Identity of the SU(1,1) and SU(2) Clebsch-Gordan coefficients coupling unitary discrete representations

Abstract

The Clebsch-Gordan (CG) coefficient coupling two positive discrete series or two negative discrete series of irreducible unitary representations (IUR) of SU(1,1) are shown to be identical to SU(2) CG coefficients. The transformations changing the indices of the SU(1,1) CG coefficients into those of the identical SU(2) CG coefficients are derived and shown to transform the SU(2) CG coefficients into SU(1,1) CG coefficients. The associated phases are discussed. General index transformations are derived and used to generate SU(2) CG coefficient symmetries, among which are some of the more abstract Regge symmetries. A simple invariance property of the intermediate SU(1,1) CG coefficients is at the base of the symmetries. The demonstrations are carried out with coexistent IUR of the tensor product groups SU(1,1)(+)SU(1,1) and SU(2)(+)SU(2) embedded in a simple encompassing group structure, the most degenerate discrete IUR of SU(2,2).