Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group

Abstract

In the present paper we show explicitly how to reduce reducible representations of the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group. We first construct a basis in which all the reducible representations are expressed as integrals of representations over masses and in which the infinitesimal generators act as in the Foldy-Shirokov realization for nonzero masses and in the Lomont-Moses realization for zero masses. However, the spin operators which appear in the Foldy-Shirokov realizations and the generators of the Euclidean group which appear in the Lomont-Moses realizations are reducible in general. Thus reducible representations are only partially reduced in this basis. On carrying out the reduction of the spin operators and the generators of the Euclidean group, we introduce a second basis such that the reducible representations are completely reduced. By changing the emphasis slightly, the methods of the present paper can be used to obtain the irreducible representations of the generators. One of us (H. E. M.) has already used the methods of the present paper to reduce the electromagnetic vector potential and, in papers which follow the present one, will show how to reduce wavefunctions in general and will also derive the Clebsch-Gordan expansion for the direct product of two massless representations of finite spin and the same sign of energy. From the work of Mautner and Mackey it is known that every reducible unitary ray representation of the proper, orthochronous, inhomogeneous Lorentz group can be reduced to a direct integral of the irreducible representations. The techniques of the present paper thus enable us to carry out the reduction explicitly.