Gauge fields on a lattice

Abstract

The modern formulation of non-Abelian lattice gauge theories is due to Wilson [Wil74]. Independently, gauge theories were discussed on a lattice by Wegner [Weg71] as a gauge-invariant extension of the Ising model and in an unpublished work by A. Polyakov in 1974 which deals mostly with Abelian theories. Placing gauge fields on a lattice provides, first, a nonperturbative regularization of ultraviolet divergences. Secondly, the lattice formulation of QCD possesses some nonperturbative terms in addition to perturbation theory. A result of this is that one has a nontrivial definition of QCD beyond perturbation theory which guarantees confinement of quarks. The lattice formulation of gauge theories deals with phase-factor-like quantities, which are elements of the gauge group, and are natural variables for quantum gauge theories. The gauge group on the lattice is therefore compact, offering the possibility of nonperturbative quantization of gauge theories without fixing the gauge. The lattice quantization of gauge theories is performed in such a way as to preserve the compactness of the gauge group. The continuum limit of lattice gauge theories is reproduced when the lattice spacing is many times smaller than the characteristic scale. This is achieved when the non-Abelian coupling constant tends to zero as it follows from the renormalization-group equation. In this chapter we consider the Euclidean formulation of lattice gauge theories. First, we introduce the lattice terminology and discuss the action of lattice gauge theory at the classical level.