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).