Compact QED$_3$ - a simple example of a variational calculation in a   gauge theory
  We apply a simple mean field like variational calculation to compact QED in 2+1 dimensions. Our variational ansatz explicitly preserves compact gauge invariance of the theory. We reproduce in this framework all the known results, including dynamical mass generation, Polyakov scaling and the nonzero string tension. It is hoped that this simple example can be a useful reference point for applying similar approximation techniques to nonabelian gauge theories. 
Ian I. Kogan and Alex Kovner