6. THE GEOMETRIC INTERPRETATION OF FEYNMAN GRAPHS 49 6.2. We should interpret these identities as follows. We should think of particles moving through space-time as equipped with an “internal clock” as the particle moves, this clock ticks at a rate independent of the time pa- rameter on space-time. The world-line of such a particle is a parameterized path in space-time, that is, a map f : R M. This path is completely arbitrary: it can go backwards or forwards in time. Two world-lines which differ by a translation on the source R should be regarded as the same. In other words, the internal clock of a particle doesn’t have an absolute starting point. If I = [0,τ] is a closed interval, and if f : I M is a path describing part of the world-line of a particle, then the energy of f is, as before, E(f) = [0,τ] df, df . In quantum field theory, everything that can happen will happen, but with a probability amplitude of eiE where E is the energy. Thus, to calculate the probability that a particle starts at the point x in space-time and ends at the point y, we must integrate over all paths f : [0,τ] M, starting at x and ending at y. We must also integrate over the parameter τ, which is interpreted as the time taken on the internal clock of the particle as it moves from x to y. This leads to the expression (in Lorentzian signature) we discussed earlier, P (x, y) = t=0 f:[0,t]→M f(0)=x,f(t)=y eiE(f). 6.3. We would like to have a similar picture for interacting theories. We will consider, for simplicity, the φ3 theory, where the fields are φ C∞(M), and the action is S(φ) = M 1 2 φ D φ + 1 6 φ3. We will discuss the heuristic picture first, ignoring the difficulties of renor- malization. At the end, we will explain how the renormalization group flow and the idea of effective interactions can be explained in the world-line point of view. The fundamental quantities one is interested in are the correlation func- tions, defined by the heuristic functional integral formula E(x1, . . . , xn) = φ∈C∞(M) eS(φ)/ φ(x1) · · · φ(xn). We would like to express these correlation functions in the world-line point of view. 6.4. The φ3 theory corresponds, in the world-line point of view, to a theory where three particles can fuse at a point in M. Thus, world-lines in the φ3 theory become world-graphs further, just as the world-lines for the free theory are parameterized, the world-graphs arising in the φ3 theory
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