Overview

To obtain some insight into nonabelian lattice gauge theory, and to gain some familiarity with it, consider again mean field theory (MFT). This time we take a more systematic approach than in chapter 30 and start from the partition function for the entire many-body system, from which an analysis of its free energy will follow. The discussion is based on [Ba74, Ba75, Ba75a, Ma89a]. This chapter is taken from [Ma89a].

We first briefly repeat a summary of the results of the previous chapter. The basic plaquette is illustrated in Fig. 32.1. The partition function is obtained from a path integral over all link variables of the exponential of the action, which is obtained from the sum over all plaquettes of a term formed from the product of the link variables around the plaquette ^[1]

(32.1)

Fig. 32.1: Basic plaquette in lattice gauge theory for SU(n).

Stated in this form, the equations constitute lattice gauge theory for any internal symmetry group SU(n).

For SU(2), as we have seen, the link variables are expressed as ^[2]

(32.2)

For SU(3) the link variables are expressed as

(32.3)

An explicit representation for the SU(3) (and higher n) matrices will not be needed for the developments in this section.

The measure for SU(2) is

(32.4)

This can also be generalized to SU(n); the specific form will again not be required for the present developments.