A \(C^*\)-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking
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 symmetry group acts then on the compact convex set of equilibrium states, which are the \(\text{KMS}_\beta\) states.
scientific article; zbMATH DE number 1370102