Multiple choices of time in quantum cosmology

TL;DR

This paper investigates how different choices of time functions in quantum cosmology affect the resulting quantum theories, revealing that such choices can lead to significant discrepancies in observable predictions.

Contribution

It develops tools to analyze the relation between quantum theories based on different time functions and demonstrates that clock choices can cause large semiclassical effects.

Findings

Quantum theories based on different clocks can be inequivalent.

Changing the clock can significantly alter the spectra of quantum observables.

Semiclassical effects induced by clock choice can be arbitrarily large.

Abstract

It is often conjectured that a choice of time function merely sets up a frame for the quantum evolution of gravitational field, meaning that all choices should be in some sense compatible. In order to explore this conjecture (and the meaning of compatibility), we develop suitable tools for determining the relation between quantum theories based on different time functions. First, we discuss how a time function fixes a canonical structure on the constraint surface. The presentation includes both the kinematical and the reduced perspective, and the relation between them. Second, we formulate twin theorems about the existence of two inequivalent maps between any two deparameterizations, a {\it formal canonical} and a {\it coordinate} one. They are used to separate the effect of choice of clock from other effects. We show, in an example, how the spectra of quantum observables are transformed under the change of clock and prove, via a general argument, the existence of choice-of-time-induced semiclassical effects. Finally, we study an example, in which we find that the semiclassical discrepancies can in fact be arbitrarily large for dynamical observables. We conclude that the values of critical energy density or critical volume in the bouncing scenarios of quantum cosmology cannot in general be at the Planck scale and always need to be given with reference to a specific time function.