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 Ii,k

eff

[L] 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 Ii,k

eff

[L] 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.