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 ( α 0 γ δ ) , 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).