. S.S.S.R.71, 1017 (1950). What we call Gel'fand-Tseitlin branching rules are given in this paper.
Weyl, H.: Classical Groups. Princeton University Press, 1939.
Thomas, L. H.: Ann. Math.42, 113 (1941);Newton, T. D.: Ann. Math.51, 730 (1950);Dixmier, J.: Bull. Soc. de France89, 9 (1960);Kihlberg, A.: Arkiv Fysik27, 373 (1964);Ström, S.: Arkiv Fysik30, 455 (1965);Kihlberg, A., andS. Ström: Arkiv Fysik31, 491 (1965).
Dirac, P. A. M.: Proc. Roy. Soc. A183, 372 (1947).
Barut, A. O., andC. Fronsdal: Proc. Roy. Soc. A287, 532 (1965).
Herman, R.: Comm. Math. Phys.3, 75 (1966).
Holman, W. J., andL. C. Biedenharn: Ann. Phys.39, 1 (1966).
Kuriyan, J. G., andE. C. G. Sudarshan: Phys. Rev.162, 1650 (1967);Mukunda, N., L. O'Raifeartaigh, andE. C. G. Sudarshan: Phys. Rev. Letters15, 1041 (1965): Phys. Letters19, 322 (1965);Mukunda, N.: Conference on Noncompact Groups in Particle Physics. New York: Benjamin 1966.
Sudarshan, E. C. G.: Proceedings of the Third Coral Gables Conference on Symmetry Principles. San Francisco and London: Freeman 1966.
Kuriyan, J. G., andPh. D. Thesis: Syracuse University (1966) unpublished.
--,N. Mukunda, andE. C. G. Sudarshan: Syracuse University. Report SU-118 IAS. Preprint (1967).
For a lucid presentation of rotation groups, seeGel'fand, I. M., R. A. Minlos, Z. Ya. Shapiro. Representations of Rotation and Lorentz Groups and the Applications. Oxford: Pergamon (1963). In the sequel we shall follow an approach very similar to this.
It is worth emphasizing that allU I R's ofSO(5, 1) are obtained by the analysis of the Master Analytic Function. We believe that we have considered all possible values of the parameters and therefore the list is complete.
Harish-Chandra: Proc. Roy. Soc. A189, 372 (1947).
Author information
Authors and Affiliations
Additional information
Work supported in part by the National Science Foundation.
Work supported in part by the U.S. Atomic Energy Commission.
Rights and permissions
About this article
Cite this article
Kuriyan, J.G., Mukunda, N. & Sudarshan, E.C.G. Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups. Commun.Math. Phys. 8, 204–227 (1968). https://doi.org/10.1007/BF01645857
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01645857