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

Abstract

This 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.

80 pages, LaTeX, 3 eps figures

References in corpus (4)

• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Moduli space and structure of noncommutative 3-spheres
• Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem
• KMS states and complex multiplication

Cited by in corpus (16)

• p-Adic Mathematical Physics
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• On Bost-Connes type systems for number fields
• Quantum Statistical Mechanics, L-series and Anabelian Geometry
• Phase transition in the Connes-Marcolli GL2-system
• Very Basic Noncommutative Geometry
• A walk in the noncommutative garden
• Von Neumann algebras arising from Bost-Connes type systems
• Noncommutative geometry and motives: the thermodynamics of endomotives
• Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra
• The notion of dimension in geometry and algebra
• Crossed products by Hecke pairs I: *-algebraic theory
• Crossed products by Hecke pairs II: C*-completions
• Bost-Connes type systems for function fields
• Arithmetic structures on noncommutative tori with real multiplication
• Examples of noncommutative manifolds: complex tori and spherical manifolds