Simple background-independent Hamiltonian quantum model

We study a formulation and probabilistic interpretation of a simple general-relativistic Hamiltonian quantum system. The system has no unitary evolution in background time. The quantum theory yields transition probabilities between measurable quantities (partial observables). These converge to the classical predictions in the $\overrightarrow{ħ}0$ limit. Our main tool is the kernel of the projector on the solutions of the Wheeler–deWitt equation, which we analyze in detail. It is a real quantity, which can be seen as a propagator that propagates “forward” as well as “backward” in a local parameter time. Individual quantum states, on the other hand, may contain only “forward propagating” components. The analysis sheds some light on the interpretation of background-independent transition amplitudes in quantum gravity.