Using Wilsonʹs renormalization group to repair symmetries

Wilsonʹs renormalization group can be used to approximate the behavior of an extremely large number of coupled degrees of freedom. A transformation runs a cutoff that limits the number of coupled states, and its action can be approximated by following the evolution of a finite number of relevant and marginal operators in most theories of physical interest. Symmetries usually constrain these operators; however, if the cutoff itself violates a symmetry, symmetry-breaking operators appear, and in general some sort of fine tuning is required to fix their strengths so that the symmetry is restored in physical quantities. I discuss a simple constraint on the renormalization group flow (coupling coherence) that can be used to isolate and repair such hidden symmetries and give several examples. These ideas have been employed to restore Lorentz and gauge symmetries in light-front QED and QCD calculations, which are also briefly discussed.