Relation between mean field theory and renormalization method?

Question

Can someone tell us the difference or relationship between mean field theory and renormalization method to deal with many body problems?

Answer 1

Mean field theory in many-body physics is the approximation where the individual particles are assumed to interact with the average (or mean field) distribution of all the other particles through some two-body force model. If the numerical procedure is iterated until the mean fields (or single particle distributions) agree on two successive iterations, it is referred to as the self-consistent field (or Hartree) approximation.

Renormalization is required when one wishes to include the effects of 2nd quantization (mean field is a 1st quantization approach). This involves including fluctuations of the force carrier field (the field responsible for the two-body interaction) about the mean field. This is a quantum field theory computation that usually leads to divergent results. For certain theories (called renormalizable) these infinite terms can be grouped together with a parameter of the theory (the coupling parameter, or the mass term for example) and interpreted as the real or dressed parameter that is experimentally measured. What is left after this renormalization is a finite contribution (frequently called a vacuum polarization) contribution.