Reduction of the Direct Product of Representations of the Poincaré Group

Abstract

We expand the direct product of two representations of the Poincaré group into representations of the Poincaré group in the general case that the factors of the direct product may have any mass, whether real, zero, or imaginary, and the total energy may be indefinite. The representations of the Poincaré group, which appear in the expansion of the direct product, have masses which run through a continuous spectrum of real and imaginary values and are irreducible in terms of the mass and sign of energy (for real mass), but are reducible in terms of the infinitesimal generators of the little groups. To obtain the expansion in terms of irreducible representations, one need only reduce the infinitesimal generators of the little groups. This reduction is carried out for the real mass components and, in principal at least, can be carried out for the infinitesimal generators of the little groups for the imaginary mass components. The factors of the direct product and the representations which appear in the expansion are expressed in terms of a particular momentum representation called ``the standard helicity representation'' which enables us to use a uniform notation for all masses, whether real, zero, or imaginary. The earlier portion of the present paper summarizes the properties of these representations.