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A geometric perspective on Algebraic Quantum Field Theory

Vincenzo Morinelli

TL;DR

This work develops a Lie-theoretic, geometric framework for Algebraic Quantum Field Theory (AQFT) built around Euler elements and abstract wedges. By extending the Brunetti–Guido–Longo (BGL) net construction to abstract wedge spaces on causal homogeneous manifolds, it links standard-subspace and Von Neumann algebra nets to wedge symmetries via $J_W$ and $\Delta_W$, anchored by the Bisognano–Wichmann property. A central result, the Euler Element Theorem, shows that BW and a localization regularity condition force the wedge generator to be an Euler element, enabling a unified treatment of regularity, localizability, and wedge-type analysis, and leading to type III$_1$ wedge algebras in this generalized setting. The approach yields nontrivial generalizations beyond second quantization, connecting AQFT with Hermitian Lie algebras, causal homogeneous spaces, and conformal/circle models, with implications for representation theory and modular theory in quantum field theory.

Abstract

In this paper we give a streamlined overview of some of the recent constructions provided with K.-H. Neeb, G. Ólafsson and collaborators for a new geometric approach to Algebraic Quantum Field Theory (AQFT). Motivations, fundamental concepts and some of the relevant results about the abstract structure of these models are here presented.

A geometric perspective on Algebraic Quantum Field Theory

TL;DR

This work develops a Lie-theoretic, geometric framework for Algebraic Quantum Field Theory (AQFT) built around Euler elements and abstract wedges. By extending the Brunetti–Guido–Longo (BGL) net construction to abstract wedge spaces on causal homogeneous manifolds, it links standard-subspace and Von Neumann algebra nets to wedge symmetries via and , anchored by the Bisognano–Wichmann property. A central result, the Euler Element Theorem, shows that BW and a localization regularity condition force the wedge generator to be an Euler element, enabling a unified treatment of regularity, localizability, and wedge-type analysis, and leading to type III wedge algebras in this generalized setting. The approach yields nontrivial generalizations beyond second quantization, connecting AQFT with Hermitian Lie algebras, causal homogeneous spaces, and conformal/circle models, with implications for representation theory and modular theory in quantum field theory.

Abstract

In this paper we give a streamlined overview of some of the recent constructions provided with K.-H. Neeb, G. Ólafsson and collaborators for a new geometric approach to Algebraic Quantum Field Theory (AQFT). Motivations, fundamental concepts and some of the relevant results about the abstract structure of these models are here presented.
Paper Structure (17 sections, 2 theorems, 77 equations)

This paper contains 17 sections, 2 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wedge regions in some spacetimes
  4. Euler elements and abstract wedges
  5. Wedge domains in causal homogeneous spaces
  6. The geometry of nets of real subspaces
  7. Scalar free fields in Algebraic Quantum Field Theory
  8. Nets of standard subspaces
  9. The Brunetti--Guido--Longo (BGL) net
  10. Euler elements, Bisognano--Wichmann property and algebra types
  11. Non-modular covariant nets of standard subspaces
  12. The Euler Element Theorem
  13. Regularity and Localizability
  14. Regularity via generating positive cone
  15. Regularity in a semidirect product
  16. ...and 2 more sections

Key Result

Theorem 2.2

MN21 Let $(U,\mathcal{H})$ be an (anti-)unitary C-positive representation of $G_{\tau_h}$ where $h\in \mathcal{E}({\mathfrak g})$. Then the BGL net satisfies (HK1)-(HK8) in Definition def:ssn.

Theorems & Definitions (3)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1