Double Categories for Operator Algebras and AQFT

Algebraic quantum field theory (AQFT), introduced by Rudolf Haag and Daniel Kastler, provides a framework for assigning an algebra of observables to regions of spacetime [1]. Categorically, an AQFT may be described as a covariant functor from a category of spacetime regions — typically open, bounded regions of Minkowski spacetime with inclusions as morphisms — into a category of operator algebras [2],

$$𝒜:\text{Regions}⟶\text{Corresponding Operator Algebras},$$

satisfying certain properties known as the Haag-Kastler axioms.

Fixing a region, one restricts to the assignment $𝒪\mapsto 𝒜(𝒪)$, i.e. an AQFT net. In this way, the 1-functorial formulation makes precise how geometric inclusions and operator-algebraic structure interact.

However, thinking of an AQFT as a functor hides a second kind of movement. Namely, what happens when — rather than enlarging a region by an inclusion $U\subseteq V$ — we want to move the same region to a new place, by identifying $U$ with a geometrically equivalent copy sitting inside $V$ via an embedding $h:U\hookrightarrow V$? For this, we turn to double categories.

To construct an AQFT as a double functor, we introduce a double category of regions in Minkowski spacetime, with inclusions as vertical maps and admissible embeddings as horizontal maps, and take as codomain Juan Orendain’s globularly generated double category of von Neumann algebras (a special class of operator algebras on Hilbert spaces), whose vertical maps are $\ast$-homomorphisms and whose horizontal maps are bimodules [3]. In recent work, we show that such a double functor can be defined, reformulate the Haag-Kastler axioms in this framework, and provide examples of the construction [4].

In addition to this double-functorial formulation of AQFT, we discuss ongoing progress toward incorporating the von Neumann type classification into this framework. Type I algebras admit nonzero minimal projections, type II admit nonzero finite projections but no minimal ones, and type III admit no nonzero finite projections [5]. Building on Penneys’ 2-categorical framework for tracial von Neumann algebras [6], we outline a tracial/semifinite refinement of the codomain designed for type I/type II phenomena. We also indicate how the type III case suggests a further refinement incorporating modular-theoretic structure, which we plan to develop in subsequent work.