Double Categorical Approaches to AQFT I: Axiomatic Setup

TL;DR

This paper introduces a double categorical framework for algebraic quantum field theory (AQFT), enabling a more structured and compatible interplay between local algebra inclusions and bimodule transformations.

Contribution

It constructs a spacetime double category and a von Neumann algebra double category, formalizing AQFT as a pseudo double functor to address functoriality issues.

Findings

Built a spacetime double category and a von Neumann algebra double category.

Formulated Haag-Kastler axioms within the double categorical setup.

Provided examples illustrating the framework's applicability.

Abstract

In operator-algebraic AQFT one routinely moves back and forth between two kinds of structure: inclusions of local algebras coming from inclusions of regions, and bimodules/intertwiners that implement the standard $L^2$-based constructions used to compare and compose observables. The obstruction to making this interplay genuinely functorial is that there are two independent compositions (restriction along inclusions and fusion/transport along bimodules) and they must be compatible on commuting spacetime diagrams, which is exactly the situation a double category is designed to encode. Part I resolves this by building a spacetime double category and a von Neumann algebra double category inspired by previous work by Orendain, and by packaging an AQFT input as a pseudo double functor whose vertical part is the Haag-Kastler net and whose squares record the required compatibilities in a well-typed way forced by commutativity. We formulate the Haag-Kastler axioms in this setup, establish the coherence needed for the construction, and work out representative examples.