Ehrenfest theorems and charge antiscreening in Abelian projected gauge theories

We derive exact relations for SU(2) lattice gauge theory in 3+1 dimensions. In terms of Abelian projection, these are the expectation values of Maxwell equations that define a new field strength operator and conserved, dynamic electric currents formed from the charged matter and ghost fields. The effect of gauge fixing is calculated, and in the maximally Abelian gauge we find antiscreening of U(1) Wilson loop source charges. We discuss the importance of these quantities in the dual superconducting vacuum mechanism of confinement.