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 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 Γ {displaystyle Gamma } is an analogue of the classical action functional S {displaystyle S} and depends on the fields of a given theory. It includes all quantum and thermal fluctuations. Variation of Γ {displaystyle Gamma } yields exact quantum field equations, for example for cosmology or the electrodynamics of superconductors. Mathematically, Γ {displaystyle Gamma } 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 Γ {displaystyle Gamma } , however, is difficult to obtain. FRG provides a practical tool to calculate Γ {displaystyle Gamma } employing the renormalization group concept. The central object in FRG is a scale-dependent effective action functional Γ k {displaystyle Gamma _{k}} often called average action or flowing action. The dependence on the RG sliding scale k {displaystyle k} is introduced by adding a regulator (infrared cutoff) R k {displaystyle R_{k}} to the full inverse propagator Γ k ( 2 ) {displaystyle Gamma _{k}^{(2)}} . Roughly speaking, the regulator R k {displaystyle R_{k}} decouples slow modes with momenta q ≲ k {displaystyle qlesssim k} by giving them a large mass, while high momentum modes are not affected. Thus, Γ k {displaystyle Gamma _{k}} includes all quantum and statistical fluctuations with momenta q ≳ k {displaystyle qgtrsim k} . The flowing action Γ k {displaystyle Gamma _{k}} obeys the exact functional flow equation ∂ k Γ k = 1 2 STr ∂ k R k ( Γ k ( 2 ) + R k ) − 1 , {displaystyle partial _{k}Gamma _{k}={frac {1}{2}}{ ext{STr}},partial _{k}R_{k},(Gamma _{k}^{(2)}+R_{k})^{-1},} derived by Christof Wetterich and Tim R. Morris in 1993. Here ∂ k {displaystyle partial _{k}} denotes a derivative with respect to the RG scale k {displaystyle k} at fixed values of the fields.The functional differential equation for Γ k {displaystyle Gamma _{k}} must be supplemented with the initial condition Γ k → Λ = S {displaystyle Gamma _{k o Lambda }=S} , where the 'classical action' S {displaystyle S} describes the physics at the microscopic ultraviolet scale k = Λ {displaystyle k=Lambda } . Importantly, in the infrared limit k → 0 {displaystyle k o 0} the full effective action Γ = Γ k → 0 {displaystyle Gamma =Gamma _{k o 0}} is obtained. In the Wetterich equation STr {displaystyle { ext{STr}}} 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 Γ k {displaystyle Gamma _{k}} 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 Γ k ( 2 ) {displaystyle Gamma _{k}^{(2)}} is the full inverse field propagator modified by the presence of the regulator R k {displaystyle R_{k}} . The renormalization group evolution of Γ k {displaystyle Gamma _{k}} can be illustrated in the theory space, which is a multi-dimensional space of all possible running couplings { c n } {displaystyle {c_{n}}} allowed by the symmetries of the problem. As schematically shown in the figure, at the microscopic ultraviolet scale k = Λ {displaystyle k=Lambda } one starts with the initial condition Γ k = Λ = S {displaystyle Gamma _{k=Lambda }=S} . As the sliding scale k {displaystyle k} is lowered, the flowing action Γ k {displaystyle Gamma _{k}} evolves in the theory space according to the functional flow equation. The choice of the regulator R k {displaystyle R_{k}} is not unique, which introduces some scheme dependence into the renormalization group flow. For this reason, different choices of the regulator R k {displaystyle R_{k}} correspond to the different paths in the figure. At the infrared scale k = 0 {displaystyle k=0} , however, the full effective action Γ k = 0 = Γ {displaystyle Gamma _{k=0}=Gamma } is recovered for every choice of the cut-off R k {displaystyle R_{k}} , 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 Γ k {displaystyle Gamma _{k}} 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.