The Bogoliubov renormalization group and solution symmetry in mathematical physics

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.

On the running coupling constant in QCD

“…Despite the little handiness of (113), one can still readily verify its limit α (2) an.it (0) = 1/β 0 ; for more general arguments concerning universality of the IR freezing value through all orders see for instance [82,83] and [87]. Moreover, the non-perturbative UV tail of analytized coupling can be estimated by expanding the two compensating terms in (113) …”

“…Evolution of QCD running coupling can be gained integrating the differential equation (2), that can be rewritten as…”

“…Such a maximum value can be shown to be given by the equation [72] α * OPT (b) The optimized third-order results for the couplingā = α s /π and the QCD correction R (3) to the ratio R e + e − . Also shown is the second-order resultR (2) . Quark thresholds are indicated by vertical lines.…”

“…By applying now the analytization recipe the main difficulty arises from the integration of the related spectral density [79] ρ (2) it (σ) = Im α (2) it (−σ) = 1 β 0…”

“…For details and bibliography on this subject we refer to standard books and reviews (see e.g. [1,2,15,31]). We have tried to emphasize the importance of exact solutions to renormalization group equations with a given number of loops in comparison with the usual iterative solutions appropriate for high energies.…”