Quantum reference frames and quantum space-time

Sergey Nikolaevich Mayburov

S.N. Maiburov

Flaminia Giacomini

In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

Ahmed Albezawi

The quantum reference frame (QRF) programme has demonstrated that the distinction between superposition and entanglement is observer-dependent: what appears as entanglement from an external perspective may appear as a definite relational state from within a quantum frame. However, QRF has operated primarily in the kinematic and spatial domain, leaving the temporal structure of observation unaddressed and providing no account of why each internal frame experiences definiteness. We propose a synthesis with Relational Retrocausal Quantum Mechanics (RRQM), in which measurement is the retroactive rewriting of an observer’s accessible history. The central result is the entanglement/measurement duality theorem: the temporal frame-change transformation from an external observer to an internal observer maps an entangled description onto a measurement-with-definite-outcome description, and this map is precisely the RRQM rewriting operator acting on history space. We formalize this through the introduction of temporal quantum reference frames (tQRFs), extending QRF from instantaneous Hilbert space into the space of observer-relative histories. The framework resolves Wigner’s friend, Bell correlations, and Schr¨odinger’s cat through a single mechanism: observer-relative temporal frames, each retroactively consistent.

Miklós Banai

Davide Mattei, Esteban Castro-Ruiz

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.