A Linear Equation for Wilson Loops

TL;DR

This paper reformulates the non-linear Makeenko-Migdal loop equation in QCD$_2$ as a linear Schrödinger-like equation with a Coulomb potential, providing a new perspective on confinement as a bound state localization in loop space.

Contribution

It introduces a linear second-order differential equation for Wilson loops in QCD$_2$, transforming the complex loop dynamics into a bound state problem.

Findings

The loop equation becomes a Schrödinger equation with Coulomb potential.

Confinement is interpreted as a bound state localization in loop space.

Wilson loops decay exponentially outside a characteristic radius.

Abstract

The Makeenko-Migdal loop equation is non-linear and first order in the area derivative, but we show that for simple loops in QCD$_2$ it is possible to reformulate this equation as a linear equation with second order derivatives. This equation is a bound state Schr\"odinger equation with a three dimensional Coulomb potential. Thus, loop dynamics leads to a surprising new picture of confinement, where this phenomenon is due to a (bound state) localization in loop space, with the Wilson loops decaying exponentially outside a characteristic radius.