180
16. Lattice Field Theory, and the Renormalization Group Revisited
themselves. The intuitive meaning of ‘irrelevant’ is clear enough: the system
will head towards a fixed point as f → ∞ whatever the initial values of the
irrelevant couplings. The critical behaviour of the system will therefore be
independent of the number and type of all irrelevant couplings, and will be
determined by the relatively few (in general) marginal and relevant couplings.
Thus all systems which flow close to the fixed point will display the same
critical exponents determined by the dynamics of these few couplings. This
explains the property of universality observed in the physics of phase transitions, whereby many apparently quite different physical systems are described
(in the vicinity of their critical points) by the same critical exponents.
Additional terms in the Hamiltonian are, in fact, generally introduced
following a renormalization transformation. In the quantum field case, we
may expect that renormalization transformations associated with a → f a,
and iterations thereof, will in general lead to an effective theory involving all
possible couplings allowed by whatever symmetries are assumed to be relevant.
Thus, if we start with a typical ‘φ
4 ’ scalar theory as given by (16.98), we
shall expect to generate all possible couplings involving φ and its derivatives.
At first sight, this may seem disturbing: after all, the original theory (in
four dimensions) is a renormalizable one, but an interaction such as Aφ
6 is
not renormalizable according to the criterion given in section 11.8 (in four
dimensions φ has mass dimension unity, so that A must have mass dimension
-2). It is, however, essential to remember that in this ‘Wilsonian’ approach to
renormalization, summations over momenta appearing in loops do not, after
one iteration a → f a, run up to the original cut-off value π/a, but only up
to the lower cut-off π/f a. The additional interactions compensate for this
change.
In fact, we shall now see how the coefficients of non-renormalizable interactions correspond precisely to irrelevant couplings in Wilson’s approach, so
that their effect becomes negligible as we iterate to scales much larger than
a. We consider continuous changes of scale characterized by a factor f , and
we discuss a theory with only a single scalar field φ for simplicity. Imagine,
therefore, that we have integrated out, in (16.97), those components of φ(x)
with a < |x| < fa. We will be left with a functional integral of the form
(16.97), but with φ(x) restricted to |x| > fa, and with additional interaction
terms in the action. In order to interpret the result in Wilson’s terms, we
must rewrite it so that it has the same general form as the original Z φ of
(16.97). A simple way to do this is to rescale distances by
x
′
x =
(16.128)
f
′
so that the functional integral is now over φ(x ) with |x
′
| > a, as in (16.97).
We now define the fixed point of the renormalization transformation to be
that in which all the terms in the action are zero, except the ‘kinetic’ piece;
this is the ‘free-field’ fixed point. Thus, we require the kinetic action to be
16. Lattice Field Theory, and the Renormalization Group Revisited
themselves. The intuitive meaning of ‘irrelevant’ is clear enough: the system
will head towards a fixed point as f → ∞ whatever the initial values of the
irrelevant couplings. The critical behaviour of the system will therefore be
independent of the number and type of all irrelevant couplings, and will be
determined by the relatively few (in general) marginal and relevant couplings.
Thus all systems which flow close to the fixed point will display the same
critical exponents determined by the dynamics of these few couplings. This
explains the property of universality observed in the physics of phase transitions, whereby many apparently quite different physical systems are described
(in the vicinity of their critical points) by the same critical exponents.
Additional terms in the Hamiltonian are, in fact, generally introduced
following a renormalization transformation. In the quantum field case, we
may expect that renormalization transformations associated with a → f a,
and iterations thereof, will in general lead to an effective theory involving all
possible couplings allowed by whatever symmetries are assumed to be relevant.
Thus, if we start with a typical ‘φ
4 ’ scalar theory as given by (16.98), we
shall expect to generate all possible couplings involving φ and its derivatives.
At first sight, this may seem disturbing: after all, the original theory (in
four dimensions) is a renormalizable one, but an interaction such as Aφ
6 is
not renormalizable according to the criterion given in section 11.8 (in four
dimensions φ has mass dimension unity, so that A must have mass dimension
-2). It is, however, essential to remember that in this ‘Wilsonian’ approach to
renormalization, summations over momenta appearing in loops do not, after
one iteration a → f a, run up to the original cut-off value π/a, but only up
to the lower cut-off π/f a. The additional interactions compensate for this
change.
In fact, we shall now see how the coefficients of non-renormalizable interactions correspond precisely to irrelevant couplings in Wilson’s approach, so
that their effect becomes negligible as we iterate to scales much larger than
a. We consider continuous changes of scale characterized by a factor f , and
we discuss a theory with only a single scalar field φ for simplicity. Imagine,
therefore, that we have integrated out, in (16.97), those components of φ(x)
with a < |x| < fa. We will be left with a functional integral of the form
(16.97), but with φ(x) restricted to |x| > fa, and with additional interaction
terms in the action. In order to interpret the result in Wilson’s terms, we
must rewrite it so that it has the same general form as the original Z φ of
(16.97). A simple way to do this is to rescale distances by
x
′
x =
(16.128)
f
′
so that the functional integral is now over φ(x ) with |x
′
| > a, as in (16.97).
We now define the fixed point of the renormalization transformation to be
that in which all the terms in the action are zero, except the ‘kinetic’ piece;
this is the ‘free-field’ fixed point. Thus, we require the kinetic action to be