A \(C^*\)-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking
type: Item
instance of: scholarly article
title: A \(C^*\)-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking
zbMATH Open document ID: 0962.11031
author: Paula Tretkoff
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2000-02-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93727
full work available at URL: jtnb11-1.html
review text: As the author pedagogically explains, in a seminal work by \textit{J.-B. Bost} and \textit{A. Connes} [Sel. Math., New Ser. 1, 412-457 (1995; Zbl 0842.46040)], motivated by previous papers of Julia, these authors developed the idea that by displaying the Riemann zeta function as the partition function of a dynamical system with spontaneous symmetry breaking taking place at the pole of the zeta function, one can gain insight into the statistics of the primes of the field of rational numbers, by using the tools of quantum statistical mechanics. They constructed the dynamical system as a one-parameter automorphism group on an appropriate Hecke algebra and this programme has provided a motivation and a guide for attempts towards the solution of the Riemann hypothesis. A generalization of this work to arbitrary global fields, using semigroup cross products, for the class number 1 case, etc., have been proposed by different authors. In the paper, the author also extends to arbitrary number fields the construction of Bost and Connes of a \(C^*\)-dynamical system with spontaneous symmetry breaking and the Riemann zeta function as partition function. The advantage of the construction in the paper with respect to these other approaches comes from viewing, in contrast to those, the ideals rather than just the principal ideals as playing the same role as the positive integers do in the approach by Bost and Connes. The dynamical system constructed has a natural symmetry group which displays the phenomenon of spontaneous symmetry breaking at the pole of the Dedekind zeta function. In physical terms, this means that for inverse temperature \(\beta\) less than 1 the temperature is high enough to create disorder in the system, so that the equilibrium state is unique and invariant under the action of the symmetry group, a phase transition occuring at \(\beta=1\). Then, when the temperature is low enough the particles of the system start to align and the symmetry is broken. The symmet…
reviewed by: Emilio Elizalde
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 11Z05
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 81R40
zbMATH DE Number: 1370102
zbMATH Keywords: Bost-Connes theory
zbMATH Keywords: quantum theory
zbMATH Keywords: quantum statistical mechanics
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: partition function
zbMATH Keywords: spontaneous symmetry breaking
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: \(C^*\)-dynamical system
zbMATH Keywords: Dedekind zeta function
MaRDI profile type: Publication
OpenAlex ID: W1967603663
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DOI: 10.5802/JTNB.236
description: scientific article; zbMATH DE number 1370102