Gribov Problem for Gauge Theories: a Pedagogical Introduction

TL;DR

This paper provides a pedagogical overview of the Gribov problem in non-Abelian gauge theories, explaining how gauge ambiguities arise in the functional integral quantization and discussing recent developments and examples.

Contribution

It offers a clear, accessible introduction to the Gribov problem, connecting differential geometry, gauge theory, and recent computational approaches.

Findings

Illustration of Gribov copies in SU(2) gauge theories

Discussion of Gribov ambiguity in general relativity

Overview of recent lattice calculations addressing the problem

Abstract

The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a non-linear differential equation, and the various solutions of such a non-linear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.