Computational Physics

15.7 Gauge field theories 515

The total action

SLQED=

∑

n;μν

Splaquette(n;μν) (15.125)

occurs in the exponent of the time-evolution operator or of the Boltzmann factor
(for field theory in imaginary time). The partition function of the Euclidean field
theory is

ZLQED(β)=

∫ 2 π

0

∏

n,μ

dθμ(n)exp(−βSLQED), (15.126)

where the product

∏

n,μis over all the links of the lattice. For low temperature
(largeβ), only values ofθclose to 0 (mod 2π) will contribute significantly to
the integral. Expanding the cosine for small angles, we can extend the integrals to
the entire real axis and obtain

SLQED(βlarge)=

∑

n,μ,ν

1

2

(

◦

∑

μν

θ(n)

) 2

. (15.127)

Usingθμ=eaAμand the fact that the lattice constantais small, we see that the
action can be rewritten as

βSLQED≈

β

4

∫

d^4 x

a^4

[a^4 e^2 Fμν(x)Fμν(x)], (15.128)

where now the summation is overallμν, whereas in the sums above (over the
plaquettes),μandνwere restricted to positive directions. TheFμνare defined in
Eq. (15.107b). Takingβ= 1 /e^2 we recover the Maxwell Lagrangian:

βSLQED≈

1

4

∫

d^4 xFμνFμν (15.129)

in the continuum limit.
What are interesting objects to study? Physical quantities are gauge-invariant,
so we search for gauge-invariant correlation functions. Gauge invariance can be
formulated in the lattice model as an invariance under a transformation defined by
a lattice functionχ(n)which induces a shift in theθμ(n):

θμ(n)→θμ(n)+

χ(n+μ)−χ(n)

a

. (15.130)

This suggests that gauge-invariant correlation functions are defined in terms of a
sum overθμover a closed path: in that case a gauge transformation does not induce
a change in the correlation function since the sum over the finite differences of the
gauge functionχ(n)over the path will always vanish. Furthermore, as correlation
functions usually contain products of variables at different sites, we consider the