Equilibrium states for the infinite ideal Bose gas

Abstract

We construct a set of equilibrium states for the ideal Bose gas at temperature β−1 and chemical potential μ. Our choice for the equilibrium states is based on the Kubo−Martin−Schwinger conditions. In particular, we define ℰβ,μ to be the set of states which satisfy the KMS conditions suitably defined. In the condensed phase a certain subset Δβ,μ of ℰβ,μ correspond to states whose mean local densities are not uniformly bounded with respect to the volume. We, therefore, propose that ℰβ,μ Δβ,μ ≡ ?β,μ corresponds to the set of equilibrium states. Then it is shown that ?β,μ contains, as special cases, the equilibrium states obtained via thermodynamical limit arguments by Araki−Woods and Lewis−Pulé. The extremal elements of the convex set ℰβ,μ are obtained explicitly.