18 1. INTRODUCTION

This expression is the renormalization group flow from infinite energy to

energy Λ. This is an infinite dimensional integral, as the field φH has un-

bounded energy.

This functional integral is renormalized using the technique of coun-

terterms. This involves first introducing a regulating parameter r into the

functional integral, which tames the singularities arising in the Feynman

graph expansion. One choice would be to take the regularized functional

integral to be an integral only over the finite dimensional space of fields

φ ∈

C∞(M)(Λ,r].

Sending r → ∞ recovers the original integral. This limit won’t exist,

but one renormalizes this limit by introducing counterterms. Counterterms

are functionals

SCT

(r, φ) of both r and the field φ, such that the limit

lim

r→∞

φH ∈C∞(M)(Λ,r]

exp

1

S(φL + φH ) −

1

SCT

(r, φL + φH )

exists. These counterterms are local, and are uniquely defined once one

chooses a renormalization scheme.

The effective action

Seff

[Λ] is then defined by this limit:

Seff

[Λ](φL) =

lim

r→∞

φH

∈C∞(M)(Λ,r]

exp

1

S(φL + φH ) −

1

SCT

(r, φL + φH )

6.2. In practise, we don’t use the energy-scale regulator r but rather

a length-scale regulator ε. The reason is the same as before: it is easier

to construct local theories using the length-scale regulator than the energy-

scale regulator. In what follows, I will ignore this rather technical point; to

make the following discussion completely accurate, the reader should replace

the energy-scale regulator r by the length-scale regulator we will use later.

The counterterms

SCT

are constructed by a simple inductive procedure,

and are local action functionals of the field φ ∈

C∞(M).

Once we have chosen such a renormalization scheme, we find a set of

counterterms

SCT

(r, φ) for any local action functional S. These countert-

erms are uniquely determined by the requirements that firstly, the r → ∞

limit above exists, and secondly, they are purely singular as a function of

the regulating parameter r.

6.3. What we see from this is that the bijection between theories and

local action functionals is not canonical, but depends on the choice of a

renormalization scheme. Thus, theorem A is the most natural formulation:

there is no natural bijection between theories and local action functionals.

Theorem A implies that the space of theories is an infinite dimensional

manifold, modelled on the topological vector space of R[[ ]]-valued local

action functionals on

C∞(M).