Abstract
An algebraic quantum field theory (AQFT) may be expressed as a functor from a category of spacetimes to a category of algebras of observables; different choices of domain and codomain categories are suitable for different types of QFT.
For the domain, a generic category $\mathsf{C}$ whose objects admit interpretation as spacetimes is not necessarily viable; often, additional constraints on the morphisms of $\mathsf{C}$ must be imposed. We introduce \emph{disjointness relations} on categories, a generalisation of the orthogonality relations of Benini, Schenkel and Woike. In any category $\mathsf{C}$ equipped with a disjointness relation, we identify a subcategory $\mathsf{D}_\mathsf{C}$ which we propose to be suitable as the domain of an AQFT. We verify that when $\mathsf{C}$ is the category of all globally hyperbolic spacetimes of dimension $d+1$ and all local isometries, equipped with the disjointness relation of spacelike separation, the specified subcategory $\mathsf{D}_\mathsf{C}$ is the commonly-used domain $\mathsf{Loc}_{d+1}$ of relativistic AQFTs. By identifying appropriate chiral disjointness relations, we construct a category $\chi\mathsf{Loc}$ suitable as domain for chiral conformal field theories (CFTs) in two dimensions. We compare this to an established AQFT formulation of chiral CFTs, and show that any chiral CFT expressed in the established formulation induces one defined on $\chi\mathsf{Loc}$.
Codomain categories for AQFT are typically chosen to be categories of complex $*$-algebras, most often also equipped with a $C^*$-norm. We apply a result of Alfsen and Shultz to demonstrate an equivalence of the category of complex $C^*$-algebras and the category of real LJB-algebras (Lie-Jordan-Banach algebras). From this, we show that AQFTs may equivalently be formulated using real Lie-Jordan algebras of observables, and argue that this reformulation describes physically relevant structure of observables more directly than its complex $*$-algebra counterpart.