Quantum limits of a space-time reference frame

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.