Prefactorization algebras of superselection sectors

TL;DR

This paper connects superselection sectors in algebraic quantum field theory with prefactorization algebras, revealing their monoidal structures and geometric origins under standard assumptions.

Contribution

It introduces a prefactorization algebra framework for superselection sectors, explaining their $ ext{E}_n$-monoidal structures via geometric and algebraic principles.

Findings

Superselection sectors form locally constant $C^*$-categorical prefactorization algebras.

The $ ext{E}_n$-monoidal structure arises from a combination of Haag duality and Lorentzian geometry.

Refinements to equivariant contexts under discrete groups are provided.

Abstract

This paper revisits the theory of superselection sectors in algebraic quantum field theory from the modern perspective of prefactorization algebras. Under the standard assumptions of Haag duality and a locally faithful vacuum representation, it is shown that every AQFT defined over a filtered orthogonal category of spacetime regions, satisfying some mild additional geometric hypotheses, has an associated locally constant $C^\ast$-categorical prefactorization algebra of superselection sectors over the same orthogonal category. In the case of double cones in the $(n\geq 2)$-dimensional Minkowski spacetime, our approach provides a conceptual explanation for the well-known $\mathbb{E}_n$-monoidal structure on the $C^\ast$-category of superselection sectors as the combination, through Dunn-Lurie additivity $\mathbb{E}_n\simeq \mathbb{E}_1\otimes \mathbb{E}_{n-1}$, of the familiar $\mathbb{E}_1$-monoidal structure from Haag duality and an $\mathbb{E}_{n-1}$-monoidal structure from Lorentzian geometry. A refinement of our results to equivariant contexts under a discrete group $G$ is also provided.