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.
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