Quantum limits of a space-time reference frame

TL;DR

This paper explores the fundamental quantum limits on defining space and time using a single quantum system as a reference frame, revealing an intrinsic uncertainty relation akin to Heisenberg's principle.

Contribution

It introduces a quantum trade-off between spatial and temporal precision within a single composite quantum system, combining quantum speed limits with relativity.

Findings

Spatial and temporal localizability are fundamentally linked.

Sharpening time precision increases position uncertainty.

An intrinsic uncertainty of the order of the Compton wavelength is revealed.

Abstract

We study the limitations for defining spatial and temporal intervals when the only available reference frame is a single composite quantum system, whose internal degrees of freedom serve as a temporal reference, a clock, and whose center of mass degrees of freedom act as a spatial reference, a rod. By combining quantum speed limits with the mass energy equivalence of special relativity, we show that spatial localizability and temporal resolution are not independent: sharpening one inevitably blurs the other. Specifically, the internal energy coherence needed for precise timekeeping affects the center of mass dynamics, enhancing position spreading during free evolution. As a result, a single composite system cannot act as a perfect quantum reference frame for both space and time, leading to a Heisenberg like uncertainty relation between spatial and temporal intervals. After analyzing this trade off from an external perspective, we formulate it in a purely relational manner, by means of covariant observables relative to the space time quantum reference frame, uncovering an additional intrinsic uncertainty of order the Compton wavelength of the frame.