Quantum-Relativistic Anomalous Proper Time

Abstract

We investigate surprising consequences of the hypothesis that gravity and proper time obey the quantum superposition principle. We show that, for a massive body prepared in a superposition of two positions and post-selected in a suitable final state, the elapsed proper time in another region of space can assume an anomalous effective value lying outside the range allowed by classical general relativity. Remarkably, the effective elapsed proper time can even become negative, corresponding to an effective evolution backward in time. We also propose an experiment to observe an anomalous effective proper time associated with the internal degrees of freedom of a quantum particle in an interferometer under Earth's gravitational field. The proposed experiment is, in principle, feasible with current technology, and its realization could provide the first experimental evidence of an interplay between quantum mechanics and general relativity beyond the effects of the Newtonian potential in flat spacetime.

The paper asks whether coherent superpositions of gravitational time dilation, combined with post-selection, can produce an effective elapsed proper time outside the range of the individual branches. It considers both a probe near a mass in a spatial superposition and an internal atomic clock traversing two heights in Earth’s classical gravitational field. Starting from weak-field evolution $e^{-iH\tau/\hbar}$, it derives the conditional probe dynamics for initial path amplitudes $\alpha,\beta$ and antisymmetric path post-selection. To first order, the evolution corresponds to $\tau_{\mathrm{ef}}=(\beta\tau_B-\alpha\tau_A)/(\beta-\alpha)$, the weak value of a branch-dependent proper-time operator. For positive real amplitudes, this lies outside $[\tau_A,\tau_B]$ and can become negative. The approximate post-selection probability is $|\beta-\alpha|^2/2$; averaging both output ports recovers the classical weighted proper time. An experimental proposal reads out the conditional internal-clock phase through Ramsey interferometry. For an optical transition near $6\times10^{14}\,\mathrm{Hz}$, a $10\,\mathrm{cm}$ average arm-height separation, $1\,\mathrm{s}$ interrogation, and $\beta=1/\sqrt{2}+0.1$, the predicted phase displacement relative to one branch is approximately $0.1$ radians. This configuration uses a classical gravitational field, not a superposition of gravitational fields. The effective-time description requires $\Delta E|\tau_{\mathrm{ef}}-\tau_A|/\hbar<1$, making the anomalous shift smaller than the clock’s single-system time uncertainty and requiring ensemble measurements. Environmental decoherence prevents the proposed macroscopic reversal interpretation. The gravitational-source-superposition scheme is acknowledged to be impractical, while maintaining atomic-clock precision within the proposed interferometer remains an experimental difficulty.