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
)), we get a continuous linear map
Definition 8.1.1. A function J has smooth first derivative if this map
extends to a continuous linear map
where D(M) is the space of distributions on M.
Lemma 8.1.2. Let Φ ∈
and suppose that J ∈
has smooth first derivative. Then so does W (Φ,J) ∈
Proof. Recall that
W (Φ,J) = log e
Thus, it suﬃces to verify two things. Firstly, that the subspace
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
for any φ ∈
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 ] ∈
one for each parametrix P , such that the following properties hold.
(1) If P, P are parametrices, then
P − P , I[P ]
= I[P ].
This expression makes sense, because P − P is a smooth function
on M × M.