A calculus in differentiable spaces and its application to loops

Details A calculus in differentiable spaces and its application to loops A calculus in differentiable spaces and its application to loops

Aug 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999cqgra..16.2697r&link_type=abstract

Classical and Quantum Gravity, Volume 16, Issue 8, pp. 2697-2708 (1999).

Mathematics

Logic

1

Scientific paper

Recently, a set of tools has been developed with the purpose of studying 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, for 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 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. Here we shall give a unified viewpoint for all of these derivatives by developing a differentiable calculus on differentiable spaces. In particular, we shall show that the loop derivative is the curvature of a canonically defined 1-form that we shall call the universal connection 1-form.

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