152
16. Lattice Field Theory, and the Renormalization Group Revisited
bation theory. As Wilson (1974, 1975) was the first to propose, one quite
natural non-perturbative way of regulating ultraviolet divergences is to approximate continuous space-time by a discrete lattice of points. Such a lattice
will introduce a minimum distance – namely the lattice spacing ‘a’ between
neighbouring points. Since no two points can ever be closer than a, there is
now a corresponding maximum momentum Λ = π/a (see following equation
(16.6)) in the lattice version of the theory. Thus the theory is automatically
ultraviolet finite from the start, without presupposing the existence of any
perturbative expansion; renormalization questions will, however, enter when
we consider the a dependence of our parameters. As long as the lattice spacing is much smaller than the physical size of the hadrons one is studying,
the lattice version of the theory should be a good approximation. Of course,
Lorentz invariance is sacrificed in such an approach, and replaced by some
form of hypercubic symmetry; we must hope that for small enough a this will
not matter. We shall discuss how simple field theories are ‘discretized’ in the
next section; scalar fields, fermion fields, and gauge fields each require their
own prescriptions.
Next, we must ask how a discretized quantum field theory can be formulated in a way suitable for numerical computation. Any formalism based on
non-commuting operators seems to be ruled out, since it is hard to see how
they could be numerically simulated. Indeed, the same would be true of ordinary quantum mechanics. Fortunately a formulation does exist which avoids
operators: Feynman’s sum over paths approach, which was briefly mentioned
in section 5.2.2. This method is the essential starting point for the lattice approach to quantum field theory, and it will be introduced in section 16.3. The
sum over paths approach does not involve quantum operators, but fermions
still have to be accommodated somehow. The way this is done is briefly
described in section 16.3: see also appendix P.
It turns out that this formulation enables direct contact to be made between quantum field theory and statistical mechanics, as we shall discuss in
section 16.3.3. This relationship has proved to be extremely fruitful, allowing
physical insights and numerical techniques to pass from one subject to the
other, in a way that has been very beneficial to both. In section 16.4 we make
a worthwhile detour to explore the physics of renormalization and of the RGE
from a lattice/statistical mechanics perspective, before returning to QCD in
section 16.5.
16.2 Discretization
16.2.1 Scalar fields
We start by considering a simple field theory involving a scalar field φ. Postponing until section 16.3 the question of exactly how we shall use it, we assume
16. Lattice Field Theory, and the Renormalization Group Revisited
bation theory. As Wilson (1974, 1975) was the first to propose, one quite
natural non-perturbative way of regulating ultraviolet divergences is to approximate continuous space-time by a discrete lattice of points. Such a lattice
will introduce a minimum distance – namely the lattice spacing ‘a’ between
neighbouring points. Since no two points can ever be closer than a, there is
now a corresponding maximum momentum Λ = π/a (see following equation
(16.6)) in the lattice version of the theory. Thus the theory is automatically
ultraviolet finite from the start, without presupposing the existence of any
perturbative expansion; renormalization questions will, however, enter when
we consider the a dependence of our parameters. As long as the lattice spacing is much smaller than the physical size of the hadrons one is studying,
the lattice version of the theory should be a good approximation. Of course,
Lorentz invariance is sacrificed in such an approach, and replaced by some
form of hypercubic symmetry; we must hope that for small enough a this will
not matter. We shall discuss how simple field theories are ‘discretized’ in the
next section; scalar fields, fermion fields, and gauge fields each require their
own prescriptions.
Next, we must ask how a discretized quantum field theory can be formulated in a way suitable for numerical computation. Any formalism based on
non-commuting operators seems to be ruled out, since it is hard to see how
they could be numerically simulated. Indeed, the same would be true of ordinary quantum mechanics. Fortunately a formulation does exist which avoids
operators: Feynman’s sum over paths approach, which was briefly mentioned
in section 5.2.2. This method is the essential starting point for the lattice approach to quantum field theory, and it will be introduced in section 16.3. The
sum over paths approach does not involve quantum operators, but fermions
still have to be accommodated somehow. The way this is done is briefly
described in section 16.3: see also appendix P.
It turns out that this formulation enables direct contact to be made between quantum field theory and statistical mechanics, as we shall discuss in
section 16.3.3. This relationship has proved to be extremely fruitful, allowing
physical insights and numerical techniques to pass from one subject to the
other, in a way that has been very beneficial to both. In section 16.4 we make
a worthwhile detour to explore the physics of renormalization and of the RGE
from a lattice/statistical mechanics perspective, before returning to QCD in
section 16.5.
16.2 Discretization
16.2.1 Scalar fields
We start by considering a simple field theory involving a scalar field φ. Postponing until section 16.3 the question of exactly how we shall use it, we assume