Quantum reference frames via transition amplitudes in timeless quantum gravity

We propose an algorithm of extracting Schrodinger theories under all viable physical time from the Einstein-Hilbert path integral, formulated as the timeless transition amplitudes (P) over cap: K -> K* between the boundary states in a kinematic Hilbert space K. Each of these Schrodinger theories refers to a certain set of quantum degrees of freedom in K as a background, with their given values specifying moments of the physical time. Restricted to these specified background values, the relevant elements of (P) over cap are transformed by the algorithm into the unitary propagator of a corresponding reduced phase space Schrodinger theory. The algorithm embodies the fundamental principle of quantum Cauchy surfaces, such that all the derived Schrodinger theories emerge from one timeless canonical theory defined by (P) over cap as a rigging map, via the relational Dirac observables referring to the corresponding backgrounds. We demonstrate its application to a Friedmann-Robertson-Walker loop quantum cosmology (FRW loop quantum cosmology) model with a massless Klein-Gordon scalar field. Recovering the famous singularity-free quantum gravitational dynamics with the background of the scalar field, we also obtain in another reference frame a modified Klein-Gordon field quantum dynamics with the background of the spatial (quantum) geometry.