3 - Gauge fields on a four-dimensional euclidean lattice

Summary

Lattice formulation, local gauge invariance, and the continuum action

In its standard formulation, lattice-gauge theory replaces the continuum spacetime of Euclidean quantum-field theory with a four-dimensional lattice of spacetime points and links. If the lattice is hypercubic, then its lattice spacing a provides an ultraviolet cutoff. Although such a spacetime cutoff is not convenient in most analytic calculations, it has several extraordinary advantages. First, it allows strong-coupling calculations both through expansions and by simulation methods. These approaches provide a framework for formulating and developing physical pictures of confinement, chiral-symmetry breaking, and other nonperturbative phenomena that are so challenging in ordinary continuum formulations of quantum-field Theory.

The various fields of the lattice version of quantum chromodynamics exist on the sites and links of the regular lattice. In the standard formulation, the fermion fields reside on the sites and the gauge fields reside on the links [10]. The fermion fields, matter fields in general, carry the color quantum number of the gauge group SU(3). It is convenient, therefore, to imagine that there is a color frame of reference at each site so that the fermion field with color index α, ψ_α, can be visualized. (This geometrical point of view dates back to the original Yang–Mills paper which introduced non-Abelian gauge fields into high-energy physics when they generalized the notion of isospin symmetry from the familiar global symmetry of nuclear physics to a local symmetry of high-energy field theory [11].)