A duality between massless particles and a charge-monopole system

Abstract

It is pointed out that the same realisation of a unitary representation of SO(4, 2) from the ladder series is used in the coordinate-space description of a charge-monopole system, (or more generally a dyon-dyon system) and in the canonical momentum-space description of a massless particle. Therefore, in the latter case a momentum-space analogue appears for the monopole vector potential, complete with its Dirac string singularity. Analogues of gauge transformations relate equivalent realisations with different locations of the momentum-space string. Quantisation of helicity replaces quantisation for the product of electric and magnetic charge. The problem of localising a charge on a monopole string is related to recent work by Flato et al. [5] on the localisability of a massless particle in momentum-space. Further, the multi-component form of the generators of SO(4, 2) for a massless particle has a dual, in coordinate space, corresponding to a charge-monopole system for a monopole of the Yang-Mills type. The question of the interpretation of the momentum analogue of the monopole field is raised.