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 Kl 1 (x, x)Kl 2 (x, y)Kl 3 (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 Kl 1 (x, x)Kl 2 (x, y)Kl 3 (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 En I[L] (f1,...,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 : En I[L] : C∞(M n ) R[[ ]].
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