3. WILSONIAN LOW ENERGY THEORIES 9 action. Thus, P is a distribution on M 2 . Away from the diagonal in M 2 , P is a distribution. The value P (x, y) of P at distinct points x, y in the space-time manifold M can be interpreted as the correlation between the value of the field φ at x and the value at y. Feynman realized that the propagator can be written as an integral P (x, y) = τ=0 e−τm2K τ (x, y)dτ where Kτ(x, y) is the heat kernel. The fact that the heat kernel can be interpreted as the transition probability for a random path allows us to write the propagator P (x, y) as an integral over the space of paths in M starting at x and ending at y: P (x, y) = τ=0 e−τm2 f:[0,τ]→M f(0)=x,f(τ)=y exp τ 0 df 2 . (This expression can be given a rigorous meaning using the Wiener measure). From this point of view, the propagator P (x, y) represents the proba- bility that a particle starts at x and transitions to y along a random path (the worldline). The parameter τ is interpreted as something like the proper time: it is the time measured by a clock travelling along the worldline. (This expression of the propagator is sometimes known as the Schwinger repre- sentation). Any reasonable action functional for a scalar field theory can be decom- posed into kinetic and interacting terms, S(φ) = Sfree(φ) + I(φ) where Sfree(φ) is the action for the free theory discussed above. From the space-time point of view on quantum field theory, the quantity I(φ) prescribes how particles interact. The local nature of I(φ) simply says that particles only interact when they are at the same point in space-time. From this point of view, Feynman graphs have a very simple interpretation: they are the “world-graphs” traced by a family of particles in space-time moving in a random fashion, and interacting in a way prescribed by I(φ). This point of view on quantum field theory is the one most closely related to string theory (see e.g. the introduction to (GSW88)). In string theory, one replaces points by 1-manifolds, and the world-graph of a collection of interacting particles is replaced by the world-sheet describing interacting strings. 3.5. Let us now briefly describe how to treat effective field theory from the world-line point of view. In the energy-scale picture, physics at scales less than Λ is described by saying that we are only allowed fields of energy less than Λ, and that the action on such fields is described by the effective action Seff[Λ]. In the world-line approach, instead of having an effective action Seff[Λ] at each energy-scale Λ, we have an effective interaction Ieff[L] at each
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