The loop derivative as a curvature

Abstract

Recently, a set of tools has been developed with the purpose of the study of Quantum Gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, of the so called path bundle of a manifold. It is well known that this topological principal bundle plays the role of a universal bundle for the reconstruction of principal bundles and their connections. The path bundle is canonically endowed with a parallel transport and associated with it important types of derivatives have been considered by several authors: the Mandelstam derivative, the connection derivative and the Loop derivative. In the present article we shall give a unified viewpoint for all these derivatives by developing a differentiable calculus on the path bundle. In particular we shall show that the loop derivative is the curvature of a canonically defined one form that we shall called the universal connection one form.