Intrinsic irreversibility of quantum systems with diagonal singularity

The work of the Brussels-Austin group on irreversibility over the last years has shown that quantum large Poincaré systems with diagonal singularity lead to an extension of quantum theory beyond the conventional Hilbert space framework and logic. We characterize the algebra of observables, the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity. We illustrate the general ideas for the Friedrichs model