Quantum Reference Frames via Transition Amplitudes in Timeless Quantum Gravity

Abstract

We propose an algorithm of extracting Schrödinger theories under all viable physical time from the Einstein-Hilbert path integral, formulated as the timeless transition amplitudes $\hat{\mathbb{P}}:\mathbb{K}\to {\mathbb{K}}^{\ast }$ between the boundary states in a kinematic Hilbert space $\mathbb{K}$. Each of these Schrödinger 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 Schrödinger theory. The algorithm embodies the fundamental principle of quantum Cauchy surfaces, such that all the derived Schrödinger 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.

Updated journal version

References in corpus (10)

• Quantum Nature of the Big Bang: Improved dynamics
• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• From black holes to white holes: a quantum gravitational, symmetric bounce
• The Superspace of Geometrodynamics
• Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I
• Inhomogeneous Loop Quantum Cosmology: Hybrid Quantization of the Gowdy Model
• Quantum-gravitational effects on gauge-invariant scalar and tensor perturbations during inflation: The slow-roll approximation
• Cosmological perturbations in Hybrid Loop Quantum Cosmology: Mukhanov-Sasaki variables
• A path-integral approach to the problem of time
• Quantum Cauchy Surfaces in Canonical Quantum Gravity

Cited by in corpus (1)

• Conformally Invariant Brans-Dicke Loop Quantum Cosmology: A Quantum Geometric Model of Linking Theory