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.
Gribov ambiguities lead to a nonperturbative failure of the BRST symmetry, among other things.
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.
The Faddeev–Popov formalism, developed by Ludvig Faddeev and Victor Popov, provides a way to deal with the gauge choice in non-Abelian theories.
The determinant of this Faddeev–Popov operator is then introduced into the path integral using ghost fields.
[note 2] If both fields obey the Landau gauge condition, we must have that and thus that the Faddeev–Popov operator has at least one zero mode.
[5] If the gauge field is infinitesimally small, this operator will not have zero modes.
The set of gauge fields where the Faddeev–Popov operator has its first zero mode (when starting from the origin) is called the "Gribov horizon".
In order to do so, he considered the ghost propagator, which is the vacuum expectation value of the inverse of the Faddeev–Popov operator.
If this operator is always positive definite, the ghost propagator cannot have poles — which is called the "no-pole condition".
[7] Deriving a perturbative expression for the ghost propagator, Gribov finds that this no-pole condition leads to a condition of the form[7][8] with N the number of colors (which is 3 in QCD), g the gauge coupling strength, V the volume of space-time (which goes to infinity in most applications), and d the number of space-time dimensions (which is 4 in the real world).
In order to impose this condition, Gribov proposed to introduce a Heaviside step function containing the above into the path integral.
The Fourier representation of the Heaviside function is: In this expression, the parameter
The integration over this Gribov parameter is then performed using the method of steepest descent.
Plugging the solution to this equation back into the path integral yields a modified gauge theory.
With the modification stemming from the Gribov parameter, it turns out that the gluon propagator is modified to[7][9] where
The ghost propagator is also modified and, at one-loop order, displays a behavior
[10] Several years later, Daniel Zwanziger also considered the Gribov problem.
Instead of considering the ghost propagator, he computed the lowest eigenvalue of the Faddeev–Popov operator as a perturbative series in the gluon field.
[11] This condition can be expressed by introducing the horizon function into the path integral (in a way analogous to how Gribov did the same) and imposing a certain gap equation on the vacuum energy of the resulting theory.
[12] This yielded a new path integral with a modified action, which is, however, nonlocal.
At leading order, the results are identical to the ones previously found by Gribov.
In order to more easily deal with the action he found, Zwanziger introduced localizing fields.
Once the action had become local, it was possible to prove that the resulting theory is renormalizable[13] — i.e. all infinities generated by loop diagrams can be absorbed by multiplicatively modifying the content (coupling constant, field normalization, Gribov parameter) already present in the theory without needing extra additions.
[16] Furthermore, the first Gribov region is convex, and all physical configurations have at least one representative inside it.
[17] In 2013 it was proven that the two formalisms — Gribov's and Zwanziger's — are equivalent to all orders in perturbation theory.
For a long time, lattice simulations seemed to indicate that the modified gluon and ghost propagators proposed by Gribov and Zwanziger were correct.
In 2007, however, computers had become sufficiently strong to probe the region of low momenta, where the propagators are most modified, and it turned out that the Gribov–Zwanziger picture is not correct.
[22] A solution to this discrepancy has been proposed, adding condensates to the Gribov–Zwanziger action.