Clebsch-Gordan problem for three-dimensional Lorentz group in the elliptic basis: I. Tensor product of continuous series

Abstract

This paper is the first of two papers devoted to the study of the Clebsch-Gordan (CG) problem for the three-dimensional Lorentz group in an elliptic (or SO(2)) basis. Here we describe the reduction of the tensor product of two unitary irreducible representations (UIRs) of the continuous series, i.e. belonging to either the principal or complementary series. The corresponding CG coefficients are defined as matrix elements of an intertwining operator between the tensor product representation and the irreducible component appearing in the decomposition. We then obtain an expression for CG coefficients in terms of a single function, namely in terms of the bilateral series with unit argument defined in the complex space of the variable . In the general case the functions are expressed in terms of two hypergeometric functions with unit argument; however, it reduces to the single function if at least one of the coupling UIRs belong to a discrete series. We derive a completeness relation for CG coefficients for all the cases under consideration.