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 ${\mathrm{\Gamma m}}_1{s}_1{\mathrm{\otimes \Gamma m}}_2{s}_2$ contains only irreducible components of the form Γ_mJ, and that Γ_mJ occurs with nonzero multiplicity only if J − (s₁+s₂) is an integer. For such J's the multiplicity of Γ_{m, J} for J≥s₁+s₂ is (2s₁+1) (2s₂+1) for each positive m. ${\mathrm{\Gamma m}}_1{s}_1{\mathrm{\otimes \Gamma s}}_2^{\mathrm{(\pm )}}$ contains only irreducible components of the form Γ_mJ, where J − (s₁+s₂) is an integer. The multiplicity of such Γ_{m, J} for J≥s₁+s₂ is (2s₁+1) for each positive m. ${\mathrm{\Gamma s}}_1^{{\mathrm{(\epsilon }}_1)}{\mathrm{\otimes \Gamma s}}_2^{{\mathrm{(\epsilon }}_2)}$ contains irreducible components of the form Γ_s^(∈) and Γ_mJ, where s = | ∈₁s₁+∈₂s₁ |, ∈ = sign (∈₁s₁+∈₂s₂) and J − (s₁+s₂) is an integer. The multiplicity of Γ_{m, J} is one for J≥(s₁+s₂) and for each positive m. The multiplicity of Γ_s^(∈) is infinite. The symmetrized squares are also analyzed. Numerous examples are given.