Kinematics of the Relativistic Two-Particle System

Abstract

Knowing the (canonical) forms that the generators of the infinitesimal transformations of P (the restricted Poincaré Group) take when operating within the Hilbert space of states of an arbitrary irreducible representation of P, such as that which affords a kinematic description of a single relativistic particle, we develop the forms in which they appear when operating within the Hilbert space of states of the (reducible) direct product representation of P which describes a system of two non-interacting relativistic particles. On introducing the Clebsch-Gordan (C-G) series of P which expresses the resolution of this latter Hilbert space into a direct integral over the Hilbert spaces wherein operate the irreducible constituents of the direct product representation, we prove that the generators operate in canonical form within each of these subspaces. If we adopt the alternative viewpoint that the C-G series of P must be written exactly so as to achieve this, our analysis may be regarded as providing a fully explicit and mathematically complete derivation of the formula for the C-G coefficients of P that appear in the C-G series. This formula has previously been suggested only on the basis of heuristic physical argument, and its proof is the principal accomplishment of the present work. One important fact which receives further clarification from the explicit nature of our analysis is the following: in order to see how the intrinsic angular momentum of the two-particle system, in a state of given linear momentum, is compounded from the relative angular momentum and intrinsic spins of the particles, one must view the state from a frame of reference wherein it appears to have zero momentum.