The Bogolyubov renormalization group

Abstract

We begin with personal notes describing the atmosphere of "Bogoliubov renormalization group" birth. Then we expose the history of RG discovery in the QFT and of the RG method devising in the mid-fifties. The third part is devoted to proliferation of RG ideas into diverse parts of theoretical physics. We conclude with discussing the perspective of RG method further development and its application in mathematical physics. Moreover, this latter English-language publication ['Russian Math. Surveys', 49:5 (1994) 155-176], due to the translator's poor qualification both in subject terminology and Russian, contains a lot (around 50) of errors distorting the author's text.By these reasons the author decided to present corrected English text. It results from the editing of the RMS publication and is adequate to the Russian preprint P2-94-310.

Renormalization group derivation of phase equations

“…He called this property functional self-similarity (FSS). The notion is almost equivalent to a group structure of the old style RG [19,18] and is a generalization of the usual self-similarity related to power laws. In this sense, the RG analysis is interpreted as a method to construct a FSS symmetric solution.…”

“…The argument in this paragraph follows a review article by Shirkov [17]. The RG is a symmetry structure with respect to the alternation of the way of giving boundary values [17][18][19]. Let us see this property in our system.…”

Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

“…The first order equation now readsẍ 1 + x 1 = −A cos(t + θ), and we choose the solution in the following form, 7) which means that the first order initial value W 1 (t 0 ) = 0 so that the lowest order value W 0 (t 0 ) approximates the exact value as closely as possible. Similarly, the second order solution may be given by…”

Renormalization-group symmetries for solutions of nonlinear boundary value problems

“…Although the renormalization group method was originally formulated for quantum field problems, we can explain its core idea by an example of a planar problem of radi-ation transfer [12,16]. We assume that the half-space x > 0 is filled with homogeneous matter and a stationary stream of particles, characterized by a number α, is falling on the boundary x = 0 of this medium.…”