Year: 2018
Type: article
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 $\stackrel{^}{\mathbb{P}}:\text{ }\text{ }\mathbb{K}\ensuremath{\rightarrow}{\mathbb{K}}^{*}$ 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 $\stackrel{^}{\mathbb{P}}$ 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 $\stackrel{^}{\mathbb{P}}$ 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.
Authors: Jerzy Lewandowski, Chun-Yen Lin
Institution: University of Warsaw
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G4294669765, G5632371392
OpenAlex ID: W2767922757