Emergent Semiclassical Time in Quantum Gravity. I. Mechanical Models

TL;DR

This paper investigates how emergent semiclassical time functions arise in quantum gravity models, using relational particle toy models to analyze the assumptions, approximations, and implications for quantum cosmology.

Contribution

It introduces a new iterative scheme for semiclassical time in quantum cosmology and compares emergent time with hidden dilational Euler time in relational models.

Findings

Two notions of emergent semiclassical WKB time are equivalent to classical Leibniz--Mach--Barbour time.

The semiclassical approach includes significant terms like back-reactions and second derivatives.

Relational particle models serve as exact solutions to test semiclassical approximations.

Abstract

Strategies intended to resolve the problem of time in quantum gravity by means of emergent or hidden timefunctions are considered in the arena of relational particle toy models. In situations with `heavy' and `light' degrees of freedom, two notions of emergent semiclassical WKB time emerge; these are furthermore equivalent to two notions of emergent classical `Leibniz--Mach--Barbour' time. I futhermore study the semiclassical approach, in a geometric phase formalism, extended to include linear constraints, and with particular care to make explicit those approximations and assumptions used. I propose a new iterative scheme for this in the cosmologically-motivated case with one heavy degree of freedom. I find that the usual semiclassical quantum cosmology emergence of time comes hand in hand with the emergence of other qualitatively significant terms, including back-reactions on the heavy subsystem and second time derivatives. I illustrate my analysis by taking it further for relational particle models with linearly-coupled harmonic oscillator potentials. As these examples are exactly soluble by means outside the semiclassical approach, they are additionally useful for testing the justifiability of some of the approximations and assumptions habitually made in the semiclassical approach to quantum cosmology. Finally, I contrast the emergent semiclassical timefunction with its hidden dilational Euler time counterpart.