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

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.