3. GENERALITIES ON FEYNMAN GRAPHS 41 3.6. One can ask, how much of this picture holds if U is replaced by an infinite dimensional vector space? We can’t define the Lebesgue measure in this situation, thus we can’t define the integral directly. However, one can still contract tensors using Feynman graphs, and one can still define the expression W (P, I), as long as one is careful with tensor products and dual spaces. (As we will see later, the singularities in Feynman graphs arise because the inverse to the quadratic forms we will consider on infinite di- mensional vector spaces do not lie in the correct completed tensor product.) Let us work over a ground field K = R or C. Let M be a manifold and E be a super vector bundle on M over K. Let E = Γ(M, E) be the super nuclear Fr´ echet space of global sections of E. Let denote the completed projective tensor product, so that E E = Γ(M ×M, E E). (Some details of the symmetric monoidal category of nuclear spaces, equipped with the completed projective tensor product, are presented in Appendix 2). Let O(E ) denote the algebra of formal power series on E , O(E ) = n≥0 Hom(E ⊗n , K)S n where Hom denotes continuous linear maps and the subscript Sn denotes coinvariants. Note that O(E ) is an algebra: direct product of distributions defines a map Hom(E ⊗n , K) × Hom(E ⊗m , K) Hom(E ⊗n+m , K). These maps induce an algebra structure on O(E ). We can also regard O(E ) as simply the completed symmetric algebra of the dual space E , that is, O(E ) = Sym (E ). Here, E is the strong dual of E , and is again a nuclear space. The completed symmetric algebra is taken in the symmetric monoidal category of nuclear spaces, as detailed in Appendix 2. As before, let O+(E )[[ ]] O(E )[[ ]] be the subspace of those functionals I which are at least cubic modulo . Let Symn E denote the Sn-invariants in E ⊗n . If P Sym2 E and I O+(E )[[ ]] then, for any stable graph γ, one can define wγ(P, I) O(E ). The definition is exactly the same as in the finite dimensional situation. Let T (γ) be the set of tails of γ, H(γ) the set of half-edges of γ, V (γ) the set of vertices of γ, and E(γ) the set of internal edges of γ. The tensor products of interactions at the vertices of γ define an element of Hom(E ⊗H(γ) , R).
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