Unitary Representations of the Homogeneous Lorentz Group in an O(2, 1) Basis

Abstract

Unitary irreducible representations of the homogeneous Lorentz group, SO(3, 1), belonging to the principal series and containing integral angular momenta, are reduced with respect to the subgroup SO(2, 1). It is found that the representation {j₀, ρ} of SO(3, 1), where j₀ is a nonnegative integer and ρ a real number, contains each representation of SO(2, 1) of the continuous class (nonexceptional and integral) twice, and each of the discrete representations $D_k^{\mathrm{(\pm )}}$ of SO(2, 1) once, for k = 1, 2, ⋯, j₀. The latter representations are absent for j₀ = 0. It is shown that the basis states of the representation $D_k^{\mathrm{(\pm )}}$ (for k ≥ 2) lie in the domain of those generators of SO(3, 1) that are outside the SO(2, 1) subalgebra, while the states of the representations $D_1^{\mathrm{(\pm )}}$ do not lie in this domain. It is further shown that from the point of view of the nature of this domain, the representations $D_1^{\mathrm{(\pm )}}$ of SO(2, 1) are very intimately connected to the continuous class representations of SO(2, 1), and that these two discrete representations act as a bridge between the remaining discrete representations on the one hand, and the continuous class representations on the other.