The Lorentz group in the oscillator realization. III. The group SO(3,1)

Abstract

Employing the boson operators of Barut and Böhm, we study the oscillator realization of the Lie algebra of the Lorentz group SO(3,1) in the coordinate representation. The construction yields a direct sum of the principal series of representations ( j0,ρ) belonging to the integral or half-integral class. The decomposition of the representation space into the eigenspaces L2j0ρ of irreducible representations leads to a two-variable second order realization of the SO(3,1) algebra acting on fj0 ρ ∈L2j0ρ. The construction is shown to be highly symmetric. While the elements of the SO(2,1) subalgebra are invariant under the pseudorotation group SO(2,2), those of the full SO(3,1) algebra are invariant under the SO(2)×SO(1,1) subgroup of SO(2,2). We use this intrinsic symmetry in the construction to identify the generalized SO(2,1)⊆SO(3,1) eigenbases with the SO(2,2) harmonics in an SO(2)×SO(1,1) basis, and thereby achieve a significant unification among results which would normally appear disconnected.