Gibbs states and their classical limit

TL;DR

This paper rigorously studies the classical limit of quantum Gibbs states using continuous bundles of C*-algebras and deformation quantization, demonstrating convergence to classical probability measures in various quantum systems.

Contribution

It establishes the existence of classical limits for Gibbs states in quantum systems via deformation quantization, including Schrödinger operators and quantum spin systems.

Findings

Classical limit of Gibbs states exists for Schrödinger operators as Planck's constant approaches zero.

The classical limit corresponds to a unique probability measure satisfying the classical KMS condition.

Results extend to mean-field quantum spin systems and large spin quantum systems.

Abstract

A continuous bundle of $C^*$-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that examines the convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold). In this manner we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schr\"{o}dinger operators in the regime where Planck's constant $\hbar$ appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS- condition. Subsequently, we conduct a similar study on the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states. Finally, a short section is devoted to single site quantum spin systems in the large spin limit.