Gribov Ambiguities
Prerequisites: Gauge Fixing. Full path (15 pages)
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
Intuitive
In a gauge theory many different field configurations describe the same physical situation. They are related by gauge transformations, and the set of all configurations related to a given one in this way is called its gauge orbit. When we compute something, for example with a path integral, we want to count each physical situation once. The usual way is to impose a gauge condition, such as the Coulomb gauge (the vector potential has zero divergence), which is meant to pick out exactly one configuration from each orbit.
Geometrically, the space of all field configurations is sliced into orbits, and the gauge condition defines a surface that should cut every orbit exactly once. In electromagnetism this works. In non-abelian gauge theories, such as the theory of the strong interaction, Gribov found in 1978 that it does not: for large enough fields the Coulomb gauge surface cuts some orbits several times, so there are distinct configurations, all satisfying the gauge condition, that are gauge copies of each other. These are called Gribov copies. Near zero field, where perturbation theory lives, the surface cuts each orbit once, which is why the problem does not show up in perturbative calculations.
A simple picture is a set of circles of different sizes and a line that should cross each circle once. A straight line through the common center of concentric circles crosses each circle twice; if we try to fix this by keeping only half of the line, we have made a choice that has to be justified. Singer later showed that for gauge theories on a compactified spacetime the problem cannot be removed by choosing a better gauge condition: there is no continuous gauge condition that picks exactly one configuration from every orbit. The ambiguity therefore comes from the topology of the space of gauge fields. It matters for non-perturbative questions such as confinement, where large field configurations are important.
Concrete
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
Abstract
There is a difficulty which arises in the cornpactified case which is not present in the more general one. This involves gauge fixing and the Gribov ambiguity. ' As yet the gauge has not been fixed and there is the possibility that fixing the gauge could affect the topology of configuration space. The object of gauge fixing is to choose one member from each class of gauge-equivalent configurations so that there will be no double counting of configurations in the path integral. In the fiber-bundle picture of Dowker, ' this corresponds to finding a section in the bundle, reducing each configuration fiber to a particular configuration in the fiber. The difficulty is that Singer' has proven that there are no global gauges when space is compactified to S . That is, there is no global section of the fiber bundle that Dowker is using. This means that the space of gauge-fixed configurations is disconnected. The answer is that the Gribov ambiguity allows for certain zero-action discontinuous evolutions. Because there are no global sections, the gauge fixing fails to always specify a unique configuration from each configuration fiber. When it does fail, there are two (or more) configurations which are related by a gauge transformation. Each of these lies on a different section of the fiber bundle and one's histories may move between the disconnected sections by way of these configurations.
Nontrivial homotopy and tunneling by topological instantons by Arlen Anderson
It was proven by Singer in "Some Remarks on the Gribov Ambiguity", provided that we compactify spacetime to the 4-sphere, that "no continuous choice of exactly one connection on each orbit can be made. Thus the Gribov ambiguity for the Coulomb gauge will occur in all other gauges. No gauge fixing is possible. "
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, , for each point C in . 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 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 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.