Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics

TL;DR

This paper explores the evolution and symmetry properties of the Renormalization Group in theoretical physics, emphasizing its connection to self-similarity and applying these concepts to boundary value problems and nonlinear optics.

Contribution

It introduces the concept of Functional Self-similarity as a generalization of RG symmetry and reviews recent progress in applying Lie group analysis to discover these symmetries.

Findings

RG symmetry relates to self-similarity in differential equations.

A regular approach using Lie group analysis helps identify RG=FS symmetries.

Application to nonlinear optics demonstrates practical utility.

Abstract

Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution. After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS). Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given. As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.