Renormalization Group - Momentum Space RG

Momentum Space RG

RG, in practice, comes in two main flavours. The Kadanoff picture explained above refers mainly to the so-called real-space RG. Momentum-space RG on the other hand, has a longer history despite its relative subtlety. It can be used for systems where the degrees of freedom can be cast in terms of the Fourier modes of a given field. The RG transformation proceeds by integrating out a certain set of high momentum (large wavenumber) modes. Since large wavenumbers are related to short length scales, the momentum-space RG results in an essentially similar coarse-graining effect as with real-space RG.

Momentum-space RG is usually performed on a perturbation expansion. The validity of such an expansion is predicated upon the true physics of our system being close to that of a free field system. In this case, we may calculate observables by summing the leading terms in the expansion. This approach has proved very successful for many theories, including most of particle physics, but fails for systems whose physics is very far from any free system, i.e., systems with strong correlations.

As an example of the physical meaning of RG in particle physics we will give a short description of charge renormalization in quantum electrodynamics (QED). Let us suppose we have a point positive charge of a certain true (or bare) magnitude. The electromagnetic field around it has a certain energy, and thus may produce some pairs of (e.g.) electrons-positrons, which will be annihilated very quickly. But in their short life, the electron will be attracted by the charge, and the positron will be repelled. Since this happens continuously, these pairs are effectively screening the charge from abroad. Therefore, the measured strength of the charge will depend on how close to our probes it may enter. We have a dependence of a certain coupling constant (the electric charge) with distance.

Momentum and length scales are related inversely according to the de Broglie relation: the higher the energy or momentum scale we may reach, the lower the length scale we may probe and resolve. Therefore, the momentum-space RG practitioners sometimes declaim to integrate out high momenta or high energy from their theories.