The inverse loop transform

TL;DR

This paper develops a method to compute the inverse of the loop transform in quantum gauge field theory, linking measures on the space of generalized connections to their characteristic functionals.

Contribution

It provides a way to explicitly compute the inverse loop transform, connecting positive linear functionals to measures on the moduli space of connections.

Findings

Derived the inverse loop transform for characteristic functionals.

Established the connection between positive linear functionals and measures.

Provided a computational framework for joint distributions in gauge theories.

Abstract

The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections modulo gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz-Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform.