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Double Categorical Approaches to AQFT I: Axiomatic Setup

Khyathi Komalan

TL;DR

This work formulates Algebraic Quantum Field Theory (AQFT) as a double categorical structure to separate region-inclusion data from inter-region correlation data. It introduces the spacetime double category $\mathbf{Mink}(M)$ and the von Neumann double category $\mathbf{vNA}$, and defines a pseudo double functor $\mathcal{F}_{\mathcal{A}}: \mathbf{Mink}(M) \to \mathbf{vNA}$ whose vertical part recovers the Haag--Kastler net while its square data encode coherence between isotony and correlation transport via Connes fusion and standard-form intertwiners $L^2(\mathcal{A}(j))$. Haag–Kastler axioms are reformulated within this framework, and representative examples illustrate the construction and the explicit square-level compatibility that is invisible in the traditional $1$-categorical setting. The paper also develops structural tools (e.g., gamma-generation, decorated horizontalization, and adjunctions) to manage the double-category data and outlines Part II’s program to extend to type I/II/III von Neumann settings with tracial refinements. The approach provides a principled, functorial interface for tracking both region-based restriction/extension and inter-regional correlations, with potential impact on modular theory and higher-categorical AQFT.

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.

Double Categorical Approaches to AQFT I: Axiomatic Setup

TL;DR

This work formulates Algebraic Quantum Field Theory (AQFT) as a double categorical structure to separate region-inclusion data from inter-region correlation data. It introduces the spacetime double category and the von Neumann double category , and defines a pseudo double functor whose vertical part recovers the Haag--Kastler net while its square data encode coherence between isotony and correlation transport via Connes fusion and standard-form intertwiners . Haag–Kastler axioms are reformulated within this framework, and representative examples illustrate the construction and the explicit square-level compatibility that is invisible in the traditional -categorical setting. The paper also develops structural tools (e.g., gamma-generation, decorated horizontalization, and adjunctions) to manage the double-category data and outlines Part II’s program to extend to type I/II/III von Neumann settings with tracial refinements. The approach provides a principled, functorial interface for tracking both region-based restriction/extension and inter-regional correlations, with potential impact on modular theory and higher-categorical AQFT.

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 -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.
Paper Structure (6 sections, 14 theorems, 167 equations)

This paper contains 6 sections, 14 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Categorical Treatments of AQFT
  3. Double Categorical AQFT
  4. Structural Properties of the AQFT Double Functor
  5. Examples
  6. Conclusion and Preview of Part II

Key Result

Theorem 3.1

It is proven that and that $\gamma\mathbb{C}$ is contained in every sub-double-category $\mathbb{D}\subseteq \mathbb{C}$ satisfying In particular, $\gamma\mathbb{C}$ is the minimal sub-double-category of $\mathbb{C}$ with the same decorated horizontalization orendain2021freeglobularlygenerateddouble.

Theorems & Definitions (52)

  • Definition 1.1: Double Category
  • Definition 3.1: Pseudo double category
  • Definition 3.2: Globular square
  • Definition 3.3: $0$-marked squares
  • Definition 3.4: Globularly generated double category
  • Definition 3.5: $\gamma$-construction
  • Definition 3.6: Decorated horizontalization $H^\ast$
  • Theorem 3.1: Minimality of $\gamma\mathbb{C}$ at fixed $H^\ast$
  • Theorem 3.2: Adjunction $Q \dashv H^\ast$ on globularly generated double categories
  • Definition 3.7: Saturation
  • ...and 42 more