E2 ¯ − Parametrization of SL(2, C)

Abstract

We consider the E2¯−parametrization of unimodular 2 × 2 matrices A ∈ SL(2, C), which is of the form A=E1e−12aσ3VE2, with V=e12iπσ2 and E=(e−12iφ0ze12iφ)∈E2¯. E2¯ is a covering of the group of Euclidean motions in the plane. We compute the correspondingly factorized matrix elements of the unitary representations of SL(2, C) in an E2¯−basis the result is given in Eq. (6). As a fringe benefit we obtain an integral transform which amounts to expansion in terms of Meijer G-functions and which generalizes the familiar Hankel transform. The results of this paper are useful, e.g., for computing vertex functions in the theory of massless particles with continuous spin.