We see that if ^ > 4 then the contribution of
is negligible in
comparison with
-h t; for ^ = 4 the correction is of relative order
log(l/T) and for ^ < 4 we have a very large powerlike correction.
Similar estimates of higher diagrams show that for Q) >
= 4 they
are irrelevant. Moreover, we see that the most singular terms arise from
the (p'^-type of interaction, and all higher powers in cp may be presumed
to be irrelevant. This is indeed the case, as will be shown later. The
detailed form of the interaction
is also irrelevant—we have seen
that it was sufficient to keep only the p^-term in the expansion of JT
All these arguments are not proofs, but they give correct guidance in
complex situations and are therefore worth mentioning.
We have arrived at the following statement. Let us consider a Q)dimensional Ising model with short ranged interaction:
^
x,x'
Take the temperature jS close to the critical one: |(jS —
^ 1* Then
all the correlation functions are the same as those for a field theory with
Lagrangian:
^
(1-25)
STATISTICAL MECHANICS AND QUANTUM FIELD THEORY
9
also defined in the :^-dimensional Euclidean space. An important point
about (1.25) is that ml must be chosen in such a way that the physical
mass mphys = x
(where A is a momentum cut-oflF). This last
condition means that we are in the critical region for (1.25). The critical
point itself corresponds to the value ml = ml at which m^^ys = 0. In
order to obtain the continuum limit (or, in other words, to renormalize
the theory) we must take the limit ml ml ^r and A^ -► oo in such a
way that m^^ys remains fixed. If this is possible, we get a rotationaly
invariant theory that does not depend on the way it has been defined in
the cut-off* region.
We also see that the theory (1.25) has two phases for ^ > 1, one with
{cp} = 0 and another, the broken symmetry phase, with {cp} # 0. In the
case ^ = 1 we have just the example we analysed in the beginning,
which is the quantum mechanics of a single particle in a potential. We
know that the ground state wave function must be even, and therefore
<(p> = 0. This conclusion is identical to the one we found when
analysing the ^ = 1 Ising model. We shall clarify this coincidence in the
chapter about instantons. How does it happen that the theory with
discrete variables (t^= ± \ appears to be equivalent to the one with a
continuum field cpl This can be understood as follows. Let us change
is negligible in
comparison with
-h t; for ^ = 4 the correction is of relative order
log(l/T) and for ^ < 4 we have a very large powerlike correction.
Similar estimates of higher diagrams show that for Q) >
= 4 they
are irrelevant. Moreover, we see that the most singular terms arise from
the (p'^-type of interaction, and all higher powers in cp may be presumed
to be irrelevant. This is indeed the case, as will be shown later. The
detailed form of the interaction
is also irrelevant—we have seen
that it was sufficient to keep only the p^-term in the expansion of JT
All these arguments are not proofs, but they give correct guidance in
complex situations and are therefore worth mentioning.
We have arrived at the following statement. Let us consider a Q)dimensional Ising model with short ranged interaction:
^
x,x'
Take the temperature jS close to the critical one: |(jS —
^ 1* Then
all the correlation functions are the same as those for a field theory with
Lagrangian:
^
(1-25)
STATISTICAL MECHANICS AND QUANTUM FIELD THEORY
9
also defined in the :^-dimensional Euclidean space. An important point
about (1.25) is that ml must be chosen in such a way that the physical
mass mphys = x
(where A is a momentum cut-oflF). This last
condition means that we are in the critical region for (1.25). The critical
point itself corresponds to the value ml = ml at which m^^ys = 0. In
order to obtain the continuum limit (or, in other words, to renormalize
the theory) we must take the limit ml ml ^r and A^ -► oo in such a
way that m^^ys remains fixed. If this is possible, we get a rotationaly
invariant theory that does not depend on the way it has been defined in
the cut-off* region.
We also see that the theory (1.25) has two phases for ^ > 1, one with
{cp} = 0 and another, the broken symmetry phase, with {cp} # 0. In the
case ^ = 1 we have just the example we analysed in the beginning,
which is the quantum mechanics of a single particle in a potential. We
know that the ground state wave function must be even, and therefore
<(p> = 0. This conclusion is identical to the one we found when
analysing the ^ = 1 Ising model. We shall clarify this coincidence in the
chapter about instantons. How does it happen that the theory with
discrete variables (t^= ± \ appears to be equivalent to the one with a
continuum field cpl This can be understood as follows. Let us change