Functional renormalization group
In theoretical physics, functional renormalization group (FRG) is an implementation of the renormalization group (RG) concept which is used in quantum and statistical field theory, especially when dealing with strongly interacting systems.
The method combines functional methods of quantum field theory with the intuitive renormalization group idea of Kenneth G. Wilson.
This technique allows to interpolate smoothly between the known microscopic laws and the complicated macroscopic phenomena in physical systems.
In this sense, it bridges the transition from simplicity of microphysics to complexity of macrophysics.
Figuratively speaking, FRG acts as a microscope with a variable resolution.
One starts with a high-resolution picture of the known microphysical laws and subsequently decreases the resolution to obtain a coarse-grained picture of macroscopic collective phenomena.
The method is nonperturbative, meaning that it does not rely on an expansion in a small coupling constant.
Mathematically, FRG is based on an exact functional differential equation for a scale-dependent effective action.
In quantum field theory, the effective action
yields exact quantum field equations, for example for cosmology or the electrodynamics of superconductors.
is the generating functional of the one-particle irreducible Feynman diagrams.
Interesting physics, as propagators and effective couplings for interactions, can be straightforwardly extracted from it.
In a generic interacting field theory the effective action
The central object in FRG is a scale-dependent effective action functional
by giving them a large mass, while high momentum modes are not affected.
from the left-hand-side and the right-hand-side respectively, due to the tensor structure of the equation.
This feature is often shown simplified by the second derivative of the effective action.
denotes a supertrace which sums over momenta, frequencies, internal indices, and fields (taking bosons with a plus and fermions with a minus sign).
This is an important simplification compared to perturbation theory, where multi-loop diagrams must be included.
is the full inverse field propagator modified by the presence of the regulator
As schematically shown in the figure, at the microscopic ultraviolet scale
evolves in the theory space according to the functional flow equation.
is not unique, which introduces some scheme dependence into the renormalization group flow.
In most cases of interest the Wetterich equation can only be solved approximately.
The choice of the suitable scheme should be physically motivated and depends on a given problem.
The expansions do not necessarily involve a small parameter (like an interaction coupling constant) and thus they are, in general, of nonperturbative nature.
Note however, that due to multiple choices regarding (prefactor-)conventions and the concrete definition of the effective action, one can find other (equivalent) versions of the Wetterich equation in the literature.
which generates n-particle interaction vertices, amputated by the bare propagators
The Wick ordering of effective interaction with respect to Green function
This operation is similar to Normal order and excludes from the interaction all possible terms, formed by a convolution of source fields with respective Green function D. Introducing some cutoff
