16.3. Representation of quantum amplitudes
171
16.3.3
Connection with statistical mechanics
Not the least advantage of the path integral formulation of quantum field
theory (especially in its lattice form) is that it enables a highly suggestive
connection to be set up between quantum field theory and statistical mechanics. We introduce this connection now, by way of a preliminary to the
discussion of renormalization in the following section.
The connection is made via the fundamental quantity of equilibrium statistical mechanics, the partition function Z defined by
(
)
∑
H
Z =
exp −
,
(16.96)
k B T
configurations
which is simply the ‘sum over states’ (or configurations) of the relevant degrees of freedom, with the Boltzmann weighting factor. H is the classical
Hamiltonian evaluated for each configuration. Consider, for comparison, the
denominator in (16.87), namely
∫
Z φ = Dφ exp(−S E ) ,
(16.97)
where
∫
∫
(
)
S E = d
4 x E L E = d
4
1
1
1
x
(
2
E
∂ τ φ)
2 + (
2
∇φ) + m
2 φ
2 + λφ
4
(16.98)
2
2
in the case of a single scalar field with mass m and self-interaction λφ
4 . The
Euclideanized Lagrangian density L E is like an energy density: it is bounded
from below, and increases when the field has large magnitude or has large
gradients in τ or x. The factor exp(−S E ) is then a sensible statistical weight
for the fluctuations in φ, and Z φ may be interpreted as the partition function
for a system described by the field degree of freedom φ, but of course in four
‘spatial’ dimensions.
The parallel becomes perhaps even stronger when we discretize space-time.
In an Ising model (see the following section), the Hamiltonian has the form
∑
H = −J
s n s n+1 ,
(16.99)
n
where J is a constant, and the sum is over lattice sites n, the system variables
taking the values ±1. When (16.99) is inserted into (16.96), we arrive at
something very reminiscent of the φ(n 1 )φ(n 1 + 1) term in (16.6). Naturally,
the effective ‘Hamiltonian’ is not quite the same – though we may note that
Wilson (1971b) argued that in the case of a φ
4 interaction the parameters can
be chosen so as to make the values φ = ±1 the most heavily weighted in S E .
Statistical mechanics does, of course, deal in three spatial dimensions, not
the four of our Euclideanized space-time. Nevertheless, it is remarkable that
171
16.3.3
Connection with statistical mechanics
Not the least advantage of the path integral formulation of quantum field
theory (especially in its lattice form) is that it enables a highly suggestive
connection to be set up between quantum field theory and statistical mechanics. We introduce this connection now, by way of a preliminary to the
discussion of renormalization in the following section.
The connection is made via the fundamental quantity of equilibrium statistical mechanics, the partition function Z defined by
(
)
∑
H
Z =
exp −
,
(16.96)
k B T
configurations
which is simply the ‘sum over states’ (or configurations) of the relevant degrees of freedom, with the Boltzmann weighting factor. H is the classical
Hamiltonian evaluated for each configuration. Consider, for comparison, the
denominator in (16.87), namely
∫
Z φ = Dφ exp(−S E ) ,
(16.97)
where
∫
∫
(
)
S E = d
4 x E L E = d
4
1
1
1
x
(
2
E
∂ τ φ)
2 + (
2
∇φ) + m
2 φ
2 + λφ
4
(16.98)
2
2
in the case of a single scalar field with mass m and self-interaction λφ
4 . The
Euclideanized Lagrangian density L E is like an energy density: it is bounded
from below, and increases when the field has large magnitude or has large
gradients in τ or x. The factor exp(−S E ) is then a sensible statistical weight
for the fluctuations in φ, and Z φ may be interpreted as the partition function
for a system described by the field degree of freedom φ, but of course in four
‘spatial’ dimensions.
The parallel becomes perhaps even stronger when we discretize space-time.
In an Ising model (see the following section), the Hamiltonian has the form
∑
H = −J
s n s n+1 ,
(16.99)
n
where J is a constant, and the sum is over lattice sites n, the system variables
taking the values ±1. When (16.99) is inserted into (16.96), we arrive at
something very reminiscent of the φ(n 1 )φ(n 1 + 1) term in (16.6). Naturally,
the effective ‘Hamiltonian’ is not quite the same – though we may note that
Wilson (1971b) argued that in the case of a φ
4 interaction the parameters can
be chosen so as to make the values φ = ±1 the most heavily weighted in S E .
Statistical mechanics does, of course, deal in three spatial dimensions, not
the four of our Euclideanized space-time. Nevertheless, it is remarkable that