Skeleton graph expansion of critical exponents in "cultural revolution" years

Kenneth Wilson's Nobel Prize winning breakthrough in the renormalization group theory of phase transition and critical phenomena almost overlapped with the violent "cultural revolution" years (1966–1976) in China. An unexpected chance in 1972 brought the author of these lines close to the Wilson–Fisher ϵ-expansion of critical exponents and eventually led to a joint paper with Lu Yu published entirely in Chinese without any English title and abstract. Even the original acknowledgment was deleted because of mentioning foreign names like Kenneth Wilson and Kerson Huang. In this article I will tell the 40-year old story as a much belated tribute to Kenneth Wilson and to reproduce the essence of our work in English. At the end, I give an elementary derivation of the Callan–Symanzik equation without referring to field theory.

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Skeleton graph expansion of critical exponents in “cultural revolution” years

Peregrinations from Physics to Phylogeny ◽

10.1142/9789814759090_0002 ◽

2016 ◽

pp. 3-27

Author(s):

Bailin Hao

Keyword(s):

Cultural Revolution ◽

Critical Exponents ◽

Graph Expansion ◽

Skeleton Graph

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Field theory: Perturbative expansion and summation methods

From Random Walks to Random Matrices ◽

10.1093/oso/9780198787754.003.0023 ◽

2019 ◽

pp. 451-470

Author(s):

Jean Zinn-Justin

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Renormalization Group ◽

Conformal Mapping ◽

Critical Exponents ◽

Perturbative Expansion ◽

Divergent Series ◽

Quantum Field ◽

Summation Methods ◽

Borel Transformation

Chapter 23 examines perturbative expansion and summation methods in field theory. In quantum field theory, all perturbative expansions are divergent series in the mathematical sense. This leads to a difficulty when the expansion parameter is not small. In the case of Borel summable series, using the knowledge of the large order behaviour, a number of summation techniques have been developed to derive convergent sequences from divergent series. Some methods apply directly on the series like Padé approximants or order–dependent mapping (the ODM method). Others involve first a Borel transformation, like the Padé–Borel method. The method of Borel transformation, suitably modified, followed by a conformal mapping, has been applied to renormalization group (RG) functions of the phi4 3 field theory and has led to precise values of critical exponents.

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Critical exponents by a skeleton graph expansion

Physics Letters A ◽

10.1016/0375-9601(73)90797-4 ◽

1973 ◽

Vol 44 (2) ◽

pp. 85-86 ◽

Cited By ~ 7

Author(s):

M.J. Stephen ◽

E. Abrahams

Keyword(s):

Critical Exponents ◽

Graph Expansion ◽

Skeleton Graph

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Kenneth Wilson — Renormalization and QCD

International Journal of Modern Physics A ◽

10.1142/s0217751x14300439 ◽

2014 ◽

Vol 29 (18) ◽

pp. 1430043 ◽

Cited By ~ 2

Author(s):

Franz J. Wegner

Keyword(s):

Field Theory ◽

Renormalization Group ◽

Critical Phenomena ◽

Gauge Invariance ◽

Great Pleasure ◽

Flow Equations ◽

Kenneth Wilson

Kenneth Wilson had an enormous impact on field theory, in particular on the renormalization group and critical phenomena, and on QCD. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and QCD, and flow equations and similarity renormalization.

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Complex Critical Exponents from Renormalization Group Theory of Earthquakes: Implications for Earthquake Predictions

Journal de Physique I ◽

10.1051/jp1:1995154 ◽

1995 ◽

Vol 5 (5) ◽

pp. 607-619 ◽

Cited By ~ 202

Author(s):

Didier Sornette ◽

Charles G. Sammis

Keyword(s):

Renormalization Group ◽

Group Theory ◽

Critical Exponents ◽

Renormalization Group Theory ◽

Earthquake Predictions

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Critical exponents and equation of state from series summation

Quantum Field Theory and Critical Phenomena ◽

10.1093/oso/9780198834625.003.0041 ◽

2021 ◽

pp. 975-991

Author(s):

Jean Zinn-Justin

Keyword(s):

Field Theory ◽

Phase Transition ◽

Superfluid Helium ◽

Critical Exponents ◽

Summation Method ◽

Perturbative Expansion ◽

Field Theories ◽

Summation Methods ◽

Universal Properties ◽

Borel Transformation

Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.

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