Phase space, fibre bundles and current algebras

Abstract

The purpose of this paper is to extend into phase space the cellular description introduced by Bohmet al. (1970) and to show how this may help to give an understanding of the current algebra approach to elementary particle phenomena. We investigate this cellular structure in phase space in some detail and show how certain features of the structure may be described in terms of the mathematics of fibre bundle theory. The frame bundle is discussed and compared with the Yang-Mills theory. As a result of this discussion we are able to introduce generalised currents which are related to the duals of the curvature forms, and these are shown to span the Lie algebra of a sub-group of the structure group of the frame bundle. We then discuss the implications of these results in terms of our cell structure. By assuming that the de Rahm cohomology, defined by the curvature forms and their duals, reflect a cohomology on the integers defined on the original cell structure, we show that the currents and ‘curvature’ can be given a meaning in terms of a discrete structure. In this case the currents only span a Lie algebra in some suitable limit, implying that a description using Lie algebras is only an approximation.