Bogoliubov Renormalization Group and Symmetry of Solution 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.

Contribution to the proceedings of conference "RG 2000" (Taxco, Mexico, Jan. 1999). To be published in Physics Reports

References in corpus (1)

• Functional self-similarity and renormalization group symmetry in mathematical physics

Cited by in corpus (19)

• An Introduction to the Nonperturbative Renormalization Group
• Constructive Methods of Invariant Manifolds for Kinetic Problems
• On the running coupling constant in QCD
• Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality
• Dark matter clustering: a simple renormalization group approach
• A hint of renormalization
• Can Renormalization Group Flow End in a Big Mess?
• How to generate a significant effective temperature for cold dark matter, from first principles
• Renormalization group inspired autonomous equations for secular effects in de Sitter space
• IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation
• Scale-Dependent Functions, Stochastic Quantization and Renormalization
• Systematic compactification of the two-channel Kondo model. III. Extended field-theoretic renormalization group analysis
• Approximate Lie symmetries and singular perturbation theory
• Novel Sets of Coupling Expansion Parameters for low-energy pQCD
• Do reductionist cures select for holistic diseases? Adaptive chronic infection, structured stress, and medical magic bullets
• Meta-Structural properties in Collective phenomena
• Consciousness, cognition, and context: extending the global neuronal workspace model
• Steps of the meta-structures project to model general processes of emergence (1)
• Exact and approximate symmetries for light propagation equations with higher order nonlinearity