# Gribov's phenomenon

In the well known textbook by Itzykson-Zuber "Quantum Field Theory" there is a discussion of the Gribov phenomenon in non-abelian gauge theories (see Section 12-2-1). To my taste, the discussion is too concise as well as the original Gribov's article from 1978 (see Section 3 there). I would be interested to get some extra explanations or a reference to more detailed exposition.

The Gribov phenomenon says that in non-abelian gauge theories the standard gauge fixing functionals (such as the Coulomb gauge $$div \vec A=0$$) have the unpleasant property that they may vanish on two distinct gauge equivalent fields.

That might happen even if the two fields are infinitesimally close. To illustrate this, Itzykson and Zuber consider time-independent field $$A^\mu$$ (taking values in a non-abelian Lie algebra) in the Coulomb gauge. An infinitesimal gauge transformation $$\delta\alpha$$ maps $$A^\mu$$ to $$B^\mu=A^\mu+\partial^\mu\delta\alpha+ [\delta\alpha,A^\mu]$$, where $$[\cdot,\cdot]$$ is the Lie bracket. Let this gauge transformation be time independent and vanishing at spatial infinity. The fact that $$B^\mu$$ also satisfies the Coulomb gauge is equivalent to $$\partial_i(\partial^i\delta\alpha+[\delta\alpha,A^i])=0.$$

The claim is that such solutions always exist with fast decay in spatial infinity. They consider the operator $$\alpha\mapsto \partial_i(\partial^i\alpha+[\alpha,A^i])$$. The authors give perturbative argument that for very small $$A^\mu$$ this operator has no bound states with positive energy $$E>0$$. Then they claim that for large $$A^\mu$$ the operator does have bound states with $$E>0$$. They conclude that for some intermediate $$A^\mu$$ there must exist a zero-energy solution. All these steps are not very clear to me.