3. WILSONIAN LOW ENERGY THEORIES 11 Figure 1. The first few expressions in the renormalization group flow from scale ε to scale L. The dotted lines indicate incoming particles. The blobs indicate interactions between these particles. The symbol Ieff[L] i,k indicates the i term in the contribution to the interaction of k particles at length- scale L. The solid lines indicate the propagation of a particle between two interactions particles are allowed to propagate with proper time between ε and L. We should think of Ieff[L] i,k as being a contribution to the interaction of k particles which come together in a region of size around L. Figure 1 shows how to express, graphically, the world-line version of the renormalization group flow. 3.6. So far in this section, we have sketched the definition of two versions of the renormalization group flow: one based on energy, and one based on length. There is a more general definition of the renormalization group flow which includes these two as special cases. This more general version is based on the concept of parametrix.

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