Please refer to the following article:
Quantum Mechanics of Gauge Fixing
Lenz F., Naus H. W. L., Ohta K. and Thies M.
Annals of Physics
Volume 233, Issue 1, July 1994, Pages 17-50
Abstract:
In the framework of the canonical Weyl gauge formulation of QED, the quantum mechanics of gauge fixing is discussed. Redundant quantum mechanical variables are eliminated by means of unitary transformations and Gauss′s law. This results in representations of the Weyl-gauge Hamiltonian which contain only unconstrained variables. As a remnant of the original local gauge invariance global residual symmetries may persist. In order to identify these and to handle infrared problems and related "Gribov ambiguities," it is essential to compactify the configuration space. Coulomb, axial, and light-cone representation of QED are derived. The naive light-cone approach is put into perspective. Finally, the Abelian Higgs model is studied; the unitary gauge representation of this model is derived and implications concerning the symmetry of the Higgs phase are discussed.
They construct a Hamiltonian (entirely in terms of physical fields) which contains a "standard Coulomb term". So for problems which are related to electrostatics this gauge (either Coulomb or axial gauge on top of A°=0) is a good starting point.
\hat{V}_C = e^2 \int d^3x\,d^3y\,\frac{\rho(x)\,\rho(y)}{|x-y|}
The charge density in the numerator is just the 0th component of the four-vector current density and looks like
\rho = \bar{\psi}\gamma^0\psi
Of course this is not the only and term in H and of course the situation is never totally static; but the static apprixomation could be a good starting point.
