Quantum Statistical Mechanics of $\mathbb{Q}$-lattices and noncommutative geometry

TL;DR

This paper explores the application of quantum statistical mechanics to ext{Q}-lattices and noncommutative geometry, focusing on the Bost-Connes and Connes-Marcolli systems and their number-theoretic implications.

Contribution

It introduces the quantum statistical mechanical frameworks of the Bost-Connes and Connes-Marcolli systems and relates them to number theory and noncommutative geometry.

Findings

The Bost-Connes system connects ext{Q}-lattice states to maximal abelian extensions of ext{Q}.

The Connes-Marcolli ext{GL}_2 system generalizes the Bost-Connes system.

These systems reveal deep links between quantum statistical mechanics and number theory.

Abstract

After recalling some basic notions of quantum statistical mechanics, we explain the Bost-Connes system that relates the structure of the maximal abelian extension of $\mathbb{Q}$ to the space of \kms states of a \cs-dynamical system. Afterwards, we study briefly the Connes-Marcolli $\text{GL}_2$-system as a generalization of the former system.