Topological Measure and Graph-Differential Geometry on the Quotient Space of Connections

TL;DR

This paper develops a diffeomorphism-invariant measure on the space of gauge-equivalent connections, enabling rigorous integral calculus and operator analysis in quantum gauge theories.

Contribution

It introduces a non-linear generalization of cylindrical measures on the quotient space of connections, ensuring well-defined operators in the associated Hilbert space.

Findings

A diffeomorphism-invariant measure is constructed.

Momentum operators are shown to be essentially self-adjoint.

The measure allows for integral calculus on the space of connections.

Abstract

(This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of the quotient space. The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly, to become essentially self adjoint operators.