scholarly journals Functional self-similarity and renormalization group symmetry in mathematical physics

1999 ◽  
Vol 121 (1) ◽  
pp. 1315-1332 ◽  
Author(s):  
V. F. Kovalev ◽  
D. V. Shirkov
Keyword(s):  
Renormalization Group ◽  
Mathematical Physics ◽  
Group Symmetry ◽  
Self Similarity

scholarly journals RENORMALIZATION GROUP SYMMETRY AND SOPHUS LIE GROUP ANALYSIS

1995 ◽  
Vol 06 (04) ◽  
pp. 503-512 ◽  
Author(s):  
DMITRIJ V. SHIRKOV
Keyword(s):  
Lie Group ◽  
Group Analysis ◽  
Mathematical Physics ◽  
The Self ◽  
Group Symmetry ◽  
Lie Group Analysis ◽  
Short Discussion ◽  
Self Similarity ◽  
Boundary Problems ◽  
Solution Property

We start with a short discussion of the content of a term Renormalisation Group in modern use. By treating the underlying solution property as a reparametrisation symmetry, we relate it with the self-similarity symmetry well-known in mathematical physics and explain the notion of Functional Self-similarity. Then we formulate a program of constructing a regular approach for discovering RG-type symmetries in different problems of mathematical physics. This approach based upon S. Lie group analysis allows one to analyse a wide class of boundary problems for different type of equations. Several examples are mentioned.

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

scholarly journals The Bogoliubov renormalization group and solution symmetry in mathematical physics

2001 ◽  
Vol 352 (4-6) ◽  
pp. 219-249 ◽  
Author(s):  
Dmitrij V Shirkov ◽  
Vladimir F Kovalev
Keyword(s):  
Renormalization Group ◽  
Mathematical Physics

scholarly journals On scaling and regular variation

2015 ◽  
Vol 97 (111) ◽  
pp. 161-174 ◽  
Author(s):  
N.H. Bingham
Keyword(s):  
Mathematical Analysis ◽  
Regular Variation ◽  
Mathematical Physics ◽  
Self Similarity ◽  
Scaling Arguments

We survey scaling arguments, both asymptotic (involving regular variation) and exact (involving self-similarity), in various areas of mathematical analysis and mathematical physics.

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