74 2. THEORIES, LAGRANGIANS AND COUNTERTERMS The adjointness properties of the differential operators D , DE 1 imply that D Kt is symmetric. The propagator for the theory is P (ε, L) = L ε D Kt ∈ E ⊗2 ⊗ A which is again symmetric. Note that unless we impose additional positivity conditions on the oper- ator DE 1 , the heat kernel Kt may not exist at t = ∞ thus, the propagator P (ε, ∞) may not exist. In almost all examples, however, the operator DE 1 is positive, and so the heat kernel K∞ does exist. If we specialize the case of the free scalar field theory on a Riemannian manifold (M, g0), then, as we have seen, E1 = C∞(M), DE 1 = g0 is the non-negative Laplacian for the metric g0. In this example the operator D is the identity. Thus, the propagator prescribed by this general definition coincides with the propagator presented in our earlier analysis of the free field theory. 13.4. As before, we can define the algebra O(E , A ) = Hom(E ⊗n , A )S n of all functionals on E with values in A . Here Hom denotes the space of continuous linear maps. The properties of the symmetric monoidal category of nuclear spaces, as detailed in Appendix 2, show that O(E , A ) = n Symn E ∨ ⊗ A . There is a subspace Ol(E , A ) ⊂ O(E , A ) of A -valued local action functionals, defined as follows. Definition 13.4.1. A functional Φ ∈ O(E , A ) is a local action func- tional if, when we expand Φ as a sum Φ = Φn of its homogeneous com- ponents, each Φn : E ⊗n → A can be written in the form Φn(e1,...,en) = k j=1 M (D1,je1) · · · (Dn,jen)dμ where dμ ∈ Densities(M) is some volume element on M, and each Di,j : E ⊗ A → C∞(M) ⊗ A is an A -linear differential operator.

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