3. GENERALITIES ON FEYNMAN GRAPHS 39

in two different ways. If we write P =

∑

u ⊗ u where u , u ∈ U, these

extra tails are labelled by u and u . Thus we find that

d

dε

∂

∂a1

· · ·

∂

∂ak

exp

(

−1W

(

P + εP , I

))

(0)

=

1

2

γ,φ

g(γ)

|Aut(γ, φ)|

wγ,φ(P, I)(a1,...,ak,u , u ).

Here the sum is over graphs γ with k + 2 tails, and an ordering φ of these

k + 2 tails. The factor of

1

2

arises because of the two different ways to order

the new tails. Comparing this to the previous expression, we find that

d

dε

∂

∂a1

· · ·

∂

∂ak

exp

(

−1W

(

P + εP , I

))

(0)

=

1

2

∂

∂a1

· · ·

∂

∂ak

∂

∂u

∂

∂u

exp

(

−1W

(P, I)

)

(0)

Since

∂P =

1

2

∂

∂u

∂

∂u

this completes the proof.

This expression makes it clear that, for all P1,P2 ∈

Sym2

U,

W (P1,W (P2,I)) = W (P1 + P2,I) .

3.5. Now suppose that U is a finite dimensional vector space over R,

equipped with a non-degenerate negative definite quadratic form Φ. Let

P ∈

Sym2

U be the inverse to −Φ. Thus, if ei is an orthonormal basis for

−Φ, P =

∑

ei ⊗ ei. (When we return to considering scalar field theories, U

will be replaced by the space

C∞(M),

the quadratic form Φ will be replaced

by the quadratic form − φ, (D +

m2)φ

, and P will be the propagator for

the theory).

The Feynman diagram expansion W (P, I) described above can also be

interpreted as an asymptotic expansion for an integral on U.

Lemma 3.5.1.

W (P, I) (a) = log

x∈U

exp

1

2

Φ(x, x) +

1

I(x + a) .

The integral is understood as an asymptotic series in , and so makes

sense whatever the signature of Φ. As I mentioned before, we use the con-

vention that the measure on U is normalized so that

x∈U

exp

1

2

Φ(x, x) = 1.

This normalization accounts for the lack of a graph with one loop and zero

external edges in the expansion.