Abstract
In order to construct the quantum field theory in a curved space with no “old” infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO 0(n, 1) to positive mass representations of both the Euclidean and Poincaré groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincaré groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
See, e.g., Salam, A.: Preprint ICTP, IC/71/108, Trieste 1971.
Fronsdal, C.: Rev. Mod. Phys.37, 221 (1965).
Inönü, E., Wigner, E.P.: Proc. Nat. Acad. Sci. U.S.39, 510 (1953).
Segal, I.E.: Duke Math. J.18, 221 (1951).
Saletan, E. J.: J. Math. Phys.2, 1 (1961).
Doebner, H. D., Melsheimer, O.: Nuovo Cimento49A, 306 (1967).
Lykhmus, J.: Contractions of Lie Groups, at the Proceedings of the 1967 Summer School on Elementary Particle Physics (held at Otepää, Estonian SSR), Vol.4. Tallin 1969.
Philips, T. O., Wigner, E. P.: De Sitter space and positive energy. In: Group Theory and Its Applications, ed. by E. M. Loebl. New York and London: Academic Press, p. 631, 1968.
Ström, S.: Ark. Fysik30, 455 (1965) see also S. Ström, Ann. Inst. Henri Poincaré13, 77 (1970).
Evans, N. T.: J. Math. Phys.8, 170 (1967).
Chakrabarti, A.: J. Math. Phys.9, 2087 (1968); see also Schwartz, F.: J. Math. Phys.12, 131 (1971).
Wolf, K. B.: J. Math. Phys.12, 197 (1971).
Pontryagin, L. S.: Topological Groups, p. 137, New York-London-Paris: Gordon and Breach, Inc., 1966.
Hermann, R.: Lie Groups for Physicists, New York: W. A. Benjamin, 1966.
Knapp, A. W., Stein, E. M.: Ann. Math.93, 489 (1971).
Sciarrino, A., Toller, M.: J. Math. Phys.8, 1252 (1967). Mukunda, N.: J. Math. Phys.8, 2210 (1967);9, 50 (1968). Ström, S.: Ark. Fys.34, 215 (1967);40, 1 (1969). Kuryan, J. G., Mukunda, N., Sudarshan, E. C. G.: J. Math. Phys.9, 2100 (1968). Macfadyen, N. W.: J. Math. Phys.12, 492 (1971).
Niederle, J.: J. Math. Phys.8, 1921 (1968). Limić, N., Niederle, J.: Ann. Inst. Henri Poincaré IXA, 327 (1968).
Hannabuss, K. C.: Proc. Cambridge Phil. Soc.70, 283 (1971).
Author information
Authors and Affiliations
Additional information
On leave of absence from the Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.
Rights and permissions
About this article
Cite this article
Mickelsson, J., Niederle, J. Contractions of representations of de Sitter groups. Commun.Math. Phys. 27, 167–180 (1972). https://doi.org/10.1007/BF01645690
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01645690