Unitary Representations of the Lorentz Groups: Reduction of the Supplementary Series under a Noncompact Subgroup

Abstract

Unitary representations of O(2, 1) belonging to the exceptional class are reduced with respect to the noncompact subgroup O(1, 1). We recover the result that the spectrum of the generator of this subgroup covers the real line twice. Unitary representations of O(3, 1) belonging to the supplementary series are reduced with respect to the noncompact subgroup O(2, 1). These representations of O(3, 1) may be labeled by a parameter ρ in the range 0 < ρ < 1. Representations corresponding to 0 < ρ ≤ ½ yield upon reduction only those representations of O(2, 1) that belong to the continuous nonexceptional class; each of these appears twice. A representation corresponding to ½ < ρ < 1, however, yields upon reduction a single representation of O(2, 1) of the exceptational class (with parameter σ = ρ − ½) and, in addition, all the representations of O(2, 1) of the nonexceptional continuous class. The exceptional representation appears only once, while the nonexceptional ones appear twice each.