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.

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