3. GENERALITIES ON FEYNMAN GRAPHS 37 3.3. Now that we have defined the functions wγ(P, I) ∈ O(U ), we will arrange them into a formal power series. Let us define W (P, I) = γ 1 |Aut(γ)| g(γ) wγ(P, I) ∈ O+(U )[[ ]] where the sum is over connected stable graphs γ, and g(γ) is the genus of the graph γ. The condition that all genus 0 vertices are at least trivalent implies that this sum converges. Figure 1 illustrates the first few terms of the graphical expansion of W (P, I). Our combinatorial conventions are such that, for all a1,...,ak ∈ U, ∂ ∂a1 · · · ∂ ∂ak W (P, I) (0) = γ,φ g(γ) |Aut| (γ, φ) wγ,φ(P, I)(a1,...,ak) where the sum is over graphs γ with k tails and an isomorphism φ : {1,...,k} ∼ T (γ), and the automorphism group Aut(γ, φ) preserves the ordering φ of the set of tails. Lemma 3.3.1. W (0,I) = I. Proof. Indeed, when P = 0 only graphs with no edges can contribute, so that W (0,I) = i,k i |Aut(vi,k)| wv i,k (0,I). Here, as before, vi,k is the graph with a single vertex of genus i and valency k, and with no internal edges. The automorphism group of vi,k is the symmetric group Sk, and we have seen that wv i,k (0,I) = k!Ii,k. Thus, W (0,I) = ∑ i,k i Ii,k, which (by definition) is equal to I. 3.4. As before, let P ∈ Sym2 U, and let us write P = ∑ P ⊗P . Define an order two differential operator ∂P : O(U ) → O(U ) by ∂P = 1 2 ∂ ∂P ∂ ∂P . A convenient way to summarize the Feynman graph expansion W (P, I) is the following. Lemma 3.4.1. W (P, I) (a) = log {exp( ∂P ) exp(I/ )} (a) ∈ O+(U )[[ ]]. The expression exp( ∂P ) exp(I/ ) is the exponential of a differential operator on U applied to a function on U thus, it is a function on U.

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