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[[ ]].
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