Gribov Ambiguity

TL;DR

This paper reviews the Gribov ambiguity in non-Abelian gauge theories, discussing how restricting to the Gribov region modifies the gluon propagator and leads to gluon confinement, highlighting fundamental issues in gauge fixing.

Contribution

It provides a comprehensive review of the Gribov problem, the restriction to the Gribov region, and the resulting implications for gluon confinement and positivity violation in quantum field theory.

Findings

Restriction to the Gribov region modifies the gluon propagator.

Gluon confinement is indicated by violation of reflection positivity.

The Gribov ambiguity affects the unitarity of gauge theories.

Abstract

Gribov ambiguity is a problem that arises when we try to single out the physical gauge degree of freedom in non-Abelian gauge theory by imposing the covariant gauge constraint. Unfortunately, the solution of the gauge constraint is not unique, thus the redundant gauge degree of freedom, called Gribov copies, remains unfixed. One of the traditional methods to partially resolve the Gribov problem is to restrict the space of gauge orbits inside the bounded region known as the Gribov region. The meaning of partially resolve is that this procedure can solve only the positivity's problem of the Faddeev-Popov operator but the Gribov copies are still there. However, on the bright side, the restriction to the Gribov region leads to the modification of the gluon propagator. Additionally, the new form of the gluon propagator yields the violation of the reflection positivity which is considered as the important axiom of the Euclidean quantum field theory. This shows that the gluon field in the Gribov region is an unphysical particle or technically confined. In this review article, we will start by discussing the traditional Faddeev-Popov method and its consequence on the proof of the unitarity of the perturbative Yang-Mills theory. Next, we will discuss the blind spot of the Faddeev-Popov quantization and study the mathematical and physical origin of the Gribov problem. Then, the method of the Gribov restriction will be elaborated. After that, we demonstrate the modification of the gluon field after restricting inside the bounded Gribov region. Finally, we show that the new form of the gluon leads to the violation of the reflection positivity axiom.