A Unified Scheme for Generalized Sectors based on Selection Criteria --Order parameters of symmetries and of thermality and physical meanings of adjunctions--

TL;DR

This paper introduces a unified framework for analyzing generalized superselection sectors in quantum physics using selection criteria, channels, and order parameters, extending existing methods to cases with spontaneous symmetry breaking.

Contribution

It extends the Doplicher-Roberts construction to spontaneous symmetry breaking scenarios and clarifies the geometric and physical meanings of order parameters and adjunctions.

Findings

Reformulation of sector structures with spontaneous symmetry breaking

Extension of Doplicher-Roberts method to new physical situations

Geometric interpretation of order parameters as classifying spaces

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 some basic aspects in measurement processes. This is achieved by the help of c-q and q-c channels: the former determines the states to be selected and to be parametrized by the order parameters, and the latter provides 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: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parametrized by the variable belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.