Decomposition of direct products of representations of the inhomogeneous Lorentz group

Abstract

The direct products of the physically significant, irreducible, unitary representations of the proper, orthochronous inhomogeneous Lorentz group are reduced. It is shown that Γm1s1⊗Γm 2s2 contains only irreducible components of the form ΓmJ, and that ΓmJ occurs with nonzero multiplicity only if J - (s1+s2) is an integer. For such J's the multiplicity of Γm1J for J≥s1+s 2 is (2s1+l)(2s2+1) for each positive m. Γm1s1⊗Γs2(±) contains only irreducible components of the form Γmj, where J - (s1+s2) is an integer. The multiplicity of such Γm, j for J>s1+S 2 is (2s1+1) for each positive m. Γs 1(ε1)⊗Γs2(ε2) contains irreducible components of the form Γs(ε) and ΓmJ, where s=|ε1s1|, ε=sign (ε1S 1+ε2s2) and J - (s1+s 2) is an integer. The multiplicity of Γm, J is one for J> (s1+s2) and for each positive m. The multiplicity of Γs(ε) is infinite. The symmetrized squares are also analyzed. Numerous examples are given.