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 the smooth interpolation 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 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 q\lesssim 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 q\gtrsim k} . The flowing action Γ k {\displaystyle \Gamma _{k}} obeys the exact functional flow equation
k ∂ k Γ k = 1 2 STr k ∂ k R k ( Γ k ( 1 , 1 ) + R k ) − 1 , {\displaystyle k\,\partial _{k}\Gamma _{k}={\frac {1}{2}}{\text{STr}}\,k\,\partial _{k}R_{k}\,(\Gamma _{k}^{(1,1)}+R_{k})^{-1},}
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