Unimodular Matrices Homomorphic to Lorentz Transformations in n ≧ 2 Spacelike Dimensions

Abstract

We generalize the usual homomorphism between 2 × 2 unimodular matrices B (Λ) and restricted 4 × 4 LORENTZ matrices Λ to the case of one timelike and n ≧ 2 spacelike dimensions. For every such n which is even (odd), this generalization associates homomorphically to each restricted (orthochronous) (n+1)-dimensional LORENTZ matrix a set of Ν × N-dimensional unimodular matrices, where N=2n/2 or 2(n-1)/2, depending on whether n is even or odd. In the case n ≧ 2, we prove the theorem that, if B (Λ) and Β (ΛT) are any two such unimodular matrices associated with Λ and its adjoint ΛT, respectively, then B (ΛT) =ω Β (Λ) t, where ω is an Nth root of unity and † means hermitean adjoint. We also prove that for n>3 one can select these two unimodular matrices so that this equation holds with ω=1, but that no such selection is possible for n=2, 3.