6. THE GEOMETRIC INTERPRETATION OF FEYNMAN GRAPHS 51

Then γ has two vertices, labelled by interactions I0,3.

The integral

f:γ→M

e−E(f)

is obtained by putting the heat kernel Kl on each edge of γ of length l, and

integrating over the position of the two vertices. Thus, we find

f:γ→M

e−E(f)

=

x,y,∈M

Kl1 (x, x)Kl2 (x, y)Kl3 (y, y).

However, the second integral, over the space of metrized graphs, does not

make sense. Indeed, the heat kernel Kl(x, y) has a small l asymptotic ex-

pansion of the form

Kl(x, y) =

l−n/2e−x−y

2/4l

lifi(x,

y).

This implies that the integral

l1,l2,l3 f:γ→M

e−E(f)

=

x,y,∈M

Kl1 (x, x)Kl2 (x, y)Kl3 (y, y)

does not converge.

This second integral is the weight attached to the graph γ in the Feyn-

man diagram expansion of the functional integral for the

1

3!

φ3

interaction.

6.6. Next, we will explain (briefly and informally) how to construct the

correlation functions from a general scale L effective interaction I[L]. We

will not need this construction of the correlation functions elsewhere in this

book. A full treatment of observables and correlation functions will appear

in (CG10).

The correlation functions will allow us to give a world-line formulation

for the renormalization group equation on a collection {I[L]} of effective

interactions: the renormalization group equation is equivalent to the state-

ment that the correlation functions computed using I[L] are independent of

L.

The correlations EI[L](f1,

n

. . . , fn) we will define will take, as their input,

functions f1,...,fn ∈

C∞(M).

Thus, the correlation functions will give a

collection of distributions on M

n:

EI[L]

n

:

C∞(M n)

→ R[[ ]].