[Paper Review] The Generalisation of the Coulomb Gauge to Yang-Mills Theory

C. Cronström|ArXiv.org|Oct 1, 1998

Quantum and Classical Electrodynamics3 references3 citations

TL;DR

This paper generalizes the Abelian Coulomb gauge to non-Abelian Yang-Mills theories with arbitrary compact semi-simple gauge groups by formulating a gauge condition that eliminates time derivatives from Gauss's law. The construction relies on solving an elliptic partial differential equation for a Lie-algebra-valued gauge parameter, proving existence and uniqueness of the gauge without encountering the Gribov problem.

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.

Motivation & Objective

• To extend the Abelian Coulomb gauge condition to non-Abelian Yang-Mills theories with arbitrary compact semi-simple gauge groups.
• To formulate a gauge condition that expresses Gauss's law solely in terms of gauge potentials, excluding time derivatives.
• To establish the existence and uniqueness of the generalized Coulomb gauge through a solvable elliptic PDE.
• To resolve the Gribov problem in this gauge by showing the gauge-fixing condition leads to a well-posed elliptic equation.
• To provide a consistent framework for canonical quantization of non-Abelian gauge theories using a non-Abelian generalization of the Coulomb gauge.

Proposed method

• Define a generalized Coulomb gauge condition using a Lie-algebra-valued gauge parameter derived from solving an elliptic PDE.
• Construct the gauge transformation that enforces the condition by solving a linear elliptic equation for the gauge parameter.
• Ensure the gauge condition reduces Gauss's law to a form involving only spatial components of the gauge potential.
• Use the properties of compact semi-simple Lie groups to ensure the elliptic operator is self-adjoint and has a unique solution.
• Demonstrate that the gauge-fixing condition is globally well-defined and free from Gribov copies due to the elliptic nature of the equation.
• Apply functional analytic methods to prove existence and uniqueness of the gauge-fixing solution in the appropriate function space.

Experimental results

Research questions

• RQ1 Can the Abelian Coulomb gauge condition be consistently generalized to non-Abelian Yang-Mills theories with arbitrary compact semi-simple gauge groups?
• RQ2 Does the generalized Coulomb gauge condition eliminate time derivatives from Gauss's law, enabling a canonical formulation?
• RQ3 Is the gauge-fixing condition globally well-defined, and does it avoid the Gribov ambiguity?
• RQ4 What type of differential equation governs the gauge transformation needed to reach the generalized Coulomb gauge?
• RQ5 Can the existence and uniqueness of the gauge condition be rigorously proven using elliptic PDE theory?

Key findings

• A consistent generalization of the Coulomb gauge to non-Abelian Yang-Mills theories is achieved via a Lie-algebra-valued gauge parameter.
• The generalized gauge condition eliminates time derivatives from Gauss's law, resulting in a purely spatial constraint.
• The existence and uniqueness of the gauge are guaranteed by solving a well-posed elliptic linear PDE for the gauge parameter.
• The Gribov problem does not arise in this formulation because the gauge-fixing condition leads to a unique solution of the elliptic equation.
• The method applies to any compact semi-simple gauge group, making it broadly applicable to standard non-Abelian gauge theories.
• The construction provides a solid foundation for canonical quantization of Yang-Mills theories without Gribov ambiguities.