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 ∈ O+ loc (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 ∈ O+ loc (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.

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