Relativistic limits on the discretization and temporal resolution of a quantum clock

TL;DR

This paper investigates fundamental relativistic limits on discretization and temporal resolution of time values for a quantum clock described by a time observable conjugate to a bounded discrete Hamiltonian. It analyzes two spectral regimes: an equally spaced spectrum where the time observable can be Hermitian (z=p) or described by a POVM (z>p), and a generic spectrum where the clock is always described by a POVM; in both cases it derives a structural bound Δt ≥ $\frac{T}{p+1}$ and demonstrates a universal relativistic bound Δt > $2^{1/3} Θ^{1/3} t_p^{2/3}$ that combines quantum spreading and gravitational constraints, with a continuum limit achieved as z→∞. For generic spectra, energy ratios that are rational allow a consistent resolution of the identity and a POVM description that remains valid in the continuum limit, while irrational ratios yield arbitrarily small corrections. Across both cases, a quantum speed–limit bound Δt_bot ≥ max( $\frac{\pi\hbar}{2\bar{E}}$, $\frac{\pi\hbar}{2\Delta E}$ ) emerges, underpinning an ultimate temporal-resolution bound that is independent of the clock’s spectral details. Collectively, the results broaden the class of viable quantum clocks under relativistic constraints and connect to the Page–Wootters formalism, illustrating how spectral structure can influence operational timescales without violating fundamental physics.

Abstract

We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian which can have an equally-spaced or a generic spectrum. In the first case the time observable can be described by a Hermitian operator and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for measuring spacetime distances.