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Possible derivation of some SO (p, q) group representations by means of a canonical realization of the SO (p, q) Lie algebra. (English) Zbl 0161.23705

Keywords:
symmetries in microphysics
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 [1] A general formula is given in H. Bacry , Space time and degrees of freedom of the elementary particle , Comm. Math. Phys. , t. 5 , 1967 , p. 97 , for the semi-simple Lie groups. Let us recall it: 1/2(d - r) = g where d, r, g denote respectively the dimensionality of the Lie group, the number of its fundamental invariants, and the number of the generators of a maximal abelian subalgebra of the enveloping algebra. This formula comes from a more general framework in Gelfand and Kirillov , Sur les corps liés avec algèbres enveloppantes des algèbres de Lie , Publications Mathématiques , n^\circ 31 (I. H. E. S.). [2] [n/2] denotes the rank of the group, namely n/2 if n is even, (n - 1)/2 if n is odd. [3] Let us note the approach of A. KIHLBERG in which he considers as subgroup the maximal compact group and enters in similar considerations ( A. Kihlberg , Arkiv für Fysik , t. 30 , 1965 , p. 121 ). Many references on the study of particular non compact rotation groups are given in this paper. We recall some particular works in ref. [4]. Zbl 0171.11703 · Zbl 0171.11703 [4] The DE SITTER group SO(4,1) has been treated thoroughly by Dixmier , in J. Bull. Soc. Math. France , t. 89 , 1961 , p. 9 . The SO(3,2) group has been investigated by J.B. Ehrman , in Proc. Camb. Phil. Soc. , t. 53 , 1957 , p. 290 . MR 140614 | Zbl 0078.29302 · Zbl 0078.29302 [5] E.P. Wigner , Ann. Math. , t. 40 , 1939 , p. 149 . See also [6]. Zbl 0020.29601 | JFM 65.1129.01 · Zbl 0020.29601 · doi:10.2307/1968551 · www.emis.de [6] F.R. Halpern and E. Branscomb , Wigner’s analysis of the unitary representations of the Poincaré group , UCRL-12359, 1965 . [7] Y. Murai , Progr. Theor. Phys. , t. 11 , 1954 , p. 441 . H. Bacry , Ann. Inst. H. Poincaré , A II , 1965 , p. 327 . [8] In the following, the latin indices take always these values. [9] R. Raczka , N. Limic and J. Niederle , Discrete degenerate representations of non compact rotation group, IC/66/2, Trieste and Continuous degenerate representations of non compact rotation groups, IC/66/18 , Trieste . · Zbl 0158.45805 [10] L. Castell , The physical aspects of the conformal group SO0(4,2), IC/67/66 , Trieste . · Zbl 0162.58604 [11] H. Bacry and J.L. Richard , Partial theoretical group treatment of the relativistic hydrogen atom , 1967 (to be published in J. Math. Phys. ). Zbl 0158.45905 · Zbl 0158.45905 · doi:10.1063/1.1705146 [12] I.M. Gel’fand , R.A. Minlos et Z.Y. Shapiro , Representations of the rotation and Lorentz groups and their applications , Pergamon Press , 1963 . For other references, see [6]. Zbl 0108.22005 · Zbl 0108.22005
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