A note on Haag duality

TL;DR

This paper clarifies and simplifies the understanding of Haag duality in free scalar quantum field theory, connecting foundational works and elucidating the proof using Tomita-Takesaki modular theory.

Contribution

It provides a clear, self-contained exposition of Haag duality in different formulations and simplifies the proof using Tomita-Takesaki theory, making it more accessible.

Findings

Clarifies Haag duality in Segal and Weyl formulations

Provides a simplified proof using Tomita operator adjoint

Connects Araki's and EO's approaches in QFT

Abstract

Haag duality is a remarkable property in QFT stating that the commutant of the algebra of observables localized in some region of spacetime is exactly the algebra associated to the causally disconnected region. It is a strong condition on the local structure and has direct consequences on entanglement measures. It was first shown to hold for a free scalar field and causal diamonds by Araki in 1964 and later by many authors in different ways. In particular, Eckmann and Osterwalder (EO) used Tomita-Takesaki modular theory to give a direct proof. However, it is not straightforward to relate this proof to the works of Araki, since they rely on two forms of the canonical commutation relations (CCR), called Segal and Weyl formulations, while EO work as starting point assumes that duality holds in the so-called ``first quantization'' in the Weyl formulation. It is our purpose to first introduce the works of Araki in a more easy-to-read but still rigorous and self-contained fashion, and show how Haag duality is stated in the Segal and Weyl formulations and in both first and second quantizations (and their immediate combination). This permits to understand the setting of the EO proof of Haag duality. There is nothing essentially new in this manuscript, with the exception of what we consider a simplification of EO proof that uses the adjoint $S^*$ of the Tomita operator $S$ instead of introducing several auxiliary operators. We hope this note will be useful for those seeking to understand where Haag duality comes from in a free scalar QFT.