Introduction

Abstract

In the previous chapter we examined various applications of the formalism of *-algebras to quantum statistical mechanics. These applications principally concerned equilibrium properties of macroscopic systems and fell into one of two categories. First, we analyzed thermodynamic limit phenomena of specific particle models by use of the Gibbs ansatz for the equilibrium states. Second, we examined the structure of a set of states, the KMS states, which appeared appropriate for the description of equilibrium. The first of these approaches was developed for the ideal quantum gases in Sections 5.1 and 5.2, while the second was contained in Sections 5.3 and 5.4. It should be reemphasized that the thermodynamic limits of the Gibbs equilibrium states of the ideal Fermi gas are exactly the KMS states of the C*-system formed by the CAR algebra and the corresponding free evolution. For the Bose gas, however, the situation is more complex because the free evolution τ is not a strongly continuous group of *-automorphisms of the CCR algebra 𝕬 and hence (𝕬, τ) does not form a C*-system. Thus, it is not evident that one can use a global C*-structure to characterize the set of equilibrium states. Nevertheless, the Gibbs equilibrium states ω are faithful states of the CCR algebra 𝕬, the free evolution τ has extensions τ̂ to the corresponding von Neumann algebras π _ω (𝕬)″, and the (π _ω (𝕬)″, τ̂) form W*-dynamical systems for which ω is τ̂-KMS.