Asymptotic behavior of Wilson loops from Schroedinger equation on the gauge group

TL;DR

This paper explores the asymptotic behavior of Wilson loops in quantum Yang-Mills theory by linking their probability distributions to solutions of the Schrödinger equation on the gauge group, revealing geometric interpretations.

Contribution

It demonstrates that Wilson loop amplitudes under certain conditions can be derived from the Schrödinger equation on the gauge group, connecting quantum gauge fields with geometric and path integral formulations.

Findings

Wilson loop amplitudes satisfy a Schrödinger equation on the gauge group.

The solution is expressed via path integrals with a geometric interpretation.

The partition function of 2D Yang-Mills theory on the surface relates to this Schrödinger equation.

Abstract

Probability distribution of non-Abelian parallel transporters on the group manifold and the corresponding amplitude are investigated for quantum Yang-Mills fields. It is shown that when the Wilson area law and the Casimir scaling hold for the quantum gauge field, this amplitude can be obtained as the solution of the free Schroedinger equation on the gauge group. Solution of this equation is written in terms of the path integral and the corresponding action term is interpreted geometrically. We also note that the partition function of two-dimensional pure Yang-Mills theory living on the surface spanned on the loop solves the obtained equation.