Unitary Representations of SL(2, C) in an E(2) Basis

Abstract

Starting from the functional representation of Gel'fand and Naimark, the unitary irreducible representations of SL(2, C) are described in a basis of the subgroup E(2)⊗D, where E(2)⊗D is the subgroup of all 2 × 2 matrices of the form (α0γδ), αδ=1. Physically, this is the subgroup into which SL(2, C) degenerates at infinite momentum and may be thought of as the 2-dimensional Euclidean group together with its dilations. Advantages to using the E(2)⊗D basis are: (1) It is convenient to calculate form factors; (2) the generators of E(2)⊗D are represented either multiplicatively or by first-order differential operators and are independent of the values of the SL(2, C) Casimir operators; (3) the principal and supplementary series of SL(2, C) are treated on the same footing and, in particular, have the same inner product; and (4) the transformation coefficients to the usual angular-momentum basis are related to Bessel functions. The E(2)⊗D is used to compute explicitly the finite matrix elements of an arbitrary Lorentz transformation and to investigate the structure of vector operators in unitary representation of SL(2, C).