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 ( −1 W ( 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 ( −1 W ( P + εP , I )) (0) = 1 2 ∂ ∂a1 · · · ∂ ∂ak ∂ ∂u ∂ ∂u exp ( −1 W (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.

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