KMS states and their classical limit

Christiaan J. F. van de Ven

Published 2022-11-03Version 1

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 formalization in which convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold) is studied. 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, a similar study is conducted for the free energy in the classical limit of mean-field quantum spin systems in the regime of large particles, and the existence of the classical limit of the relevant Gibbs states is discussed.