Essential Properties of the Vacuum Sector for a Theory of Superselection Sectors

TL;DR

This paper extends the analysis of superselection sectors in quantum field theory without the spectrum condition, establishing a categorical framework and conditions for conjugates, with implications for curved spacetime theories.

Contribution

It generalizes DHR analysis by developing a categorical approach to superselection sectors without the spectrum condition, including criteria for conjugate sectors.

Findings

Established a symmetric tensor C*-category of bimodules for local excitations.

Provided necessary and sufficient conditions for the existence of conjugates.

Suggested the importance of the reference representation's properties in curved spacetime contexts.

Abstract

As a generalization of DHR analysis, the superselection sectors are studied in the case of absence of the spectrum condition for the reference representation. Considered a net of local observables in the 4-dimensional Minkowski spacetime, we show that it is possible to associate to a set of representations, that are local excitations of a reference one fulfilling Haag duality, a symmetric tensor $\mathrm{C}^*$-category $\Bim$ of bimodules of the net, with subobjects and direct sums. The existence of conjugates is studied introducing an equivalent formulation of the theory in terms of the presheaf associated with the observable net. This allows us to find, under the assumption that the local algebras in the reference representation are properly infinite, necessary and sufficient conditions for the existence of conjugates. Moreover, we present several results that suggest how the mentioned assumption on the reference representation can be considered essential also in the case of theories in curved spacetimes.