Poisson Brackets of Normal-Ordered Wilson Loops

TL;DR

This paper develops a classical framework for Yang-Mills theory using deformation quantization of Wilson loop variables, establishing a Poisson algebra structure and demonstrating an isomorphism with a previously introduced algebra.

Contribution

It introduces a Poisson algebra for normal-ordered Wilson loops in large-N Yang-Mills theory and shows its equivalence to a Weyl-ordered algebra via a Poisson morphism.

Findings

Established a Poisson algebra for normal-ordered Wilson loops.

Proved the equivalence of this algebra with a Weyl-ordered algebra.

Used topological graph theory concepts in the analysis.

Abstract

We formulate Yang-Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of $\hbar$) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum operators, we find, using ideas closly related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them.