Manifestly covariant forms for Lorentz group unitary representations

Abstract

We show that the unitary irreducible representations (UIRs) of the proper homogeneous Lorentz group SO(3,1) can be realized in a manifestly covariant form, using Hilbert spaces of functions on suitable geometrical configurations of vectors in space time. The concept and properties of isotropic lines, which are generators of spacelike hyperboloids, and of stability groups of lightlike vectors, are crucial to the constructions. Both principal and supplementary Series UIRs are treated.