172
16. Lattice Field Theory, and the Renormalization Group Revisited
quantum field theory in three spatial dimensions appears to have such a close
relationship to equilibrium statistical mechanics in four spatial dimensions.
One insight we may draw from this connection is that, in the case of pure
gauge actions (16.47) or (16.48), the gauge coupling is seen to be analogous
to an inverse temperature, by comparison with (16.96). One is led to wonder
whether something like transitions between different ‘phases’ exist, as coupling
constants (or other parameters) vary – and, indeed, such changes of ‘phase’
can occur.
A second point is somewhat related to this. In statistical mechanics, an
important quantity is the correlation length ξ, which for a spin system may
be defined via the spin-spin correlation function
∑
G(x) = ‹s(x)s(0)› =
s(x)s(0)e
−H/kBT ,
(16.100)
all s(x)
where we are once more reverting to a continuous x variable. For large |x|,
this takes the form
(
)
1
x
G(x)
exp
−| |
∝
.
(16.101)
|x|
ξ(T )
The Fourier transform of this (in the continuum limit) is
˜
G(k
2
(
) ∝ k
2
)
+ ξ
−
1
2 (T )
− ,
(16.102)
as we learned in section 1.3.3. Comparing (16.100) with (16.87), it is clear
that (16.100) is proportional to the propagator (or Green function) for the field
s(x); (16.102) then shows that ξ
−1 (T ) is playing the role of a mass term m.
Now, near a critical point for a statistical system, correlations exist over very
large scales ξ compared to the inter-atomic spacing a; in fact, at the critical
point ξ(T c ) ∼ L, where L is the size of the system. In the quantum field
theory, as indicated earlier, we may regard a
−1 as playing a role analogous to
a momentum cut-off Λ, so the regime ξ ≫ a is equivalent to m ≪ Λ, as was
indeed always our assumption. Thus studying a quantum field theory this way
is analogous to studying a four-dimensional statistical system near a critical
point. This shows rather clearly why it is not going to be easy: correlations
over all scales will have to be included. At this point, we are naturally led to
the consideration of renormalization in the lattice formulation.
16.4 Renormalization, and the renormalization group,
on the lattice
16.4.1 Introduction
In the continuum formulation which we have used elsewhere in this book,
fluctuations over short distances of order Λ
−1 generally lead to divergences
16. Lattice Field Theory, and the Renormalization Group Revisited
quantum field theory in three spatial dimensions appears to have such a close
relationship to equilibrium statistical mechanics in four spatial dimensions.
One insight we may draw from this connection is that, in the case of pure
gauge actions (16.47) or (16.48), the gauge coupling is seen to be analogous
to an inverse temperature, by comparison with (16.96). One is led to wonder
whether something like transitions between different ‘phases’ exist, as coupling
constants (or other parameters) vary – and, indeed, such changes of ‘phase’
can occur.
A second point is somewhat related to this. In statistical mechanics, an
important quantity is the correlation length ξ, which for a spin system may
be defined via the spin-spin correlation function
∑
G(x) = ‹s(x)s(0)› =
s(x)s(0)e
−H/kBT ,
(16.100)
all s(x)
where we are once more reverting to a continuous x variable. For large |x|,
this takes the form
(
)
1
x
G(x)
exp
−| |
∝
.
(16.101)
|x|
ξ(T )
The Fourier transform of this (in the continuum limit) is
˜
G(k
2
(
) ∝ k
2
)
+ ξ
−
1
2 (T )
− ,
(16.102)
as we learned in section 1.3.3. Comparing (16.100) with (16.87), it is clear
that (16.100) is proportional to the propagator (or Green function) for the field
s(x); (16.102) then shows that ξ
−1 (T ) is playing the role of a mass term m.
Now, near a critical point for a statistical system, correlations exist over very
large scales ξ compared to the inter-atomic spacing a; in fact, at the critical
point ξ(T c ) ∼ L, where L is the size of the system. In the quantum field
theory, as indicated earlier, we may regard a
−1 as playing a role analogous to
a momentum cut-off Λ, so the regime ξ ≫ a is equivalent to m ≪ Λ, as was
indeed always our assumption. Thus studying a quantum field theory this way
is analogous to studying a four-dimensional statistical system near a critical
point. This shows rather clearly why it is not going to be easy: correlations
over all scales will have to be included. At this point, we are naturally led to
the consideration of renormalization in the lattice formulation.
16.4 Renormalization, and the renormalization group,
on the lattice
16.4.1 Introduction
In the continuum formulation which we have used elsewhere in this book,
fluctuations over short distances of order Λ
−1 generally lead to divergences