How to switch between relational quantum clocks

TL;DR

The paper tackles the problem of relational time in quantum systems by developing a systematic method to switch between quantum clocks using a clock-choice-neutral Dirac quantization, complemented by quantum symmetry reduction to clock-specific reduced theories. It demonstrates the approach on the parametrized non-relativistic particle, including careful regularization of the time-of-arrival observable and explicit maps linking Dirac- and reduced-quantized descriptions, thereby establishing quantum clock covariance via a clock-neutral Hilbert space. A key finding is that quantum uncertainties generally induce discontinuities or non-globality in relational dynamics when switching clocks, revealing a quantum relativity of clock readings. The work frames a general pathway for quantum cosmology and quantum gravity, highlighting that a complete relational theory requires both the neutral Dirac structure and the clock-specific reduced theories, with implications for how general covariance might be realized in quantum gravity.

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.

Figures (2)

• Figure 1: Schematic phase space setup for changing between relational evolutions relative to clocks $t$ and $q$.
• Figure 2: Overview of the various steps of the Dirac quantization, the three reduced quantizations, as well as their links. All steps are explained in detail in the main text. $\mathcal{P}_\pm$ are the right/left mover reduced phase spaces and $\mathcal{H}_\pm$ their quantizations, while $\mathcal{P}_{q|t}$ and $\mathcal{H}_{q|t}$ are the reduced phase and Hilbert space relative to clock $t$, respectively. $\mathcal{H}_{\rm kin}$ and $\mathcal{H}_{\rm phys}$ are the kinematical and physical Hilbert space of the Dirac quantization, respectively. Mapping from $\mathcal{H}_{\rm phys}$ to the reduced Hilbert spaces involves a trivialization $\mathcal{T}_t$ or $\mathcal{T}_q$ of the constraint to the chosen clock and a subsequent projection onto the classical gauge fixing conditions.