10 - Introduction to the renormalization group

Summary

The initial formulation of the renormalization group appeared in the study of quantum electrodynamics (QED) in the high-energy limit where the momenta of the particles (electrons, positrons, photons) were much larger than the rest mass of the electron (Gell-Mann and Low, 1954). It is then natural to neglect the electron mass from the beginning (rather than calculating Feynman diagrams and taking their high-energy limit). However, the theory with massless electrons has to be handled with some care: the usual definition of the renormalized charge, normally defined as the electron—photon vertex in the limit of vanishing momenta, leads to an infrared divergence. Then the definition of the renormalized charge has to be modified as being the value of the same vertex but at some arbitrary non-zero momenta. This introduces a new length scale to the theory (the inverse of an arbitrary momentum in units ħ = c = 1). Physical quantities should not depend on this arbitrariness, and different charges of the electron, related to different scales of definition, lead to the same physics.

It was only in the beginning of the 1970s that the study of dilatation invariance in field theory, extended by K. Wilson to the vicinity of the critical point and similar physical problems, threw light on the true meaning of the renormalization group.