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.