Editing Renormalization group (section)

==Momentum space==
Renormalization groups, in practice, come in two main "flavors". 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 analogous coarse-graining effect as with real-space RG.

Momentum-space RG is usually performed on a [[perturbation theory|perturbation]] expansion. The validity of such an expansion is predicated upon the actual physics of a system being close to that of a [[free field]] system. In this case, one may calculate observables by summing the leading terms in the expansion. 
This approach has proved 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,  consider an overview of ''charge renormalization'' in [[quantum electrodynamics]] (QED). 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 virtual electron-positron pairs (for example). Although virtual particles annihilate very quickly, during their short lives the electron will be attracted by the charge, and the positron will be repelled. Since this happens uniformly everywhere near the point charge, where its electric field is sufficiently strong, these pairs effectively create a screen around the charge when viewed from far away. The measured strength of the charge will depend on how close our measuring probe can approach the point charge, bypassing more of the screen of virtual particles the closer it gets. Hence a ''dependence of a certain coupling constant (here, the electric charge) with distance scale''.

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 claim to ''integrate out'' high momenta or high energy from their theories.