Renormgroup Symmetries

Abstract

The notion “RenormGroup Symmetry” (RGS) originated in mathematical physics at the beginning of the nineties [8] (see also reviews [12, 13, 14]) as a result of joining up the notion “symmetry group” as applied to differential equations and “renormgroup”, i.e., symmetry group of a particular solution. In its turn, the notion of Renormalization Group, or briefly RenormGroup (RG), was imported to mathematical physics from theoretical physics, namely from quantum field theory [16, 17, 4, 1]. In quantum field theory renormgroup was based upon finite (Dyson) transformations and appeared as a continuous group in a usual mathematical sense. This group was then successfully used in developing a regular method of improving approximate perturbation solutions, the renormgroup method [2, 3]. In transferring renormgroup concept to mathematical physics problems [8] the aim was ultimately the same as in quantum field theory — to improve the perturbation theory solutions and to correct the behavior of these solutions in the vicinity of a singularity. In mathematical physics we usually deal with the problems, based on systems of differential equations or integro-differential equations, the symmetry of which can be found using computational methods of modern group analysis. In problems of mathematical physics this feature appeared as decisive in creating renormgroup algorithm (see e.g. [12, 13, 14]) which has united renormgroup ideology of quantum field theory with a regular way of symmetry construction for solutions of boundary value problems.