Generalized Measures in Gauge Theory

TL;DR

This paper constructs generalized measures on the space of connections and gauge transformations in gauge theory, providing a rigorous foundation for measures analogous to Lebesgue and Haar measures using loop and lattice techniques.

Contribution

It introduces a framework for defining well-behaved generalized measures on connection and gauge transformation spaces, extending previous measures and invariance properties.

Findings

Constructed generalized measures on A and Ga spaces.

Proved invariance of the uniform measure under automorphisms.

Showed how to obtain gauge-invariant measures via averaging with Haar measure.

Abstract

Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure'' on the space A of connections on P and for ``Haar measure'' on the group Ga of gauge transformations. More precisely, we define algebras of ``cylinder functions'' on the spaces A, Ga, and A/Ga, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures on A, Ga, and A/Ga in terms of graphs embedded in M. We use this characterization to construct generalized measures on A and Ga, respectively. The ``uniform'' generalized measure on A is invariant under the group of automorphisms of P. It projects down to the generalized measure on A/Ga considered by Ashtekar and Lewandowski in the case G = SU(n). The ``generalized Haar measure'' on Ga is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure on A against generalized Haar measure gives a gauge-invariant generalized measure on A.