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.
