The master analytic function and the Lorentz group. III. Coupling of continuous representations of O(2,1)

Abstract

The Clebsch–Gordan problem for continuous representations belonging to the principal series of O(2,1) is treated by a method developed previously for the coupling of a discrete and a continuous representation. The values of the complex variable x occurring in the fundamental differential equation of the problem are restricted to lie on the unit circle, and the Clebsh–Gordan coefficients are identified with the Fourier coefficients of solutions of this equation. If j belongs to the discrete class there is only one acceptable solution of the second order equation. But, if j1,j2,j all belong to the continuous class any two independent solutions of the equation give a possible Clebsch–Gordan series. The problem of orthogonalizing the solutions in the latter case is solved and the normalization factor is determined using the Sturm–Liouville theory of differential equations. The Clebsch–Gordan coefficients generated by an orthogonal pair of solutions become automatically orthogonal. To determine the j values appearing in the reduction, a product state xm2 is expanded in a series of the coupled states gjm(x) by means of the Burchnall–Chaundy formula followed by the Sommerfeld–Watson transformation.