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

We discuss the Bisognano–Wichmann property for localPoincaré covariant nets of standard subspaces. We provide a sufficient algebraic condition on the covariant representation ensuring the Bisognano–Wichmann and the duality properties without further assumptions on the net. We call it 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 on the relation between the Bisognano–Wichmann property and the split property in the standard subspace setting.