Interplay between Approximation Theory and Renormalization Group

The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameter...

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Bibliographic Details
Published in:Physics of particles and nuclei 2019-03, Vol.50 (2), p.141-209
Main Author: Yukalov, V. I.
Format: Article
Language:English
Subjects:
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
Particle and Nuclear Physics
PERTURBATION THEORY
Physics
Physics and Astronomy
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
RENORMALIZATION
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Summary:The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is Optimized Perturbation Theory now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is Self-Similar Approximation Theory allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.
ISSN:1063-7796
1531-8559
DOI:10.1134/S1063779619020047