20 1. INTRODUCTION

7.2. As our definition of renormalizability only makes sense on

Rn,

we

will now restrict to considering scalar field theories on

Rn.

We want to mea-

sure

Seff

[Λ] as Λ → ∞, after we have changed units. Define

RGl(Seff

[Λ])

by

RGl(Seff

[Λ])(φ) =

Seff [l−2Λ](Rl(φ))

Thus,

RGl(Seff

[Λ]) is the effective action

Seff [l2Λ],

but measured in units

that have been rescaled by l.

We can use the map RGl to implement precisely the definition of renor-

malizability suggested above.

Definition 7.2.1. A theory

{Seff

[Λ]} is renormalizable if

RGl(Seff

[Λ])

grows at most logarithmically as l → 0.

7.3. It turns out that the map RGl defines a flow on the space of theories.

Lemma 7.3.1. If

{Seff

[Λ]} satisfies the renormalization group equation,

then so does {RGl(Seff [Λ]}.

Thus, sending

{Seff

[Λ]} →

{RGl(Seff

[Λ])}

defines a flow on the space of theories: this is the local renormalization group

flow.

Recall that the choice of a renormalization scheme leads to a bijection

between the space of theories and Lagrangians. Under this bijection, the

local renormalization group flow acts on the space of Lagrangians. The con-

stants appearing in a Lagrangian (the coupling constants) become functions

of l; the dependence of the coupling constants on the parameter l is called

the β function. Renormalizability means these coupling constants have at

most logarithmic growth in l.

The local renormalization group flow RGl, as l → 0, can be interpreted

geometrically as focusing on smaller and smaller regions of space-time, while

always using units appropriate to the size of the region one is considering.

In energy terms, applying RGl as l → 0 amounts to focusing on phenomena

of higher and higher energy.

The logarithmic growth condition thus says the theory doesn’t break

down completely when we probe high-energy phenomena. If the effective

actions displayed polynomial growth, for instance, then one would find that

the perturbative description of the theory wouldn’t make sense at high en-

ergy, because the terms in the perturbative expansion would increase with

the energy.

7.4. The definition of renormalizability given above can be viewed as a

perturbative approximation to an ideal non-perturbative definition.

Definition 7.4.1 (Ideal definition). A non-perturbative theory is renor-

malizable if, as we flow the theory under RGl and let l → 0, we converge to

a fixed point.