7. RENORMALIZABILITY 19
7. Renormalizability
We have seen that the space of theories is an infinite dimensional man-
ifold, modelled on the space of R[[ ]]-valued local action functionals on
C∞(M).
A physicist would find this unsatisfactory. Because the space of theories
is infinite dimensional, to specify a particular theory, it would take an infinite
number of experiments. Thus, we can’t make any predictions.
We need to find a natural finite-dimensional submanifold of the space
of all theories, consisting of “well-behaved” theories. These well-behaved
theories will be called renormalizable.
7.1. An old-fashioned viewpoint is the following:
A local action functional (or Lagrangian) is renormalizable if
it has only finitely many counterterms:
SCT
(r) =
finite
fi(r)SiCT
In general, this definition picks out a finite dimensional subspace of the in-
finite dimensional space of theories. However, it is not natural: the spe-
cific counterterms will depend on the choice of renormalization scheme,
and therefore this definition may depend on the choice of renormalization
scheme.
More fundamentally, any definitions one makes should be directly in
terms of the only physical quantities one can measure, namely the low-energy
effective actions
Seff
[Λ]. Thus, we would like a definition of renormalizabil-
ity using only the
Seff
[Λ].
The following is the basic idea of the definition we suggest, following
Wilson and others.
Definition 7.1.1 (Rough definition). A theory, defined by effective ac-
tions
Seff
[Λ], is renormalizable if the
Seff
[Λ] don’t grow too fast as Λ ∞.
However, we must measure
Seff [Λ] in units appropriate to energy scale Λ.
For instance, if
Seff
[1] is measured in joules, then
Seff [103]
should be
measured in kilo-joules, and so on.
However, this change of units only makes sense on
Rn.
Since we can
identify energy with
length−2,
changing the units of energy amounts to
rescaling
Rn.
In addition, the field φ
C∞(Rn)
can have its own energy
(which should be thought of as giving the target of the map φ :
Rn
R
some weight). Once we incorporate both of these factors, the procedure of
changing units (in a scalar field theory) is implemented by the map
Rl :
C∞(Rn)

C∞(Rn)
φ(x)
l1−n/2φ(l−1x).
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