Lattice formulations of quantum field theory (QFT) are not unique. Differential structures in the continuum theory have many counterparts on lattices which, in the limit \(a \to 0\) of vanishing lattice spacing \(a\), smoothly reproduce these structures. The overriding criterion for a good lattice formulation is that, in this limit, the lattice theory naturally reproduces in all detail “its” continuum theory. This considerable increase in complication is the price paid to have the great advantage of finite-lattice QFT; namely well-defined, computable path integrals.
The importance of understanding the continuum limit is particularly acute for gauge theories, where gauge fixing plays a crucial role in the continuum theory. There exist many gauges. What does gauge fixing translate into on a lattice? This paper begins with the correlation functions of \(U(1)\) gauge theory (pure electromagnetism) on a 3D cubic periodic lattice (i.e. a lattice torus \(T^3_a\)) formulated using the Villain lattice action. It is shown that, as \(a \to 0\) and with appropriate vanishing of the lattice coupling, these correlation functions smoothly become the corresponding correlation functions for the continuum torus \(T^3\) in the \(R\) gauge. Elsewhere the author shows the same thing in 4D.
For the entire collection see [Zbl 0816.00038].