# The Bisognano-Wichmann property on nets of standard subspaces, some sufficient conditions

## TL;DR

This paper explores the Bisognano-Wichmann property in quantum field theory, providing algebraic conditions for its validity in nets of standard subspaces and examining its relation to other properties like duality and the split property.

## Contribution

It introduces a sufficient algebraic condition called modularity for the Bisognano-Wichmann property, applicable to certain covariant nets, and discusses cases where the property fails.

## Key findings

- Modularity condition ensures Bisognano-Wichmann and Duality properties.
- The property holds for direct integrals of scalar massive and massless representations.
- A class of massive nets does not satisfy the Bisognano-Wichmann property.

## Abstract

We discuss the Bisognano-Wichmann property for local Poincar\'e covariant nets of standard subspaces. We give a sufficient algebraic condition on the covariant representation ensuring the Bisognano-Wichmann and Duality properties without further assumptions on the net called modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property. Furthermore, we give an outlook in the standard subspace setting on the relation between the Bisognano-Wichmann property and the Split property.