Differentiable vectors and sharp momentum states of helicity representations of the Poincaré group

Abstract

This paper discusses some mathematical difficulties in handling sharp momentum eigenvectors for a massless helicity representation of the Poincaré group, related to the non-nuclearity of the space of differentiable vectors, and to the existence of singularities in the Lorentz group generators. A simple characterization of the nuclear space of differentiable vectors of the extension of the representation to a representation of the conformal group is given in terms of functions on the space R4− {0}. Using a fibration of this space over the forward light cone (in momentum space), the singularity in the generators is shown to be related to the fact that the standard presentation of the helicity representations should be reformulated in terms of nontrivial sections over the light cone. The problem is partly identical with the one encountered in the study of an electron in a magnetic monopole field, the generator singularity taking the place of the Dirac string.