Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.09841 · Physical Review A

General-relativistic particle trajectories via quantum mechanical weak values and the Schwarzschild-Alcubierre spacetime

Joshua Foo, Cameron Bellamy, Timothy C. Ralph

Published May 2, 2025• 3 citations
arXivHTMLPDF

Abstract

We show that the average trajectories of relativistic quantum particles in Schwarzschild spacetime, obtained via quantum mechanical weak measurements of momentum and energy, are equivalent to the predicted flow lines of probability current in curved spacetime quantum theory. We subsequently demonstrate that these trajectories correspond exactly to classical null geodesics in a hybrid Schwarzschild-Alcubierre spacetime. This threefold equivalence demonstrates how quantum theory in curved spacetime can be formulated via operationally defined measurements, and that such a theory may be interpreted deterministically, in the spirit of hidden-variable models such as Bohmian mechanics, through the novel connection to an underlying ``guiding metric.''

DOI
10.1103/physreva.111.052201

Related papers

Relativistic Bohmian trajectories of photons via weak measurements

Joshua Foo, Estelle Asmodelle, Austin P. Lund +1 more

Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has mainly been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here, we present an approach for constructing the relativistic Bohmian-type velocity field of single particles. The advantage of our proposal is that it is operational in nature, grounded in weak measurements of the particle's momentum and energy. We apply our weak measurement formalism to obtain the relativistic spacetime trajectories of photons in a Michelson-Sagnac interferometer. The trajectories satisfy quantum-mechanical continuity and the relativistic velocity addition rule. We propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.

arXiv:2108.05169•Nature Communications•Jul 11, 2022•26 citations

Observation of relativistic Bohmian dynamics

Yunfei Wang, Hui Wang, Tong Zhang +5 more

Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.

arXiv:2509.11609•Sep 15, 2025•0 citations

Quantum Gravity Without Metric Quantization: From Hidden Variables to Hidden Spacetime Curvatures

Mohamed Hatifi

Bohmian mechanics offers a deterministic alternative to conventional quantum theory through well-defined particle trajectories. While successful in nonrelativistic contexts, its extension to curved spacetime-and hence quantum gravity-remains unresolved. Here, we develop a covariant extension of Bohmian mechanics in curved spacetime that removes the need for metric quantization. From a Lagrangian formulation, we derive a generalized guidance equation in which Bohmian trajectories generate hidden curvature, replacing metric superposition with a statistical ensemble constrained by Heisenberg uncertainty, offering a novel perspective on quantum gravity. Consequently, in our approach, measuring the gravitational potential at a point unveils a pre-existing trajectory and its associated curvature-a departure from the observer-centric paradigm of standard quantum mechanics-providing an alternative in which gravitational effects emerge from deterministic quantum trajectories rather than wavefunction collapse. Numerical simulations in Robertson-Walker and cigar soliton spacetimes reveal that while quantum interference is curvature-sensitive, Zitterbewegung remains invariant, distinguishing fundamental quantum effects. Moreover, deviations from the Born rule in inhomogeneous spacetimes are observed and suggest gravity-induced quantum non-equilibrium. This new approach has far-reaching implications for the role of determinism and potential observational signatures of quantum non-equilibrium in cosmology.

arXiv:2502.08421•Feb 12, 2025•1 citation

Stochastic quantum mechanics trajectories near Schwarzschild horizon black holes

Juan S. Jerez‐Rodriguez, Eric S. Escobar-Aguilar, Tonatiuh Matos

Abstract This work explores the possibility of applying stochastic quantum mechanics to curved spacetimes, with an emphasis on the Schwarzschild black hole. After reviewing the fundamental concepts of this approach, the quantum stochastic equations are extended to curved spacetime using a fully covariant treatment. Subsequently, the Klein-Gordon equation is solved for scalar perturbations, and the resulting stochastic trajectories are analyzed by varying parameters such as angular momentum, particle frequency, and computational integration time. In conclusion, we find that the trajectories are influenced by gravitational fluctuations in spacetime and that, depending on the variation of the fundamental parameters, different types of stochastic trajectories are obtained.

The European Physical Journal C•May 19, 2026•1 citation

Operator-Level Quantum-Classical Correspondence in Relativistic Quantum Theory and Curved Spacetime

Pankaj Sheoran, Gopal Kashyap, Sanjay Siwach

In a recent work \cite{Shaikh:2026nsi}, it was shown that for a non-relativistic particle, the operator equations of motion can be cast in the same differential form as Newton's equation of motion. This establishes a correspondence between quantum operator dynamics and classical mechanics at the level of the equations of motion. In this work, we investigate whether this correspondence extends to quantum field theory and linearized gravity. We explicitly show that the Heisenberg equations of motion for quantum field operators reproduce the corresponding classical equations of motion for a broad class of field theories. The Klein-Gordon equation, Maxwell's equations, Yang-Mills equations, and linearized Einstein equations are shown to arise at the operator level without taking the limit $\hbar\rightarrow0$. We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.

gr-qcCross-listedarXiv:2609.25360•Sep 23, 2026

Measurement-based Lorentz-covariant Bohmian trajectories of interacting photons

Joshua Foo, Austin P. Lund, Timothy C. Ralph

In a recent article [Nat. Commun. 13, 2 (2022)], we devised a method of constructing the Lorentz-covariant Bohmian trajectories of single photons via weak measurements of the photon's momentum and energy. However, whether such a framework can consistently describe multiparticle interactions remains to be seen. Here we present a nontrivial generalization of our framework to describe the relativistic Bohmian trajectories of two interacting photons (this interaction arising from the symmetrization of the two-particle wave function) exhibiting nonclassical interference due to their indistinguishability. We begin by deriving the average velocity fields of the indistinguishable photons using a conditional weak measurement protocol, with detectors that are agnostic to the identity of the respective photons. We demonstrate a direct correspondence between the operationally derived trajectories and those obtained using a position- and time-symmetrized multiparticle Klein-Gordon wave function, whose dynamics are manifestly Lorentz covariant. We propose a space-time metric that depends nonlocally on the positions of both particles as a curvature-based interpretation of the resulting trajectories. Contrary to prior expectations, our results demonstrate a consistent trajectory-based interpretation of relativistic multiparticle dynamics in quantum theory.

arXiv:2309.16011•Physical Review A•Feb 21, 2024•10 citations
We show that the average trajectories of relativistic quantum particles in Schwarzschild spacetime, obtained via quantum mechanical weak measurements of momentum and energy, are equivalent to the predicted flow lines of probability current in curved spacetime quantum theory. We subsequently demonstrate that these trajectories correspond exactly to classical null geodesics in a hybrid Schwarzschild-Alcubierre spacetime. This threefold equivalence demonstrates how quantum theory in curved spacetime can be formulated via operationally defined measurements, and that such a theory may be interpreted deterministically, in the spirit of hidden-variable models such as Bohmian mechanics, through the novel connection to an underlying ``guiding metric.''
Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has mainly been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here, we present an approach for constructing the relativistic Bohmian-type velocity field of single particles. The advantage of our proposal is that it is operational in nature, grounded in weak measurements of the particle's momentum and energy. We apply our weak measurement formalism to obtain the relativistic spacetime trajectories of photons in a Michelson-Sagnac interferometer. The trajectories satisfy quantum-mechanical continuity and the relativistic velocity addition rule. We propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.
Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.
Bohmian mechanics offers a deterministic alternative to conventional quantum theory through well-defined particle trajectories. While successful in nonrelativistic contexts, its extension to curved spacetime-and hence quantum gravity-remains unresolved. Here, we develop a covariant extension of Bohmian mechanics in curved spacetime that removes the need for metric quantization. From a Lagrangian formulation, we derive a generalized guidance equation in which Bohmian trajectories generate hidden curvature, replacing metric superposition with a statistical ensemble constrained by Heisenberg uncertainty, offering a novel perspective on quantum gravity. Consequently, in our approach, measuring the gravitational potential at a point unveils a pre-existing trajectory and its associated curvature-a departure from the observer-centric paradigm of standard quantum mechanics-providing an alternative in which gravitational effects emerge from deterministic quantum trajectories rather than wavefunction collapse. Numerical simulations in Robertson-Walker and cigar soliton spacetimes reveal that while quantum interference is curvature-sensitive, Zitterbewegung remains invariant, distinguishing fundamental quantum effects. Moreover, deviations from the Born rule in inhomogeneous spacetimes are observed and suggest gravity-induced quantum non-equilibrium. This new approach has far-reaching implications for the role of determinism and potential observational signatures of quantum non-equilibrium in cosmology.
Abstract This work explores the possibility of applying stochastic quantum mechanics to curved spacetimes, with an emphasis on the Schwarzschild black hole. After reviewing the fundamental concepts of this approach, the quantum stochastic equations are extended to curved spacetime using a fully covariant treatment. Subsequently, the Klein-Gordon equation is solved for scalar perturbations, and the resulting stochastic trajectories are analyzed by varying parameters such as angular momentum, particle frequency, and computational integration time. In conclusion, we find that the trajectories are influenced by gravitational fluctuations in spacetime and that, depending on the variation of the fundamental parameters, different types of stochastic trajectories are obtained.
In a recent work \cite{Shaikh:2026nsi}, it was shown that for a non-relativistic particle, the operator equations of motion can be cast in the same differential form as Newton's equation of motion. This establishes a correspondence between quantum operator dynamics and classical mechanics at the level of the equations of motion. In this work, we investigate whether this correspondence extends to quantum field theory and linearized gravity. We explicitly show that the Heisenberg equations of motion for quantum field operators reproduce the corresponding classical equations of motion for a broad class of field theories. The Klein-Gordon equation, Maxwell's equations, Yang-Mills equations, and linearized Einstein equations are shown to arise at the operator level without taking the limit ℏ→0\hbar\rightarrow0ℏ→0. We then extend the analysis to curved spacetime and show that the quantum field operators satisfy the corresponding covariant field equations at the operator level. In the weak-field approximation, we show that the Heisenberg evolution of the metric-perturbations reproduces the linearized Einstein equations at the operator level. For a quantum particle in curved spacetime, the Heisenberg equations for position and momentum reproduce classical geodesic motion in the semiclassical limit, thereby extending the operator-level quantum-classical correspondence from fields to particle dynamics.
In a recent article [Nat. Commun. 13, 2 (2022)], we devised a method of constructing the Lorentz-covariant Bohmian trajectories of single photons via weak measurements of the photon's momentum and energy. However, whether such a framework can consistently describe multiparticle interactions remains to be seen. Here we present a nontrivial generalization of our framework to describe the relativistic Bohmian trajectories of two interacting photons (this interaction arising from the symmetrization of the two-particle wave function) exhibiting nonclassical interference due to their indistinguishability. We begin by deriving the average velocity fields of the indistinguishable photons using a conditional weak measurement protocol, with detectors that are agnostic to the identity of the respective photons. We demonstrate a direct correspondence between the operationally derived trajectories and those obtained using a position- and time-symmetrized multiparticle Klein-Gordon wave function, whose dynamics are manifestly Lorentz covariant. We propose a space-time metric that depends nonlocally on the positions of both particles as a curvature-based interpretation of the resulting trajectories. Contrary to prior expectations, our results demonstrate a consistent trajectory-based interpretation of relativistic multiparticle dynamics in quantum theory.