56 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

For any L 0, the propagator P (0,L) is a parametrix. In the alternative

definition of a quantum field theory presented in this section, we can use

any parametrix as the propagator.

Note that if P, P are two parametrices, the difference P − P between

them is a smooth function. We will give the set of parametrices a partial

order, by saying that

P ≤ P

if Supp(P ) ⊂ Supp(P ). For any two parametrices P, P , we can find some

P with P P and P P .

8.1. Before we introduce the alternative definition of quantum field

theory, we need to introduce a technical notation. Given any functional

J ∈

O(C∞(M

)), we get a continuous linear map

C∞(M)

→

O(C∞(M

))

φ →

dJ

dφ

.

Definition 8.1.1. A function J has smooth first derivative if this map

extends to a continuous linear map

D(M) →

O(C∞(M

)),

where D(M) is the space of distributions on M.

Lemma 8.1.2. Let Φ ∈

C∞(M)⊗2

and suppose that J ∈

O+(C∞(M

))[[ ]]

has smooth first derivative. Then so does W (Φ,J) ∈

O+(C∞(M

))[[ ]].

Proof. Recall that

W (Φ,J) = log e

∂P eJ/

.

Thus, it suﬃces to verify two things. Firstly, that the subspace

O(C∞(M

))

consisting of functionals with smooth first derivative is a subalgebra; this is

clear. Secondly, we need to check that ∂Φ preserves this subalgebra. This is

also clear, because ∂Φ commutes with

d

dφ

for any φ ∈

C∞(M)).

8.2. The alternative definition of quantum field theory is as follows.

Definition 8.2.1. A quantum field theory is a collection of functionals

I[P ] ∈

O+(C∞(M

))[[ ]],

one for each parametrix P , such that the following properties hold.

(1) If P, P are parametrices, then

W

(

P − P , I[P ]

)

= I[P ].

This expression makes sense, because P − P is a smooth function

on M × M.