Transformation from a Linear Momentum to an Angular Momentum Basis for Relativistic Particles of Nonzero Mass and Any Spin

Abstract

The infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear-momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In a previous paper we showed how the two bases were related for representations corresponding to zero mass and any finite spin. In the present paper we show how the two bases are related for representations corresponding to nonzero mass and any spin. Thus this paper and the preceding one enable us to expand relativistic plane waves into relativistic spherical waves and vice-versa for particles of any spin and any mass. In the previous paper we used the relation between the linear- and angular-momentum bases to integrate the infinitesimal generators in the angular-momentum basis and thereby obtain closed expressions which show how the angular momentum of particles of zero mass and any finite spin transform under changes of frame of reference. Similar use could be made of the transformation of the present paper. Results of such use will be given in a later paper.