Unitary Irreducible Representations of SU(2, 2). I

Abstract

This paper, which is the first of three, is concerned with the general properties of the noncompact group SU(2, 2), the Lie algebra of which is isomorphic to the Dirac algebra. In the course of our study of the unitary representations, we first obtained all the finite dimensional irreducible representations. The associated Young diagrams are shown to have simple properties; the degenerate Young diagrams always denote degenerate representations. Through a theorem of Harish-Chandra, which relates the finite representations to the unitary representations in the discrete series, we are able to obtain explicitly all the unitary infinite-dimensional irreducible representations in this series, both degenerate and non-degenerate. The notion of multiplicity for nondegenerate representations is introduced and discussed in connection with a new operator F3, which is required for a complete labeling of states.