From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices

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dc.creatorConnes, Alain
dc.creatorMarcolli, Matilde
dc.date2004-04-06
dc.date.accessioned2026-07-07T05:07:13Z
dc.date.available2026-07-07T05:07:13Z
dc.descriptionThis is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field.
dc.description80 pages, LaTeX, 3 eps figures
dc.identifierhttps://arxiv.org/abs/math/0404128
dc.identifierhttp://arxiv.org/abs/math/0404128
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70771
dc.subjectNumber Theory
dc.subjectStatistical Mechanics
dc.subjectMathematical Physics
dc.subjectOperator Algebras
dc.subjectQuantum Algebra
dc.subject58B34, 46L55, 11F03, 11G18, 11M06, 82B26, 82B10
dc.titleFrom Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
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