Symmetry and equilibrium states

Abstract

Within the general framework ofC*-algebra approach to mathematical foundation of statistical mechanics, we prove a theorem which gives a natural explanation for the appearance of the chemical potential (as a thermodynamical parameter labelling equilibrium states) in the presence of a symmetry (under gauge transformations of the first kind). As a symmetry, we consider a compact abelian groupG acting as *-automorphisms of aC*-algebra$\mathfrak{A}$ (quasi-local field algebra) and commuting (elementwise) with the time translation automorphisms ϱ _t of$\mathfrak{A}$. Under a technical assumption which is satisfied by examples of physical interest, we prove that the set of all extremal ϱ _t -KMS states ϕ (pure phases) ofG-fixed-point subalgebra$\mathfrak{A}^G$ (quasi-local observable algebra) of$\mathfrak{A}$ satisfying a certain faithfulness condition is in one-to-one correspondence with the set of all extremalG-invariant ϱ _t ·α _t -KMS states ϕ⁻ of$\mathfrak{A}$ with α varying over one-parameter subgroups ofG (the specification of α being the specification of the chemical potential), where the correspondence is that the restriction of ϕ⁻ to$\mathfrak{A}^G$ is ϕ.