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).