Equivalence of a class of Wigner coefficients of S U (1,1) with those of S U (2)

Abstract

The purpose of this note is to establish the relation (λ1λ2μ1μ2‖ΛM)nc= (−1)μ2−λ2〈 (1/2) M+λ1−λ2−1), (1/2)(M+λ2−λ1−1), (1/2)(μ1−μ2+λ1+λ2−1), (1/2)(μ2−μ1+λ1+λ2−1) ‖Λ−1,λ1+λ2−1〉, where the left-hand side is a Wigner coefficient of the noncompact group SU (1,1) and in the right-hand side appears a standard Wigner coefficient of SU (2). The parameters λ1, λ2, Λ characterize unitary irreducible representations in the positive discrete series of SU (1,1), and thus they take positive integer or half-integer values. The other parameters are restricted by μ1=λ1,λ1+1,λ1+2,⋅⋅⋅, μ2=λ2,λ2+1,λ2+2,⋅⋅⋅, M=Λ,Λ+1,Λ+2,⋅⋅⋅. Besides we have μ1+μ2=M and Λ=λ1+λ2,λ1+λ2+1,⋅⋅⋅. A similar result is obtained [cf. our Eq. (3.28)] for Wigner coefficients involving unitary irreducible representations of the negative discrete series of SU (1,1).