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S8d. Renormalization and coarse graining
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In QFT, there are two different scales, one on the bare level and one
on the renormalized level, and the meaning of the renormalization
group is slightly different from that in statistical mechanics.
On the statistical mechanics level, there is the cutoff beyond which
one cannot (or does not want to) observe anything. This effective
cutoff is a parameter Lambda in an effective theory defined by coarse
graining.
The effective theory depends on E: For different values of E you get a
_different_ effective theory, though their low energy predictions are
essentially the same. This is expressed by the Wilson flow, described
by renormalization group equations that relate the parameters
g(Lambda,mu) in the different effective theories such that some key
low energy observables mu keep the same values.
The number of such key observables (i.e, the dimension of mu)
equals the number of parameters in the effective theory
(i.e, the dimension of g); most other observables are different
at different cutoffs (though only slightly if they are observable at
low energy), because of the coarse graining done when lowering
the cutoff scale Lambda.
In QFT, the above is mimicked on the _bare_ level. The cutoff is a
large energy Lambda beyond which the bare interaction is modified to
be able to get a meaningful limit; this corresponds to coarse-graining.
The resulting bare theory with cutoff Lambda is a well-defined
effective theory and behaves precisely as described above.
To define the renormalized theory, one needs, in addition to the
cutoff, renormalization conditions defining the bare parameters in
terms of renormalized parameters q.
These conditions depend on a renormalization scale E figuring in the
equations defining the renormalization conditions. Because of the
dimensional nature of momentum, there always has to be such a
parameter E, no matter which renormalization procedure is followed.
In QFT, one usually refers to a mass scale M, which is the same as
E=Mc^2 in units such that c=1. Then M is the constant needed in the
renormalization conditions to relate certain computable expressions
to the renormalized parameters. This is discussed at length in
the QFT book by Peskin and Schroeder, Section 12.2, for a massless
Phi^4 theory, and in Section 12.5 for the general case. (For an online
source, see, e.g., equations (90-(11) of hep-th/9804079.
M is introduced there without comment, the role of M is described
later, after (20).) In the following, I continue to use E in place of M.
Thus the bare parameters are functions g(Lambda,q,E) of the cutoff
Lambda, the renormalized parameters q, and the renormalization scale E.
The renormalization group equations in the statistical mechanics
sense (the Wilson flow) would describe how g(Lambda,q,E) changes as
the cutoff Lambda is altered. However, in QFT, this is of no physical
interest. Indeed, Lambda is completely eliminated from considerations:
The renormalized theory is obtained at fixed E by letting the cutoff
Lambda go to infinity. This has the effect that the bare parameters
become meaningless, since the limit
lim_{Lambda to inf} g(Lambda,q,E)
does not exist. At this stage it becomes obvious that all bare objects
are unphysical.
Although nonphysical, the renormalization group equations in
Lambda are an important tool in the _construction_ of QFTs, where the
limit of all correlation functions must be shown to exist in a
suitable topology, and the absence
of divergences shown. In the weakest topology, based on the
ultrametric norm and corresponding to perturbation theory at all
orders, this is shown rigorously in a nice book
M. Salmhofer,
Renormalization: An Introduction,
Springer, Berlin 1999.
Unfortunately, this topology is too weak to give the existence of
the correlation functions as functions; they are only shown to exist
as formal power series.
All expressions of the theory that survive the limit, in particular
all n-point correlation functions, n=1,2,3,...,
describe observable physics. They can therefore be expressed as
functions of q and E only, whose detailed form comes from the
standard theory. However, there is a little twist since the scale E
can be chosen arbitrarily, hence cannot be measurable.
In terms of a fixed set of physical parameters mu (measurable
under well-defined experimental conditions), we can predict mu
by some function of q and E, mu=mu(q,E). Solving for q, we can
express q in terms of mu and E,
q=q_ren(mu,E).
But the exact renormalized result of a physical prediction P(q,E)
must be completely independent of E, uniquely determined by the
physical parameters mu. Thus we get the so-called Callan-Symanzik
equations
d/dE P(q_ren(mu,E),E) = 0.
They are the renormalization group equations of interest in quantum
field theory.
In contrast to the Wilson flow, however, the sliding scale in the
Callan-Symanzik flow is the renormalization scale E and _not_ the
cutoff Lambda (which at this stage is already infinite). Moreover,
since observable physics is completely independent of the
renormalization scale E, the latter has no intuitive 'physical'
interpretation.
There is no relation between the two flows, except by analogy.
The Wilson flow is needed to _get_ the renormalized theory
at fixed renormalization conditions, the Callan-Symanzik flow
describes what happens when you _change_ these conditions.