Gibbs states and their classical limit

Abstract

A continuous bundle of $C^{\ast }$-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 Schrödinger operators in the regime where Planck’s constant $\hslash$ 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.