42 2. THEORIES, LAGRANGIANS AND COUNTERTERMS We can contract this tensor with the element of E ⊗2E(γ) given by the tensor product of the propagators the result of this contraction is wγ(P, I) ∈ Hom(E ⊗T (γ) , R). Thus, one can define W (P, I) = γ 1 |Aut(γ)| wγ(P, I) ∈ O+(E )[[ ]] exactly as before. The interpretation in terms of differential operators works in this situa- tion too. As in the finite dimensional situation, we can define an order two differential operator ∂P : O(E ) → O(E ). On the direct factor Hom(E ⊗n , K)S n = Symn E ∨ of O(E ), the operator ∂P comes from the map Hom(E ⊗n , K) → Hom(E ⊗n−2 , K) given by contracting with the tensor P ∈ E ⊗2. Then, W (P, I) = log {exp( ∂P ) exp(I/ )} as before. 4. Sharp and smooth cut-offs 4.1. Let us return to our scalar field theory, whose action is of the form S(φ) = − 1 2 φ, (D +m2)φ + I(φ). The propagator P is the kernel for the operator (D +m2)−1. There are several natural ways to write this propagator. Let us pick a basis {ei} of C∞(M) consisting of orthonormal eigenvectors of D, with eigenvalues λi ∈ R≥0. Then, P = i 1 λ2 i + m2 ei ⊗ ei. There are natural cut-off propagators, where we only sum over some of the eigenvalues. For a subset U ⊂ R≥0, let PU = i such that λ i ∈U 1 λ2 i + m2 ei ⊗ ei. Note that, unlike the full propagator P , the cut-off propagator PU is a smooth function on M × M as long as U is a bounded subset of R≥0.

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