Beyond semiclassical time

TL;DR

This paper demonstrates that the Born-Oppenheimer approximation in quantum gravity yields a unitary, gauge-fixed, relational quantum theory at least up to the next-to-leading order in minisuperspace models, clarifying unitarity issues.

Contribution

It shows that the semiclassical time approach in quantum gravity is unitary and gauge-fixed, framing it as a relational quantum theory with consistent transition amplitudes.

Findings

Unitarity is preserved up to next-to-leading order.

The inner product measure relates to the Faddeev-Popov determinant.

The approach is a relational quantum theory.

Abstract

We show that the usual Born-Oppenheimer type of approximation used in quantum gravity, in which a semiclassical time parameter emerges from a weak-coupling expansion of the Wheeler-DeWitt constraint, leads to a unitary theory at least up to the next-to-leading order in minisuperspace models. As there are no unitarity-violating terms, this settles the issue of unitarity at this order, which has been much debated in the literature. Furthermore, we also show that the conserved inner product is gauge-fixed in the sense that the measure is related to the Faddeev-Popov determinant associated with the choice of semiclassical time as a reparametrization gauge. This implies that the Born-Oppenheimer approach to the problem of time is, in fact, an instance of a relational quantum theory, in which transition amplitudes can be related to conditional probabilities.