The kinematics of the angular momentum of a particle

The way a particle changes its angular momentum under «homogeneous Lorentz transformations is well-known classically. The object of the present paper and a later one is to consider the problem quantuni-mechanically for particles of any mass and any spin. In the present paper we shall consider in detail the case where a particle has a definite angular momentum in one frame of reference and we shall calculate the probability distribution of angular momentum in a frame translated with respect to the original frame. Tn a later paper we shall treat the case where the two frames of reference are moving with respect to one another. The basic mathematical tool is the form for the infinitesimal generators of the inhomogeneous Lorentz group devised by Lomont and Moses in which the Hamiltonian, square of the angular momentum,z-component of angular momentum, and helicity are diagonal. The present paper and the projected one are important in multiple scattering problems, for it is possible using the results to take into account, to a certain degree at least, the effect of the selection rules. These rules are almost always ignored in multiple scattering problems. For example, it is shown that when the density of a gas is sufficiently low, radiative cooling goes on at a much faster rate when selection rules are taken into account than when they are ignored.