Testing gravity with genuine quantum systems is on the agenda of many experimental groups worldwide. Such programmes presuppose a well defined scheme according to which the coupling of quantum matter to the classical gravitational field is determined. Such schemes do not exist. Ideally, they should be complete (i.e. account for all terms, say in a given PN-order) and generally applicable (i.e. without a priori restrictions on the quantum states the matter is assuming). But what are the hard principles on which such a scheme can be based?
In the first part I will report on some recent work in which the Hamiltonian of an electromagnetically bound 2-particle "atom” in a static Eddington–Robertson parametrised post-Newtonian gravitational field is “derived” to order (1/c)-squared. In the second part will I turn to the Schrödinger-Newton equation as semiclassical model for gravitational self-coupling of quantum systems.
