8. AN ALTERNATIVE DEFINITION 55

this is a choice that only has to be made once, and then it applies to all

theories on all manifolds.

Theorem B. Let us fix a renormalization scheme.

Then, we find a section of each torsor T

(n+1)

→ T

(n),

and so a bijection

between the set of perturbative quantum field theories and the set of local

action functionals I ∈

Oloc(C∞(M +

))[[ ]]. (Recall the superscript + means

that I must be at least cubic modulo .)

7.2. We will first prove theorem B, and deduce theorem A (which is the

more canonical formulation) as a corollary.

In one direction, the bijection in theorem B is constructed as follows. If

I ∈

Oloc(C∞(M +

))[[ ]] is a local action functional, then we will construct a

canonical series of counterterms

ICT

(ε). These are local action functionals,

depending on a parameter ε ∈ (0, ∞) as well as on . The counterterms

are zero modulo , as the tree-level Feynman graphs all converge. Thus,

ICT

(ε) ∈

Oloc(C∞(M

))[[ ]] ⊗

C∞((0,

∞)) where ⊗ denotes the completed

projective tensor product.

These counterterms are constructed so that the limit

lim

ε→0

W

(

P (ε, L),I −

ICT

(ε)

)

exists. This limit defines the scale L effective interaction I[L].

Conversely, if we have a perturbative QFT given by a collection of ef-

fective interactions I[l], the local action functional I is obtained as a cer-

tain renormalized limit of I[l] as l → 0. The actual limit doesn’t exist;

to construct the renormalized limit we again need to subtract off certain

counterterms.

A detailed proof of the theorem, and in particular of the construction of

the local counterterms, is given in Section 10.

8. An alternative definition

In the previous section I presented a definition of quantum field theory

based on the heat-kernel cut-off. In this section, I will describe an alter-

native, but equivalent, definition, which allows a much more general class

of cut-offs. This alternative definition is a little more complicated, but is

conceptually more satisfying. One advantage of this alternative definition is

that it does not rely on the heat kernel.

As before, we will consider a scalar field theory where the quadratic term

of the action is

1

2

φ(D

+m2)φ.

Definition 8.0.1. A parametrix for the operator D

+m2

is a distribu-

tion P on M ×M, which is symmetric, smooth away from the diagonal, and

is such that

((D

+m2)

⊗ 1)P − δM ∈

C∞(M

× M)

is smooth; where δM refers to the delta distribution along the diagonal in

M × M.