Construction of quantum Dirac observables and the emergence of WKB time

TL;DR

The paper addresses the problem of time in generally covariant quantum theories by constructing gauge-invariant Dirac observables through a canonical, FP-inspired framework and a Rieffel-induced inner product. It develops a systematic classical-quantum framework for relational dynamics, and demonstrates how evolving constants yield unitary evolution on the physical Hilbert space. The relativistic particle and Kasner cosmology serve as detailed exemplars, showing how WKB time emerges in the nonrelativistic limit and how quantum singularities can be avoided. Overall, the work clarifies the connection between Dirac observables and semiclassical time, offering a concrete path toward relational, gauge-invariant quantum cosmology.

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.