Spacetime Quantum Reference Frames and superpositions of proper times

TL;DR

This work develops a spacetime generalization of quantum reference frames by treating reference frames as quantum systems with clocklike internal degrees of freedom. It presents a covariant, timeless model for $N$ relativistic particles in a weak gravitational field, constrained by first-class operators, and derives relational dynamics from the perspective of a chosen particle via a Quantum Locally Inertial Frame transformation ${\hat{\mathcal{T}}}_1$. The formalism reveals extended spacetime symmetries under QRF changes, and predicts quantum superpositions of gravitational redshift and special-relativistic time dilation between different QRFs, including explicit limiting cases (Galilean, SR, Newtonian) and a measurement scenario that yields time-delocalised events. These results offer a concrete path toward fully relational physics in nonclassical spacetimes and have potential experimental implications in interferometric tests probing quantum gravitational time effects.

Abstract

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 QRF, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of describing the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the evolution parameter 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. Concretely, we consider 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, keeping the proper time in the local frame of the particle. This fully relational construction shows how the remaining particles evolve from the perspective of the QRF and includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we observe a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations in the QRF.

Figures (2)

• Figure 1: We consider a system of $N$ relativistic quantum particles in a weak gravitational field produced by a mass $M$. Each particle has a quantum state representing its position in the spacetime diagram, as well as a clock state (the hands of the clocks) which keeps the proper time in the particle's frame. We introduce a timeless model, in which the global state of the $N$ particles and of the mass $M$ is "frozen", and is described by an $N$-particle quantum state in spacetime. Concretely, this means that the quantum state of the external variables can be, for instance, in a quantum superposition of coordinate times $x^0$ and spatial coordinates $\mathbf{x}$. Classically, the position in spacetime and the velocity of a particle influence its proper time due to relativistic time dilation. When the particle is in a quantum superposition of positions or velocities, the proper time displayed by its internal clock is also in a quantum superposition from an external perspective. While, at the global level, the system does not evolve, the dynamical evolution is recovered in terms of the relational variables between one of the particles, which is chosen as the quantum reference frame (QRF), and the rest of the particles. To transform to the QRF of one of the particles, we first map the initial spacetime coordinates to the relational spacetime coordinates from the perspective of the particle chosen as the QRF. We then find a transformation which makes the metric locally inertial at the origin of the QRF. Finally, we use the proper time of the particle to parametrise the dynamics of the remaining particles from the perspective of the chosen QRF. In such QRF, the dynamical evolution of the remaining particles is described in terms of a Hamiltonian operator and is parametrised by the clock's proper time, which is just a classical parameter in the clock's rest frame.
• Figure 2: We depict the particular situation in which particle $2$ ($C_2$ in the picture) evolves in a superposition of two semi-classical trajectories from the perspective of particle $1$ ($C_1$ in the picture). In the most general case, the state of particle $2$ from the point of view of particle $1$ is arbitrary. We describe a measurement performed at time $\tau_2^*$ in the frame of clock $2$ from the point of view of $C_1$. a) The measurement, which is localised in time in the frame of $C_2$, appears delocalised in time in the frame $C_1$. In particular, it is performed in a superposition of proper times $\tau_1'$ and $\tau_1"$ in the proper time of $C_1$. The two times $\tau_1'$ and $\tau_1"$ are related to $\tau_2^*$ by the expressions $\tau_1' = \Delta_{12}^{-1}(W_1)\tau_2^*$ and $\tau_1" = \Delta_{12}^{-1}(W_2)\tau_2^*$, where $\Delta_{12}^{-1}(W_i)$, $i=1,2$, encodes the special-relativistic time dilation or the gravitational redshift evaluated on the worldline $W_{i}$. b) In general, $C_1$ "sees" the clock $C_2$ as ticking in a superposition of times, depending on the state of the external degrees of freedom. Specifically, this effect can be understood as a superposition of special-relativistic time dilations, when the particles move at relativistic velocities in a Minkowski background, or as a superposition of gravitational redshifts, when the particles move slowly in a weak gravitational field.