Overview

We turn next to a development of lattice gauge theory for Yang-Mills nonabelian gauge groups. SU(2) is considered as a specific example since in this case it is relatively simple to deal explicitly with all of the required matrices. The discussion is based on [Wi74, Cr83, Re83, Kh89].

31.1 Internal Space

One now has an additional internal space for each field variable. The quantity A is a vector in this internal isospin space with three components . The situation for the basic plaquette is illustrated in Fig. 31.1. The phase angle ? _ji is also defined to be a vector in this internal space

(31.1)

Fig. 31.1: Basic plaquette and illustration of the field A as a vector in the internal isospin space in SU(2) nonabelian lattice gauge theory.

As before the subscript (ji) indicates the connected sites.

Recall that the modification of the covariant derivative required to go from the abelian QED theory to this Yang-Mills theory is

(31.2)

In contrast to the simple phases of the abelian theory of QED, we are thus motivated to introduce the link variables as 2 2 SU(2) matrices

(31.3)

Here the internal matrix structure is again denoted by a bar under the symbol. Substitution of Eq. (31.1) leads to

(31.4)

The contribution to the action is defined to be the trace (tr) of the matrix product in this internal space taken around a plaquette (Fig. 31.1)

(31.5)

The ordering of the matrices...