Unitary Irreducible Representations of SU(2, 2). III. Reduction with Respect to an Iso-Poincaré Subgroup

The unitary irreducible representations of SU(2, 2), the covering group of the conformal group, are reduced with respect to an iso-Poincaré subgroup E(3, 1). Explicit representations of the 15 generators of SU(2, 2) in terms of differential operators in the function space of |η,ξ,ε;s,t,m〉, |ω,ξ,ε;r,t,m〉, and |ξ,ε;t,m〉, the basis vectors for ``timelike,'' ``spacelike,'' and ``lightlike'' UIR's of E(3, 1), respectively, are given. The matrix elements between a basis vector in the maximal compact subgroup SU(2) × SU(2) × U(1) and a basis vector in E(3, 1) are calculated for all 14 different classes of UIR's of SU(2, 2). We find that two classes are reducible with respect to ``lightlike'' representations, two classes are reducible with respect to ``timelike'' representations, eight classes are reducible with respect to ``spacelike'' representations, and two classes contain both ``timelike'' and ``spacelike'' representations of E(3, 1).