A New Approach to the Quantisation of Paths in Space-Time

Abstract

A discrete path in space-time can be considered as a series of applications of the translation subgroup of the Poincare group. If there is a local mapping from this translation group into a neighbourhood of the identity of a quantum Weyl algebra fibre bundle, then the whole classical path can be lifted into the fibre bundle to form a unique quantum field as a section through the fibres. Under the further assumptions of scale relativity, we also show that a discrete closed loop in space time, corresponding to two classical paths sharing the same end points, is renormalisable and the finite limit has anomalous dimension equal to the fractal dimension. We end by introducing the possibility of a ‘push forward’ connection on the bundle O(D) of Ehresmann type.