Gribov copies

In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold (of the bundle as a whole) and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish (as in the Landau or Lorenz gauge). The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity (named after Vladimir Gribov).

Gribov ambiguities lead to a nonperturbative failure of the BRST symmetry, among other things.

A way to resolve the problem of Gribov ambiguity is to restrict the relevant functional integrals to a single Gribov region whose boundary is called a Gribov horizon. Still one can show that this problem is not resolved even when reducing the region to the first Gribov region. The only region for which this ambiguity is resolved is the fundamental modular region (FMR).

When doing computations in gauge theories, one usually needs to choose a gauge. Gauge degrees of freedom do not have any direct physical meaning, but they are an artifact of the mathematical description we use to handle the theory in question. In order to obtain physical results, these redundant degrees of freedom need to be discarded in a suitable way

In Abelian gauge theory (i.e. in QED) it suffices to simply choose a gauge. A popular one is the Lorenz gauge ${\displaystyle \partial _{\mu }A_{\mu }=0}$, which has the advantage of being Lorentz invariant. In non-Abelian gauge theories (such as QCD) the situation is more complicated due to the more complex structure of the non-Abelian gauge group.

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