Computational Physics

15.4 Algorithms for lattice field theories 477
In the final sections of this chapter, simulation methods for gauge field theories
(QED, QCD) will be discussed.

15.4 Algorithms for lattice field theories

We start by reviewing the scalar Euclideanφ^4 field theory inddimensions in more
detail. The continuum action is

SE=

1

2

∫

ddx[∂μφ(x)∂μφ(x)+m^2 φ^2 (x)+gφ^4 (x)] (15.41)

(the subscript E stands for Euclidean). Forg=0, we have the free field theory,
describing noninteracting spinless bosons. Performing a partial integration using
Green’s first identity, and assuming vanishing fields at infinity, we can rewrite the
action as

SE=

1

2

∫

ddx[−φ(x)∂μ∂μφ(x)+m^2 φ^2 (x)+gφ^4 (x)]. (15.42)

The scalar field theory can be formulated on a lattice by replacing derivatives
by finite differences. We can eliminate the dependence of the lattice action on the
lattice constant by rescaling the field, mass and coupling constant according to

φˆn=ad/^2 −^1 φ(an); mˆ=am; gˆ=a^4 −dg. (15.43)

For the four-dimensional case,d =4, we have already given these relations in
the previous section. Later we shall concentrate on the two-dimensional case, for
which the fieldφis dimensionless. In terms of the rescaled quantities, the lattice
action reads:

SELattice=

1

2

∑

n

[

−

∑

μ

φˆnφˆn+μ+( 2 d+ˆm^2 )φˆ^2 n+ˆgφˆn^4

]

. (15.44)

The argumentsnare vectors inddimensions with integer coefficients and the sum
overμis over all positive and negative Cartesian directions. The action (15.44)
is the form which we shall use throughout this section and it will henceforth be
denoted asS. From now on we shall omit the carets from the quantities occurring in
the action(15.44). As we shall simulate the field theory in the computer, we must
make the lattice finite – the linear size isL.
We now describe the analytical solution of this lattice field theory for the case
g=0 (free field theory). The free field theory action is quadratic and can be written
in the form

SE=

1

2

∑

nl

φnKnlφl, (15.45)