Almost Ideal Clocks in Quantum Cosmology: A Brief Derivation of Time

TL;DR

This paper develops a formalism for quantizing time in systems with an 'almost ideal clock,' deriving a complete external time quantum mechanics framework and analyzing its limits and failures near singularities.

Contribution

It introduces a novel formalism for quantizing time using correlations with an almost ideal clock, extending previous models and analyzing its limitations.

Findings

The formalism successfully derives external time quantum mechanics.

The ideal clock limit relates to Klein-Gordon and Newton-Wigner formalisms.

The approach fails near singularities in Bianchi models.

Abstract

A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.