15.7 Gauge field theories 517Zis the vacuum partition function. Note that the fact thatTis taken large guarantees
that the ground state of the Hamiltonian is projected out in the simulation.
By varying the couplingβ, different results for the value of the Wilson loop
correlation function are found. For large loops we have either the ‘area law’:
W(C)=exp(Const.×Area within loop), (15.134)or the ‘perimeter law’ :
W(C)=exp(Const.×Perimeter of loop), (15.135)with additional short-range corrections. Let us consider the area law. In that case we
findV(R)∝R, which means that the two particles cannot be separated: pulling them
infinitely far apart requires an infinite amount of energy. We say that the particles
areconfined. On the other hand, the perimeter law says thatV(R)is a constant (it
is dominated by the vertical parts of lengthT), up to corrections decaying to zero
for largeR(for the confined case, these corrections can be neglected). Ford= 4
one finds after working out the dominant correction termV(R)∝−(e^2 /R), i.e.
Coulomb’s law [42]. We see that the lattice gauge theory incorporates two different
kinds of gauge interactions: confined particles and electrodynamics. The analysis
in which the fermions are kept at fixed positions corresponds to the fermions having
an infinite mass. It is also possible to allow for motion of the fermions by allowing
the loops of arbitrary shape, introducing gamma-matrices in the resulting action.
The procedure in which the fermions are kept fixed is called ‘quenched QED’ – in
quenched QED, vacuum polarisation effects (caused by the fact that photons can
create electron–positron pairs) are not included.
We know that electrodynamics does not confine electrons: the lattice gauge theory
in four dimensions has two phases, a low-temperature phase in which the interac-
tions are those of electrodynamics, and a high-temperature phase in which the
particles are confined [44] (‘temperature’ is inversely proportional to the coupling
constantβ). The continuum limit of electrodynamics is described by the low-
temperature phase of the theory. Why have people been interested in putting QED
on a grid? After all, perturbation theory works very well for QED, and the lattice
theory gives us an extra phase which does not correspond to reality (for QED). The
motivation for studying lattice gauge theories was precisely this latter phase: we
know that quarks, the particles that are believed to be the constituents of mesons
and hadrons, are confined: an isolated quark has never been observed. Lattice gauge
theory provides a mechanism for confinement! Does this mean that quarks are part
of the same gauge theory as QED, but corresponding to the high-temperature phase?
No: there are reasons to assume that a quark theory has a more complex structure
than QED, and moreover, experiment has shown that the interaction between quarks
vanishes when they come close together, in sharp contrast with the confining phase