# 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.