Differential geometry on the space of connections via graphs and projective limits

Abstract

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a “universal home” for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as “floating lattices” in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G : differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.