Скачать или смотреть Functional renormalization group

In theoretical physics, functional renormalization group is an implementation of the renormalization group 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 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.


derived by Christof Wetterich in 1993 and Tim R. Morris in 1994. 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. 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.




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


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