The Generalisation of the Coulomb Gauge to Yang-Mills Theory

TL;DR

This paper extends the Coulomb gauge condition from Abelian to non-Abelian Yang-Mills theories, demonstrating a straightforward generalisation that simplifies the Gauss law and resolves the Gribov problem.

Contribution

It introduces a new generalisation of the Coulomb gauge for Yang-Mills theory that avoids the Gribov ambiguity by solving an elliptic PDE for gauge fixing.

Findings

Existence and uniqueness depend on an elliptic PDE.

The generalised gauge reduces Gauss law to gauge potentials only.

The Gribov problem does not occur in this generalisation.

Abstract

I consider the problem of generalising the Abelian Coulomb gauge condition to the non-Abelian Yang-Mills theory, with an arbitrary compact and semi-simple gauge group. It is shown that a straightforward generalisation exists, which reduces the Gauss law into a form involving the gauge potentials only, but not their time derivatives. The existence and uniqueness of the generalised Coulomb gauge is shown to depend on an elliptic linear partial differential equation for a Lie-algebra valued quantity, which defines the gauge transform by means of which the generalised Coulomb gauge condition is realised. Thus the Gribov problem is actually non-existent in this case.