An algebraic condition for the Bisognano-Wichmann Property

TL;DR

This paper establishes an algebraic criterion ensuring the Bisognano-Wichmann property for certain quantum field theory nets, showing it is weaker than the Split property and identifying cases where it fails.

Contribution

It introduces a new algebraic condition on covariant representations that guarantees the Bisognano-Wichmann property without additional assumptions.

Findings

The condition applies to direct integrals of scalar massive and massless representations.

The Bisognano-Wichmann property is shown to be weaker than the Split property in these cases.

A class of massive nets not satisfying the property is also identified.

Abstract

The Bisognano-Wichmann property for local, Poincar\'e covariant nets of standard subspaces is discussed. We present a sufficient algebraic condition on the covariant representation ensuring Bisognano-Wichmann and Duality properties without further assumptions on the net. Our modularity condition holds for direct integrals of scalar massive and massless representations. We conclude that in these cases the Bisognano-Wichmann property is much weaker than the Split property. Furthermore, we present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property.