Invariant states in statistical mechanics

Abstract

Mathematically we consider a C*-algebra {Mathematical expression}, acted upon by the group T of space-translations, which has an asymptotic abelian property. We analyse invariant states over {Mathematical expression}. Physically this programme can be considered as a kinematical study of equilibrium states in statistical mechanics. Each invariant state can be uniquely decomposed into elementary invariant states (E-states). These elementary states have, amongst other characteristics, the physical property that space-averages of local observables are constants in the corresponding representations. In an E-state the discrete spectrum SD of space-translations is additive which gives rise to the classification EI, EII, and EIII corresponding to the three possibilities that SD contains one point, a lattice of points, or a set with accumulation points. An EII-state can be uniquely decomposed into states (L-states) having a symmetry with respect to a closed subgroup TL of (SD and TL are reciprocal lattices). L-states have properties with respect to TL analogous to the properties of EI-states with respect to T. The decomposition into L-states is the inverse process of 'homogenizing' a lattice state by smearing it over a lattice distance. The mathematical methods which we employ have more general applications. Show more