Construction of quantum Dirac observables and the emergence of WKB time

Understand

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.