A Unified Scheme for Generalized Sectors based on Selection Criteria. I. Thermal situations, unbroken symmetries and criteria as classifying categorical adjunctions

Abstract

A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to the some basic aspects in measurement processes. This is achieved by the help of \textit{c} $\to$ \textit{q} and \textit{q} $\to$ \textit{c} channels, the former of which determines the states to be selected and to be parametrized by the order parameters defined as the spectrum of the centre constituting the superselection sectors, and the latter of which provides, as classifying maps, the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown, as will be reported in a succeeding paper.