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

TL;DR

This paper uses noncommutative geometry to analyze Q-lattices, revealing how quantum statistical mechanics, modular Hecke algebras, and L-functions are interconnected within a unified framework, especially focusing on phase transitions and symmetry breaking.

Contribution

It provides a comprehensive description of phase transitions and symmetry breaking in a noncommutative geometric setting related to Q-lattices and modular forms.

Findings

Identification of multiple phase transitions in dimension two

Description of the classical Shimura variety as the zero-temperature limit

Connection between Galois actions and symmetry in the system

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.