Functional Renormalization Group

Abstract

The functional renormalization group is a particular implementation of the renormalization group concept which combines functional methods of quantum field theory with the renormalization group idea of Kenneth Wilson. It interpolates smoothly between the known microscopic laws and the complex macroscopic phenomena in physical systems. The renormalization group is formulated directly for a continuum field theory—no lattice regularization is required. The flow from microscopic to macroscopic scales is given by technically demanding flow equations. We derive the Polchinski equation for the scale-dependent Schwinger functional and the Wetterich equation for the scale-dependent effective action. We use the latter to calculate the effective potential, ground state energy and energy gap of quantum mechanical systems. Next we consider scalar fields and calculate the flow of the effective potential and several critical exponents in the local potential approximation. Then we present an exact solution for the scale-dependent effective potential of O(N) models and the critical exponents in the large-N limit. All numerical results were obtained with the matlab/octave programs listed at the end of the chapter.KeywordsCritical ExponentGround State EnergyFlow EquationAnharmonic OscillatorWave Function RenormalizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.