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.
The flow equation for the effective action.
In quantum field theory, the effective action formula_1 is an analogue of the classical action functional formula_2 and depends on the fields of a given theory. It includes all quantum and thermal fluctuations. Variation of formula_1 yields exact quantum field equations, for example for cosmology or the electrodynamics of superconductors. Mathematically, formula_1 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 formula_1, however, is difficult to obtain. FRG provides a practical tool to calculate formula_1 employing the renormalization group concept.
The central object in FRG is a scale-dependent effective action functional formula_7 often called average action or flowing action. The dependence on the RG sliding scale formula_8 is introduced by adding a regulator (infrared cutoff) formula_9 to the full inverse propagator formula_10. Roughly speaking, the regulator formula_11 decouples slow modes with momenta formula_12 by giving them a large mass, while high momentum modes are not affected. Thus, formula_7 includes all quantum and statistical fluctuations with momenta formula_14. The flowing action formula_15 obeys the exact functional flow equation
formula_16
derived by Christof Wetterich and Tim R. Morris in 1993. Here formula_17 denotes a derivative with respect to the RG scale formula_8 at fixed values of the fields. Furthermore, formula_19 denotes the functional derivative of formula_15 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.
The functional differential equation for formula_7 must be supplemented with the initial condition formula_22, where the "classical action" formula_2 describes the physics at the microscopic ultraviolet scale formula_24. Importantly, in the infrared limit formula_25 the full effective action formula_26 is obtained. In the Wetterich equation formula_27 denotes a supertrace which sums over momenta, frequencies, internal indices, and fields (taking bosons with a plus and fermions with a minus sign). The exact flow equation for formula_15 has a one-loop structure. This is an important simplification compared to perturbation theory, where multi-loop diagrams must be included. The second functional derivative formula_29 is the full inverse field propagator modified by the presence of the regulator formula_11.
The renormalization group evolution of formula_15 can be illustrated in the theory space, which is a multi-dimensional space of all possible running couplings formula_32 allowed by the symmetries of the problem. As schematically shown in the figure, at the microscopic ultraviolet scale formula_24 one starts with the initial condition formula_34.
As the sliding scale formula_8 is lowered, the flowing action formula_15 evolves in the theory space according to the functional flow equation. The choice of the regulator formula_11 is not unique, which introduces some scheme dependence into the renormalization group flow. For this reason, different choices of the regulator formula_11 correspond to the different paths in the figure. At the infrared scale formula_39, however, the full effective action formula_40 is recovered for every choice of the cut-off formula_11, and all trajectories meet at the same point in the theory space.
In most cases of interest the Wetterich equation can only be solved approximately. Usually some type of expansion of formula_7 is performed, which is then truncated at finite order leading to a finite system of ordinary differential equations. Different systematic expansion schemes (such as the derivative expansion, vertex expansion, etc.) were developed. 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.
Functional renormalization-group for Wick-ordered effective interaction.
Contrary to the flow equation for the effective action, this scheme is formulated for the effective interaction
formula_49
which generates n-particle interaction vertices, amputated by the bare propagators formula_50;
formula_51 is the "standard" generating functional for the n-particle Green functions.
The Wick ordering of effective interaction with respect to Green function formula_52 can be defined by
formula_53.
where formula_54 is the Laplacian in the field space. 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 formula_46 the Polchinskii equation
formula_56
where
<math>\Delta _}^{12}\mathcal {V}_\Lambda ^{(1)}\mathcal {V}_\Lambda ^{(2)}=\frac{1}{2}\left( {\frac