Classical clocks measure proper time along their worldline, and Riemannian geometry provides tools for predicting the time shown by clocks in both flat and curved spacetimes. Common approaches to time in quantum systems, based for instance on wave functions or density matrices, tend to obscure this geometric property at the quantum level. Here, a new framework is demonstrated for perturbing the classical path-length functional to include quantum degrees of freedom within a modified Riemannian geometry. In this framework, a quantum clock travels on geodesics of a family of spacetimes deformed by parameters specifying the clock's quantum state. Detailed derivations provide potentially testable corrections to gravitational time-dilation in Schwarzschild spacetime that scale with the ratio of the clock's Compton wavelength to its wave packet's spatial extent.
Classical clocks measure proper time along their worldline, and Riemannian geometry provides tools for predicting the time shown by clocks in both flat and curved spacetimes. Common approaches to time in quantum systems, based for instance on wave functions or density matrices, tend to obscure this geometric property at the quantum level. Here, a new framework is demonstrated for perturbing the classical path-length functional to include quantum degrees of freedom within a modified Riemannian geometry. In this framework, a quantum clock travels on geodesics of a family of spacetimes deformed by parameters specifying the clock's quantum state. Detailed derivations provide potentially testable corrections to gravitational time-dilation in Schwarzschild spacetime that scale with the ratio of the clock's Compton wavelength to its wave packet's spatial extent.
Abstract At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.
A quantum clock cannot be modeled as a point mass moving along a single geodesic if it is in a state with nonzero position fluctuations. Instead, it is an extended object subject to tidal forces and a superposition of time dilations at different altitudes. Here, a geometrical formulation of quantum mechanics is used to show that additional quantum properties representing correlations between different directions imply a non-Riemannian geometrical structure experienced by a quantum clock. A specific version of Finsler geometry parameterized by entropy and purity of the state provides a novel setting for a combination of quantum and gravitational effects. A crucial ingredient is given by a new parameterization of quantum-information properties related to second-order moments of a state and may also be useful in other applications.
At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.
Building on the framework developed by Perche [Phys. Rev. D 106, 025018 (2022)], we study two localized nonrelativistic quantum particles propagating along timelike geodesics in a curved spacetime background. Each particle is coupled to a quantum clock that operationally records the time spent in a prescribed spatial region. We compute the covariance of the resulting time observables for separable and entangled two-particle states, comparing flat and curved backgrounds. We then reformulate the protocol as a Bell-like experiment and show that the Bell parameter can acquire a curvature-induced correction. In particular, a protocol calibrated to saturate the classical bound in flat spacetime can be driven above this bound in curved spacetime for entangled states. We focus on two-dimensional curved backgrounds in which the local tidal term induces an effective harmonic potential in the Fermi-frame description. Our results show that spacetime curvature can modify operationally defined quantum correlations and suggest entangled quantum clocks as probes of spacetime curvature.
We propose that time is not a fundamental background but emerges from the quantum uncertainties of position, momentum, and energy. This is captured by the postulate Δτ=γΔEΔxΔp,where Δτ is the proper time interval, Δx,Δp,ΔE are the standard quantum uncertainties, and γ=1/1−v2/c2 is the Lorentz factor associated with the average velocity v. Using the standard wave‑packet relation ΔE=vΔp, our equation reduces exactly to the special relativistic proper time Δτ=Δx/(γv), demonstrating consistency with all existing tests. However, for non‑classical quantum states where ΔE=vΔp (e.g., a superposition of widely separated momenta), our equation predicts new measurable effects: (i) a significant modification of the decay length of unstable particles (muons), (ii) a dependence of clock rates on the spatial spread Δx even in flat spacetime, and (iii) a quantum correction to gravitational time dilation when the Lorentz factor is replaced by the general relativistic factor Γ. These predictions are falsifiable and provide a concrete experimental window into the emergent nature of time.
A proper time observable for a quantum clock is introduced and it is found that the proper time read by one clock conditioned on another clock reading a different proper time obeys classical time dilation in accordance with special relativistic kinematical time dilation. Here, we extend this proposal to a weak gravitational field in order to investigate whether the weak equivalence principle holds for quantum matter. We find that for general quantum states the quantum time dilation in a weak gravitational field obeys a similar gravitational time dilation found in classical relativity. However, unlike the special relativistic case, the time dilation involves the external time (a background coordinate time at the observer on the Earth) as well as the proper times of two clocks. We also investigate a quantum time dilation effect induced by a clock in a superposition of wave packets localized in momentum space or in position space and propose the setup to observe the gravitational effect in the quantum interference effect in the time dilation.