How to switch between relational quantum clocks

TL;DR

This paper develops a systematic method for switching between different quantum clocks within a relational quantum framework, addressing the multiple choice problem in quantum gravity and illustrating it with a parametrized particle model.

Contribution

It introduces a general approach for transitioning between quantum reference systems, linking Dirac quantized and deparametrized theories, applicable to quantum cosmology.

Findings

Method enables consistent clock switching in quantum theories.

Illustration with parametrized particle demonstrates practical implementation.

Quantum uncertainties cause discontinuities in relational dynamics during clock switching.

Abstract

Every clock is a physical system and thereby ultimately quantum. A naturally arising question is how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a pressing issue in view of the multiple choice problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent approach to switching quantum reference systems (arXiv:1809.00556, arXiv:1809:05093), we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle's position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method suggests a new perspective on the multiple choice problem, indicating that it is rather a multiple choice feature of a complete relational quantum theory, taken as the conjunction of Dirac quantized and quantum deparametrized theories. Precisely this conjunction permits one to consistently switch between different temporal reference systems which is a prerequisite for a quantum notion of general covariance. Finally, we show that quantum uncertainties lead to discontinuity in the relational dynamics when switching clocks.