The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis

Abstract

The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.