Indecomposable representations of the Lorentz algebra in an angular momentum basis

Abstract

Indecomposable (i.e. reducible but not completely reducible) representations of the (complex) algebra so(3,1) of the Lorentz group are analysed. A master representation of so(3,1) is obtained on the space of its universal enveloping algebra Omega . The basis chosen for so(3,1) is the 'angular momentum' basis, and for Omega a 'natural basis' with respect to the angular momentum basis of so(3,1). The master representation induces representations on a space Omega - which is obtained from Omega as quotient space modulo certain ideals. These representations have the property that rho (h3) is diagonal. A change of basis is made from the natural basis to the angular momentum basis and the representations on Omega - are analysed in this new basis. The indecomposable (as well as the irreducible) representations which are obtained have the property that their so(3) content consists of infinite-dimensional indecomposable and irreducible so(3) representations. Certain of the indecomposable so(3,1) representations have additional invariant subspaces which lead to quotient spaces which are of finite dimension. These quotient spaces then carry the familiar finite-dimensional representations of the Lorentz algebra. It is shown that the standard theory of representations of the Lorentz algebra is contained in this analysis as a special case of representations which are induced on certain quotient spaces of the space Omega +.