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)|

wvi,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 wvi,k (0,I) = k!Ii,k. Thus, W (0,I) =

∑

i,k

iIi,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.