514 Computational methods for lattice field theories
nn+n+ n++Figure 15.6. A lattice plaquette at sitenwith sidesμandνused in(15.122).calculations, so it is assumed throughout this section). This is less straightforward
than in the scalar field case as a result of the greater complexity of the QED theory.
We work in a space-time dimensiond=4. We first consider the discretisation of
the photon gauge field and describe the inclusion of fermions below. An important
requirement is that the gauge invariance should remain intact. Historically, Wegner’s
Ising lattice gauge theory[40]showed the way to the discretisation of continuum
gauge theories. We now describe the lattice formulation for QED which was first
given by Wilson[41], and then show that the continuum limit for strong coupling
is the conventional electromagnetic gauge theory.
We introduce the following objects, living on thelinksμof a square lattice with
sites denoted byn:
Uμ(n)=exp[ieaAμ(n)]=exp[iθμ(n)] (15.121)where we have defined the dimensionless scalar variablesθμ=eaAμ. The action
on the lattice is then written as a sum over all plaquettes, where each plaquette
carries an action (see Figure 15.6):
Splaquette(n;μν)=Re[ 1 −Uμ(n)Uν(n+μ)
×Uμ∗(n+μ+ν)Uν∗(n+ν)]. (15.122)Note that the effect of complex conjugation is a sign-reversal of the variableθμ. The
Us are orientation-dependent:Uμ(n)=U−†μ(n). The constant 1 has been included
in (15.122) to ensure that the total weight of a configuration with allθ-values being
equal to zero vanishes. Note that the integration overθis over a range 2π,soit
does not diverge, in contrast to the original formulation, where the gauge must be
fixed in order to prevent the path integral on a finite lattice from becoming infinite
(see also the remark after Eq. (15.111)). The plaquette action can also be written as
Splaquette(n;μν)=◦
∑
μν[ 1 −cos(θμν(n))] (15.123)where the argument of the cosine is the sum over theθ-variables around the plaquette
as inFigure 15.6:
θμν(n)=θμ(n)+θν(n+μ)−θμ(n+μ+ν)−θμ(n+ν). (15.124)