The master analytic function and the Lorentz group. II. The Clebsch–Gordan problem for O(2,1)

Abstract

The Clebsch–Gordan coefficients of the noncompact group O(2,1) representing Lorentz transformations in three-dimensional space–time are calculated in the compact O(2) basis. Considerable simplification is achieved by introducing a variable x and replacing all algebraic equations by differential equations. The coupled state appears in the theory as a solution of an ordinary differential equation reducible to the hypergeometric equation by a simple substitution. The coefficients in the Taylor–Laurent expansion of this solution in powers of x are shown to be identical with the Clebsch–Gordan coefficients. The inverse expansion, obtained by the use of certain identities for the hypergeometric function and the Sommerfeld–Watson transformation, yields the normalization factor and the values of j appearing in the reduction.