Construction of quantum Dirac observables and the emergence of WKB time

TL;DR

This paper presents a method to construct gauge-invariant Dirac observables in reparametrization-invariant systems, demonstrating how emergent WKB time arises and applying it to quantum cosmology and singularity avoidance.

Contribution

It introduces a new method for constructing gauge-invariant operators and explains the emergence of WKB time from invariant transition amplitudes.

Findings

Invariant operators evolve unitarily with respect to chosen time.

WKB time emerges from weak-coupling expansion of transition amplitudes.

Application to Bianchi I model shows singularity avoidance.

Abstract

We describe a method of construction of gauge-invariant operators (Dirac observables or ``evolving constants of motion'') from the knowledge of the eigenstates of the gauge generator in time-reparametrization invariant mechanical systems. These invariant operators evolve unitarily with respect to an arbitrarily chosen time variable. We emphasize that the dynamics is relational, both in the classical and quantum theories. In this framework, we show how the ``emergent Wentzel-Kramers-Brillouin time'' often employed in quantum cosmology arises from a weak-coupling expansion of invariant transition amplitudes, and we illustrate an example of singularity avoidance in a vacuum Bianchi I (Kasner) model.