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, wγ 3 (P (ε, L),I)(a) = l∈[ε,L] x,y∈M a(x)2K l (x, y)a(y)2 = a2, L ε e−l D a2 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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