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The renormalization group: A probabilistic view

Группа перенормировки: вероятностная точка зрения

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Il Nuovo Cimento B (1971-1996)

Summary

Following a line of thought recently initiated by Bleher and Sinai, we show that renormalization group ideas can be viewed as natural generalizations of concepts and methods well known in probability theory. In particular a connection is established with limit theorems and the theory of stable distributions. Our point of view throws new light on the relationship between the Wilson and the Gell-Mann and Low approach.

Riassunto

Seguendo una linea di pensiero recentemente proposta da Bleher e Sinai, si mostra che l'idea di gruppo di rinormalizzazione può essere considerata come la naturale generalizzazione di concetti e metodi ben noti nella teoria della probabilità. In particolare si stabilisce un legame tra gruppo di rinormalizzazione, teoremi limite e teoria delle distribuzioni stabili. Il nostro punto di vista chiarisce e mette in nuova luce il rapporto tra il metodo di Wilson e quello di Gell-Mann e Low.

Резюме

Следуя методу Блехера и Синаи, мы показываем, что идеи группы перенормировки могут быть использованы, как естественное обобщение концепций и методов, хорошо известных в теории вероятностей. В частности, устанавливается связь с предельными теоремами и теорией устойчивых распределений. Предложенное рассмотрение проливает новый свет на соотношение между подходами Вильсона и Гелл-мана и Лоу.

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References

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  3. For example we do not agree with the statement contained inM. E. Fisher:The renormalization group in the theory of magnetism, preprint (1973): «It should be remarked that the relation [of the Wilson approach] to the renormalization group in field theory isnot as close as the use of the same word might suggest».

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  21. I am indebted to Dr.A. L. Stella for a stimulating discussion on this point.

  22. A more systematic approach is discussed byG. Gallavotti andH. J. F. Knops: to be published.

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Author notes

  1. G. Jona-Lasinio

    Present address: Istituto di Fisica dell'Università, Roma

Authors and Affiliations

  1. Istituto di Fisica dell'Università, Padova

    G. Jona-Lasinio

  2. Gruppo Nazionale di Struttura della Materia, Padova

    G. Jona-Lasinio

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  1. G. Jona-Lasinio
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Additional information

A preliminary version of this paper was presented in lectures at the University of Rome and at the regional meeting on phase transitions in Vienna on February 1974.

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Jona-Lasinio, G. The renormalization group: A probabilistic view. Nuov Cim B 26, 99–119 (1975). https://doi.org/10.1007/BF02755540

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