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)Sn
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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