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Abstract

We propose an algorithm of extracting Schr\"odinger theories under all viable
physical time from the Einstein-Hilbert path integral, formulated as the
timeless transition amplitudes $\hat{\mathbb{P}}:\mathbb{K} \to \mathbb{K}^*$
between the boundary states in a kinematic Hilbert space $\mathbb{K}$. Each of
these Schr\"odinger theories refers to a certain set of quantum degrees of
freedom in $\mathbb{K}$ as a background, with their given values specifying
moments of the physical time. Restricted to these specified background values,
the relevant elements of $\hat{\mathbb{P}}$ are transformed by the algorithm
into the unitary propagator of a corresponding reduced phase space
Schr\"odinger theory. The algorithm embodies the fundamental principle of
quantum Cauchy surfaces, such that all the derived Schr\"odinger theories
emerge from one timeless canonical theory defined by $\hat{\mathbb{P}}$ as a
rigging map, via the relational Dirac observables referring to the
corresponding backgrounds. We demonstrate its application to a FRW loop quantum
cosmological 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.