Scale-Dependent Functions, Stochastic Quantization and Renormalization

Scale-Dependent Functions, Stochastic Quantization and Renormalization

We consider a possibility to unify the methods of regularization, such as the renormalization group method, stochastic quantization etc., by the extension of the standard field theory of the square-integrable functions φ(b) ∊ L²(Rd) to the theory of functions that depend on coordinate b and resoluti...

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Date:2006
Main Author: Altaisky, M.V.
Format: Article
Language:English
Published: Інститут математики НАН України 2006
Series:Symmetry, Integrability and Geometry: Methods and Applications
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Scale-Dependent Functions, Stochastic Quantization and Renormalization / M.V. Altaisky // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 27 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling oai:nasplib.isofts.kiev.ua:123456789-1461802025-02-23T17:17:50Z Scale-Dependent Functions, Stochastic Quantization and Renormalization Altaisky, M.V. We consider a possibility to unify the methods of regularization, such as the renormalization group method, stochastic quantization etc., by the extension of the standard field theory of the square-integrable functions φ(b) ∊ L²(Rd) to the theory of functions that depend on coordinate b and resolution a. In the simplest case such field theory turns out to be a theory of fields φa(b,·) defined on the affine group G: x′ = ax+b, a > 0, x, b ∊ Rd, which consists of dilations and translation of Euclidean space. The fields φa(b,·) are constructed using the continuous wavelet transform. The parameters of the theory can explicitly depend on the resolution a. The proper choice of the scale dependence g = g(a) makes such theory free of divergences by construction The author is thankful to the referee for useful comments and references. 2006 Article Scale-Dependent Functions, Stochastic Quantization and Renormalization / M.V. Altaisky // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 27 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37E20; 42C40; 81T16; 81T17 https://nasplib.isofts.kiev.ua/handle/123456789/146180 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We consider a possibility to unify the methods of regularization, such as the renormalization group method, stochastic quantization etc., by the extension of the standard field theory of the square-integrable functions φ(b) ∊ L²(Rd) to the theory of functions that depend on coordinate b and resolution a. In the simplest case such field theory turns out to be a theory of fields φa(b,·) defined on the affine group G: x′ = ax+b, a > 0, x, b ∊ Rd, which consists of dilations and translation of Euclidean space. The fields φa(b,·) are constructed using the continuous wavelet transform. The parameters of the theory can explicitly depend on the resolution a. The proper choice of the scale dependence g = g(a) makes such theory free of divergences by construction
format Article
author Altaisky, M.V.
spellingShingle Altaisky, M.V.
Scale-Dependent Functions, Stochastic Quantization and Renormalization
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Altaisky, M.V.
author_sort Altaisky, M.V.
title Scale-Dependent Functions, Stochastic Quantization and Renormalization
title_short Scale-Dependent Functions, Stochastic Quantization and Renormalization
title_full Scale-Dependent Functions, Stochastic Quantization and Renormalization
title_fullStr Scale-Dependent Functions, Stochastic Quantization and Renormalization
title_full_unstemmed Scale-Dependent Functions, Stochastic Quantization and Renormalization
title_sort scale-dependent functions, stochastic quantization and renormalization
publisher Інститут математики НАН України
publishDate 2006
citation_txt Scale-Dependent Functions, Stochastic Quantization and Renormalization / M.V. Altaisky // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 27 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT altaiskymv scaledependentfunctionsstochasticquantizationandrenormalization
first_indexed 2025-07-22T04:13:29Z
last_indexed 2025-07-22T04:13:29Z
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