Overview

So far the discussion of lattice gauge theory has been based on a gauge-invariant treatment of the nonlinear boson (gluon) couplings in a Yang-Mills nonabelian local gauge theory. In this chapter we discuss the extension of LGT to include fermions (quarks). This chapter is based on [Wi74, Ko75, Ba76a, Su77, Wi77, Dr76, Dr76a, Ko83, Dm89, La03] much of it is taken from [Dm89].

The free Dirac lagrangian density is given by

(36.1)

The second line is an equivalent form to be used in the action where a partial integration can be carried out.

In QED, a theory with U(1) local gauge invariance, the lagrangian density takes the form

(36.2)

In the euclidian metric of LGT one calculates the action

(36.3)

Here the four-vectors (x , A ) are taken as

(36.4)

In analogy, for the Dirac gamma matrices in the euclidian metric we will make the replacement

(36.5)

Thus in the following we shall use

(36.6)

36.1 Fermions in U(1) Lattice Gauge Theory

Associate fermion fields with each site as indicated in Fig. 36.1.

Fig. 36.1: Associate fermion fields with each site.

These fermion fields are taken to be Grassmann variables, that is, they are anticommuting c-numbers (see [Se86, Wa92] and Probs. 36.1 4).

Recall from chapter 30 that

(36.7)

Here the second ? _links/site goes over the positive coordinate directions at each site (Fig. 36.2).

Fig. 36.2: Positive coordinate directions at each site.

This expression gives a complete enumeration of the terms...