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 is an analogue of the classical action functional and depends on the fields of a given theory. It includes all quantum and thermal fluctuations. Variation of yields exact quantum field equations, for example for cosmology or the electrodynamics of superconductors. Mathematically, 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 , however, is difficult to obtain. FRG provides a practical tool to calculate employing the renormalization group concept.

The central object in FRG is a scale-dependent effective action functional often called average action or flowing action. The dependence on the RG sliding scale is introduced by adding a regulator (infrared cutoff) to the full inverse propagator . Roughly speaking, the regulator decouples slow modes with momenta by giving them a large mass, while high momentum modes are not affected. Thus, includes all quantum and statistical fluctuations with momenta . The flowing action obeys the exact functional flow equation

derived by Christof Wetterich and Tim R. Morris in 1993. Here denotes a derivative with respect to the RG scale at fixed values of the fields. The functional differential equation for must be supplemented with the initial condition , where the "classical action" describes the physics at the microscopic ultraviolet scale . Importantly, in the infrared limit the full effective action is obtained. In the Wetterich equation 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 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 is the full inverse field propagator modified by the presence of the regulator .

The renormalization group evolution of can be illustrated in the theory space, which is a multi-dimensional space of all possible running couplings allowed by the symmetries of the problem. As schematically shown in the figure, at the microscopic ultraviolet scale one starts with the initial condition .

As the sliding scale is lowered, the flowing action evolves in the theory space according to the functional flow equation. The choice of the regulator is not unique, which introduces some scheme dependence into the renormalization group flow. For this reason, different choices of the regulator correspond to the different paths in the figure. At the infrared scale , however, the full effective action is recovered for every choice of the cut-off , 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 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.

Aspects of functional renormalization

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

which generates n-particle interaction vertices, amputated by the bare propagators ; is the "standard" generating functional for the n-particle Green functions.

The Wick ordering of effective interaction with respect to Green function can be defined by

.

where 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 the Polchinskii equation

takes the form of the Wick-ordered equation

where

Applications

The method was applied to numerous problems in physics, e.g.:

See also

References

Papers

    Pedagogic reviews

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