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).
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