6. THE GEOMETRIC INTERPRETATION OF FEYNMAN GRAPHS 47

This graph is a tree; the integrals associated to graphs which are trees always

admit ε → 0 limits. Indeed,

wγ3 (P (ε, L),I)(a) =

l∈[ε,L] x,y∈M

a(x)2Kl(x, y)a(y)2

=

a2,

L

ε

e−l Da2

dl

and the second expression clearly admits an ε → 0 limit.

All of the calculations above are for the interaction I(φ) =

1

3!

φ3.

For

more general interactions I, one would have to apply a differential operator

to both a and Kl(x, y) in the integrands.

6. The geometric interpretation of Feynman graphs

From the functional integral point of view, Feynman graphs are just

graphical tools which help in the perturbative calculation of certain func-

tional integrals. In the introduction, we gave a brief account of the world-line

interpretation of quantum field theory. In this picture, Feynman graphs de-

scribe the trajectories taken by some interacting particles. The length scale

version of the renormalization group flow becomes very natural from this

point of view.

As before, we will work in Euclidean signature; Lorentzian signature

presents significant additional analytical diﬃculties. We will occasionally

comment on the formal picture in Lorentzian signature.

6.1. Let us consider a massless scalar field theory on a compact Rie-

mannian manifold M. Thus, the fields are

C∞(M) and the action is

S(φ) = −

1

2

M

φ D φ.

The propagator P (x, y) is a distribution on M

2,

which can be expressed as

a functional integral

P (x, y) =

φ∈C∞(M)

eS(φ)φ(x)φ(y).

Thus, the propagator encodes the correlation between the values of the fields

φ at the points x and y.

We will derive an alternative expression of the propagator as a one-

dimensional functional integral.