16.4. Renormalization, and the renormalization group, on the lattice
173
in the limit Λ → ∞, which are controlled (in a renormalizable theory) by
the procedure of renormalization. Such divergent fluctuations turn out, in
fact, to affect a renormalizable theory only through the values of some of
its parameters and, if these parameters are taken from experiment, all other
quantities become finite, even as Λ → ∞. This latter assertion is not easy
to prove, and indeed is quite surprising. However, this is by no means all
there is to renormalization theory: we have seen the power of ‘renormalization group’ ideas in making testable predictions for QCD. Nevertheless,
the methods of chapter 15 were rather formal, and the reader may well
feel the need of a more physical picture of what is going on. Such a picture was provided by Wilson (1971a) (see also Wilson and Kogut 1974), using the ‘lattice + path integral’ approach. Another important advantage
of this formalism is, therefore, precisely the way in which, thanks to Wilson’s work, it provides access to a more intuitive way of understanding renormalization theory. The aim of this section is to give a brief introduction
to Wilson’s ideas, so as to illuminate the formal treatment of the previous
chapter.
In the ‘lattice + path integral’ approach to quantum field theory, the
degrees of freedom involved are the values of the field(s) at each lattice site,
as we have seen. Quantum amplitudes are formed by integrating suitable
quantities over all values of these degrees of freedom, as in (16.87) for example.
From this point of view, it should be possible to examine specifically how the
‘short distance’ or ‘high momentum’ degrees of freedom affect the result. In
fact, the idea suggests itself that we might be able to perform explicitly the
integration (or summation) over those degrees of freedom located near the
cutoff Λ in momentum space, or separated by only a lattice site or two in
co-ordinate space. If we can do this, the result may be compared with the
theory as originally formulated, to see how this ‘integration over short-distance
degrees of freedom’ affects the physical predictions of the theory. Having done
this once, we can imagine doing it again – and indeed iterating the process,
until eventually we arrive at some kind of ‘effective theory’ describing physics
in terms of ‘long-distance’ degrees of freedom.
There are several aspects of such a programme which invite comment.
First, the process of ‘integrating out’ short-distance degrees of freedom will
obviously reduce the number of effective degrees of freedom, which is necessarily very large in the case ξ ≫ a, as envisaged above. Thus it must be
a step in the right direction. Secondly, the above sketch of the ‘integrating
out’ procedure suggests that, at any given stage of the integration, we shall
be considering the system as described by parameters (including masses and
couplings) appropriate to that scale, which is of course strongly reminiscent
of RGE ideas. And thirdly, we may perhaps anticipate that the result of
this ‘integrating out’ will be not only to render the parameters of the theory
scale-dependent, but also, in general, to introduce new kinds of effective interactions into the theory. We now consider some simple examples which we
hope will illustrate these points.
173
in the limit Λ → ∞, which are controlled (in a renormalizable theory) by
the procedure of renormalization. Such divergent fluctuations turn out, in
fact, to affect a renormalizable theory only through the values of some of
its parameters and, if these parameters are taken from experiment, all other
quantities become finite, even as Λ → ∞. This latter assertion is not easy
to prove, and indeed is quite surprising. However, this is by no means all
there is to renormalization theory: we have seen the power of ‘renormalization group’ ideas in making testable predictions for QCD. Nevertheless,
the methods of chapter 15 were rather formal, and the reader may well
feel the need of a more physical picture of what is going on. Such a picture was provided by Wilson (1971a) (see also Wilson and Kogut 1974), using the ‘lattice + path integral’ approach. Another important advantage
of this formalism is, therefore, precisely the way in which, thanks to Wilson’s work, it provides access to a more intuitive way of understanding renormalization theory. The aim of this section is to give a brief introduction
to Wilson’s ideas, so as to illuminate the formal treatment of the previous
chapter.
In the ‘lattice + path integral’ approach to quantum field theory, the
degrees of freedom involved are the values of the field(s) at each lattice site,
as we have seen. Quantum amplitudes are formed by integrating suitable
quantities over all values of these degrees of freedom, as in (16.87) for example.
From this point of view, it should be possible to examine specifically how the
‘short distance’ or ‘high momentum’ degrees of freedom affect the result. In
fact, the idea suggests itself that we might be able to perform explicitly the
integration (or summation) over those degrees of freedom located near the
cutoff Λ in momentum space, or separated by only a lattice site or two in
co-ordinate space. If we can do this, the result may be compared with the
theory as originally formulated, to see how this ‘integration over short-distance
degrees of freedom’ affects the physical predictions of the theory. Having done
this once, we can imagine doing it again – and indeed iterating the process,
until eventually we arrive at some kind of ‘effective theory’ describing physics
in terms of ‘long-distance’ degrees of freedom.
There are several aspects of such a programme which invite comment.
First, the process of ‘integrating out’ short-distance degrees of freedom will
obviously reduce the number of effective degrees of freedom, which is necessarily very large in the case ξ ≫ a, as envisaged above. Thus it must be
a step in the right direction. Secondly, the above sketch of the ‘integrating
out’ procedure suggests that, at any given stage of the integration, we shall
be considering the system as described by parameters (including masses and
couplings) appropriate to that scale, which is of course strongly reminiscent
of RGE ideas. And thirdly, we may perhaps anticipate that the result of
this ‘integrating out’ will be not only to render the parameters of the theory
scale-dependent, but also, in general, to introduce new kinds of effective interactions into the theory. We now consider some simple examples which we
hope will illustrate these points.