Richard, J. L. 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 Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 8, 301-309 (1968). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 Documents Keywords:symmetries in microphysics PDF BibTeX XML Cite \textit{J. L. Richard}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 8, 301--309 (1968; Zbl 0161.23705) Full Text: Numdam EuDML OpenURL References: [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 [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 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.