Beyond semiclassical time: dynamics in quantum cosmology

TL;DR

This paper reviews two approaches to defining Hilbert space and evolution in quantum cosmology models with local time-reparametrization invariance, highlighting their gauge fixing, unitarity, and relational interpretation.

Contribution

It compares invariant relational observables and perturbative Hilbert space construction methods in quantum cosmology, clarifying their gauge fixing and unitarity properties.

Findings

Both approaches use gauge-fixed inner products via Faddeev-Popov procedure.

Unitarity of effective Schrödinger evolution is established perturbatively.

A conditional probability interpretation of physical states is possible.

Abstract

We review two approaches to the definition of the Hilbert space and evolution in mechanical theories with local time-reparametrization invariance, which are often used as toy models of quantum gravity. The first approach is based on the definition of invariant relational observables, whereas the second formalism consists of a perturbative construction of the Hilbert space and a weak-coupling expansion of the Hamiltonian constraint, which is frequently performed as part of the Born-Oppenheimer treatment in quantum cosmology. We discuss in which sense both approaches exhibit an inner product that is gauge-fixed via an operator version of the usual Faddeev-Popov procedure, and, in the second approach, how the unitarity of the effective Schr\"{o}dinger evolution is established perturbatively. We note that a conditional probability interpretation of the physical states is possible, so that both formalisms are examples of quantum mechanics with a relational dynamics.