Classical-quantum hybrid canonical dynamics and its difficulties with special and general relativity

We discuss the Hamiltonian hybrid coupling between a classical and a quantum subsystem. If applicable to classical gravity coupled to quantized matter, this hybrid theory might realize a captivating ``postquantum'' alternative to full quantum gravity. We summarize the nonrelativistic hybrid dynamics in improved formalism adequate to Hamiltonian systems. The mandatory decoherence and diffusion terms become divergent in special and general relativistic extensions. It is not yet known if any renormalization method might reconcile Markovian decoherence and diffusion with relativity. Postquantum gravity could previously only be realized in the Newtonian approximation. We argue that pending problems of the recently proposed general relativistic postquantum theory will not be solved if Markovian diffusion/decoherence are truly incompatible with relativity.