Invariant states in statistical mechanics

Abstract

Mathematically we consider aC*-algebra $$\mathfrak{A}$$ , acted upon by the groupT of space-translations, which has an asymptotic abelian property. We analyse invariant states over $$\mathfrak{A}$$ . 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 anE-state the discrete spectrum S D of space-translations is additive which gives rise to the classificationE I,E II, andE III corresponding to the three possibilities that S D contains one point, a lattice of points, or a set with accumulation points. AnE II-state can be uniquely decomposed into states (L-states) having a symmetry with respect to a closed subgroupT L of (S D and T L are reciprocal lattices).L-states have properties with respect toT L analogous to the properties ofE I-states with respect toT. The decomposition intoL-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.