Covariance algebras in field theory and statistical mechanics

Abstract

Starting from aC*-algebra$\mathfrak{A}$ and a locally compact groupT of automorphisms of$\mathfrak{A}$ we construct a “covariance algebra”$\mathfrak{A}_1^T$ with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of$\mathfrak{A}$ i.e. *-representations of$\mathfrak{A}$ in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra$\mathfrak{A}_1^T \left( V \right)$ yielding the *-representations of$\mathfrak{A}$ in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of$\mathfrak{A}_1^T \left( V \right)$. Finally we consider the covariance algebra$\mathfrak{A}_1^T$ built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.