A \(C^*\)-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking. (English) Zbl 0962.11031

Cohen, Paula B.

J. Théor. Nombres Bordx. 11, No. 1, 15-30 (1999).

As the author pedagogically explains, in a seminal work by J.-B. Bost and 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.

Reviewer: Emilio Elizalde (Barcelona)

MSC:

11M41	Other Dirichlet series and zeta functions
46L55	Noncommutative dynamical systems
11Z05	Miscellaneous applications of number theory
82C10	Quantum dynamics and nonequilibrium statistical mechanics (general)
81R40	Symmetry breaking in quantum theory

Keywords:

Bost-Connes theory; quantum theory; quantum statistical mechanics; Riemann zeta function; partition function; spontaneous symmetry breaking; Riemann hypothesis; \(C^*\)-dynamical system; Dedekind zeta function

Citations:

Zbl 0842.46040; Zbl 0692.58060

References:

[1]	Arledge, J., Laca, M. and Raeburn, I., Semigroup crossed products and Hecke algebras arising from number fields, Doc. Mathematica2 (1997) 115-138. · Zbl 0940.47062
[2]	Bost, J-B. and Connes, A., Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series), 1 (1995), 411-457. · Zbl 0842.46040
[3]	Connes, A., Formule de trace en géométrie non commutative et hypothèse de Riemann, C. R. Acad. Sci. Paris t.323, Série 1 (Analyse), (1996) 1231-1236. · Zbl 0864.46042
[4]	Harari, D. and Leichtnam, E., Extension du phénomène de brisure spontanée de symétrie de Bost-Connes au cas des corps globaux quelconques, Selecta Mathematica, New Series 3 (1997), 205-243. · Zbl 0924.46051
[5]	Julia, B., Statistical Theory of Numbers, in Number Theory and Physics, Les Houches Winter School, J-M. Luck, P. Moussa, M. Waldschmidt eds., (1990), 276-293. · Zbl 0727.11033
[6]	Laca, M., Semigroups of * -endomorphisms, Dirichlet series and Phase Transitions, J. Functional Analysis, to appear. · Zbl 0957.46039
[7]	Laca, M. and Raeburn, I., Semigroup crossed products and the Toeplitz algebras of non-abelian groups, J. Functional Analysis139 (1996), 415-440. · Zbl 0887.46040
[8]	Laca, M. and Raeburn, I., A semigroup crossed product arising in number theory, J. London Math. Soc., to appear. · Zbl 0922.46058
[9]	Lang, S., Algebraic Number Theory, Second Edition, Springer-Verlag, BerlinHeidelbergNew YorkTokyo, 1994. · Zbl 0811.11001
[10]	Neukirch, J., Class Field Theory, Grund. der math. Wissen. 280, Springer-Verlag, BerlinHeidelbergNew YorkTokyo, 1980. · Zbl 0587.12001
[11]	Nica, A., C* - algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory27 (1992), 17-52. · Zbl 0809.46058