Why is it interesting?

In electrodynamics, Coulomb gauge is a simple example of a gauge that gives a one-to-one mapping between field strength and potential. This is not the case in the non-Abelian theory, where the Gribov ambiguity shows that there are configurations of field strengths that allow several distinct choices of Coulomb gauge potentials, as well as other configurations that cannot be brought into Coulomb gauge at all [85, 236]. These issues are often swept under the rug when working in perturbation theory, but cannot be ignored when considering potentials of order 1/g, for which the commutator term in the field strength is comparable in size to the derivative terms. As suggested by Eq. (10.8), it is precisely such potentials with which we will be concerned.

page 201 in Classical Solutions in Quantum Field Theory by Erik Weinberg

Layman

Student

It was proven by Singer in "Some Remarks on the Gribov Ambiguity" that "no continuous choice of exactly one connection on each orbit can be made. Thus the Gribov ambiguity for the Coloumb gauge will occur in all other gauges. No gauge fixing is possible. "

In our discussion of gauge fixing and the Fadeev-Popov procedure for functional quantization of gauge theories, we have seen that there could be the problem of Gribov ambiguity. This refers to the fact that there could be different field configurations which obey the same gauge-fixing condition, but which are related by a gauge transformation. This means that we are unable to find a “good gauge fixing” where the gauge-fixing condition chooses one and only one representative configuration for all potentials which are gauge equivalent.

page 334 in Quantum Field Theory: A Modern Perspective by V. P. Nair

Researcher

This Gribov ambiguity can be given a precise mathematical characterization in the language of fiber bundles.

[…]

We can think of gauge fixing as follows. When we do gauge fixing, we choose a representative potential A (obeying some gauge fixing condition) for each physical configuration. Thus we are specifying the physical configuration C and a gauge transformation g associated to it which takes it into A. We have an assignment of a point on the fiber, namely, $G_C$ , for each point C in $\mathcal{C}$. In other words, gauge fixing is the choice of a section for the bundle A. If we can choose a section globally, then we have the splitting $A = \mathcal{C} × G_\star$ globally. There is no problem with gauge fixing and no Gribov ambiguity. The existence of the Gribov problem is thus equivalent to the statement that the bundle ($G_∗$ , A, \mathcal{C}$) is nontrivial and does not have a global section. This gives a precise characterization of the Gribov problem as a topological property of the bundle of gauge potentials.

page 335 in Quantum Field Theory: A Modern Perspective by V. P. Nair

What are the physical implications of Gribov ambiguities?

The BRST method used to deal with the gauge symmetry of perturbative Yang-Mills theory does not appear to generalize to the full non-perturbative theory, for a rather fundamental reason. This was first pointed out by Neuberger back in 1986 (Phys. Lett. B, 183 (1987), p337-40.), who argued that, non-perturbatively, the phenomenon of Gribov copies implies that expectation values of gauge-invariant observables will vanish.

http://www.math.columbia.edu/~woit/wordpress/?p=2876

Examples

Example1

Example2:

History