The Bisognano-Wichmann property for non-unitary Wightman conformal field theories

TL;DR

This paper establishes a non-unitary version of the Bisognano-Wichmann property for nets of smeared Wightman fields, extending algebraic quantum field theory tools beyond traditional Hilbert space methods.

Contribution

It develops a broadly applicable non-unitary framework for the Bisognano-Wichmann property using Wightman axioms, bypassing Hilbert space techniques.

Findings

Proves a non-unitary version of the Bisognano-Wichmann property.

Demonstrates Haag duality for nets of smeared Wightman fields.

Extends algebraic quantum field theory methods to non-unitary theories.

Abstract

The Bisognano-Wichmann and Haag duality properties for algebraic quantum field theories are often studied using the powerful tools of Tomita-Takesaki modular theory for nets of operator algebras. In this article, we study analogous properties of nets of algebras generated by smeared Wightman fields, for potentially non-unitary theories. In light of recent work constructing Wightman field theories for (non-unitary) M\"obius vertex algebras, we obtain a broadly applicable non-unitary version of the Bisognano-Wichmann property. In this setting we do not have access to the traditional tools of Hilbert space functional analysis, like functional calculus. Instead, results analogous to those of Tomita-Takesaki theory are derived `by hand' from the Wightman axioms. As an application, we demonstrate Haag duality for nets of smeared Wightman fields.