Quantum Reference Frames via Transition Amplitudes in Timeless Quantum Gravity

TL;DR

This paper introduces an algorithm to derive Schr"odinger theories with physical time from the timeless path integral in quantum gravity, demonstrated on a cosmological model, revealing new insights into quantum dynamics and reference frames.

Contribution

It presents a novel algorithm to extract Schr"odinger theories from the Einstein-Hilbert path integral using quantum Cauchy surfaces, unifying different quantum reference frames in a timeless quantum gravity setting.

Findings

Successfully applied to a loop quantum cosmology model.

Recovered singularity-free quantum gravitational dynamics.

Derived modified Klein-Gordon dynamics with quantum geometric backgrounds.

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.