⚛️ quantum physics

Observation of relativistic Bohmian dynamics

Using weak measurements in a single-photon interferometer, this study experimentally reconstructs relativistic Bohmian trajectories that align with theoretical predictions and exhibit superluminal velocities and negative effective squared-mass near destructive interference, thereby establishing an empirical foundation for investigating quantum motion in relativistic spacetime.

Original authors: Yun-Fei Wang, Hui Wang, Tong Zhang, Yi-Teng Ye, Xiao-Yu Wang, Ming-Cheng Chen, Chao-Yang Lu, Jian-Wei Pan

Published 2026-09-30✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Yun-Fei Wang, Hui Wang, Tong Zhang, Yi-Teng Ye, Xiao-Yu Wang, Ming-Cheng Chen, Chao-Yang Lu, Jian-Wei Pan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For nearly a century, physicists have relied on quantum mechanics to predict the behavior of the smallest particles in the universe with astonishing precision. Yet, a fundamental question has lingered: what is actually happening to a particle between the moment it is launched and the moment it is detected? The standard view suggests that until measured, a particle exists only as a cloud of probabilities, with no definite path. An alternative perspective, known as Bohmian mechanics, proposes a different reality. In this view, particles do follow definite, deterministic paths, guided by a "pilot wave" that evolves according to the laws of quantum physics. While this idea works well for slow-moving particles, it has been incredibly difficult to test when particles move at speeds approaching that of light, or when the strange rules of relativity come into play. Reconciling the idea of a fixed path with the flexible nature of space and time has remained a theoretical puzzle, with no experimental evidence to show if such paths even exist in the relativistic world.

A team of researchers at the University of Science and Technology of China has now taken a significant step toward solving this puzzle by observing these elusive paths in a single-photon experiment. Using a sophisticated setup involving a double-slit interferometer, the scientists tracked the average movement of individual photons as they traveled through an interference pattern. They did not simply watch where the photons landed; instead, they used a technique called weak measurement to gently probe the energy and momentum of the photons without destroying their delicate state. By combining thousands of these gentle probes, they reconstructed a map of the average flow of the photons, revealing a velocity field that describes how the particles move through space and time.

The results of this experiment align closely with theoretical predictions for relativistic Bohmian dynamics. The reconstructed paths showed that the photons do not travel in straight lines but bend and curve as they navigate the interference pattern created by the two slits. Most strikingly, the researchers found that in regions where the light waves canceled each other out—a phenomenon known as destructive interference—the calculated speed of the photon flow appeared to exceed the speed of light. Furthermore, in these same regions, the mathematical value representing the photon's effective mass squared became negative. In the language of relativity, a negative mass squared usually implies a type of motion that is "spacelike," which is distinct from the normal "timelike" motion of massive objects.

It is crucial to understand that these findings do not mean the photons are actually breaking the cosmic speed limit or carrying information faster than light. The speed measured here is a local flow velocity, describing the average movement of the ensemble of photons at a specific point, rather than the speed of a signal traveling from one place to another. Just as a wave on a rope can move faster than the individual water molecules that make it up, this reconstructed flow can exhibit superluminal characteristics without violating the principles of relativity. The experiment confirms that the interference pattern does more than just change how bright the light appears; it actively reshapes the local rules of motion for the particles, organizing them into distinct regions of normal and unusual kinematic behavior.

The study also provided a direct look at a concept known as the quantum potential, a force-like term in Bohmian mechanics that arises from the shape of the wave itself rather than from any external interaction. The researchers found that the regions where the effective mass squared turned negative corresponded exactly to the areas where the wave amplitude was changing most rapidly. This suggests that the interference pattern acts as a landscape that guides the particles, creating zones where their behavior mimics that of particles with imaginary mass, even though the photons themselves remain massless. The physical rest mass of the photon never changes; it is the local description of its motion that shifts based on the surrounding wave structure.

By successfully reconstructing these trajectories, the team has established an experimental foundation for investigating the physical meaning of quantum paths in a relativistic setting. Their work demonstrates that the strange, non-intuitive features predicted by Bohmian mechanics, such as superluminal flow and negative effective mass parameters, are not just mathematical curiosities but observable features of the quantum world. While the interpretation of these paths as the literal trajectory of a point particle remains a subject of debate, the data provides a concrete benchmark for testing theories of motion in relativistic spacetime. The experiment opens a new window into understanding how quantum mechanics and relativity might coexist, offering a clearer picture of the hidden dynamics that govern the behavior of light.

Technical Summary: Observation of Relativistic Bohmian Dynamics

Problem and Context
Bohmian mechanics (or de Broglie–Bohm pilot-wave theory) offers a deterministic, nonlocal interpretation of quantum mechanics where particles follow trajectories guided by a wavefunction. While this framework has been successfully extended to non-relativistic regimes and experimentally verified using weak measurements, reconciling it with special relativity remains a fundamental challenge. The standard Schrödinger equation is not Lorentz covariant, and the nonlocal dependence of Bohmian dynamics on entangled configurations complicates its integration with relativistic spacetime. Although theoretical proposals for relativistic guidance laws exist (e.g., based on the scalar Klein-Gordon equation), their predicted dynamics—specifically the existence of a local relativistic velocity field and associated kinematic properties—have remained experimentally untested.

Methodology
The authors experimentally reconstruct relativistic Bohmian trajectories for single photons using an operational weak-value formulation. The experiment employs a double-slit interferometer with the following key components:

Source: High-purity single photons are generated via an InAs/GaAs quantum dot embedded in a tunable polarized microcavity ( $g^{(2)}(0) = 0.04$ ).
Weak Measurement Scheme: Photon polarization serves as the "pointer." Single photons are prepared in a linear polarization state $|+\rangle$ . A birefringent lithium niobate (LN) plate with a 1 $\mu$ m-wide slit is positioned at the focal plane of the lens system. As photons pass through, their polarization undergoes a small rotation $\delta\phi$ coupled to a linear combination of the photon's momentum and energy weak values.
Data Acquisition: To resolve the three underlying weak values ( $\langle \hat{p}_{x,w} \rangle$ , $\langle \hat{p}_{z,w} \rangle$ , and $\langle \hat{E}_w \rangle$ ) from a single polarization readout, the experiment utilizes five distinct measurement configurations. These involve rotating the optical axes of the LN plates relative to the beam.
Reconstruction: The raw photon counts (H and V polarization) across 8,326 spatial sampling points are jointly fitted using a Poisson likelihood model and a calibrated optical response. This yields the photon-count-weighted weak values of energy and momentum.
Trajectory Calculation: The relativistic velocity field is defined as the ratio of the momentum weak value to the energy weak value: $v(x, z) = \langle \hat{p}_w \rangle / \langle \hat{E}_w \rangle$ . Average trajectories are then reconstructed by integrating this velocity field using the fourth-order Runge–Kutta method.

Key Results

Reconstructed Trajectories: The experimentally reconstructed average trajectories align with theoretical predictions for relativistic Bohmian flow. A distinct bending of trajectories is observed in regions of destructive interference.
Superluminal Local Velocities: In the vicinity of destructive interference, the reconstructed local flow velocity exceeds the speed of light ( $c$ ). The authors clarify that this is a local flow velocity, not a signal-carrying velocity, and thus does not violate relativistic causality or allow superluminal information transfer.
Effective Squared Mass: Using the weak values, the authors calculate the effective squared-mass parameter, defined as $\bar{m}^2_{\text{eff}} = (\langle \hat{E}_w \rangle)^2 - |\langle \hat{p}_w \rangle|^2$ $\overset{m}{ˉ}_{eff}^{2} = (⟨ \hat{E}_{w} ⟩)^{2} - ∣ ⟨ \overset{p}{^}_{w} ⟩ ∣^{2}$ .
- In constructive interference regions, $\bar{m}^2_{\text{eff}}$ is positive and bounded by $E_0^2$ (timelike motion).
- In destructive interference regions, $\bar{m}^2_{\text{eff}}$ becomes negative (spacelike motion), coinciding with the regions of superluminal reconstructed velocity.
- For the massless scalar field used, this parameter corresponds to the quantum potential. The sign change indicates a local modification of the on-shell condition for the reconstructed flow, without implying a change in the physical rest mass of the photon (which remains zero).

Significance and Claims
The paper establishes an experimental basis for investigating the physical meaning of quantum trajectories within a relativistic framework. By jointly reconstructing energy and momentum weak values, the authors demonstrate that:

Relativistic Bohmian flow can be operationally defined and measured.
Interference patterns do not merely redistribute photon intensity but organize the reconstructed flow into distinct timelike and spacelike domains.
The observed negative effective squared mass and superluminal velocities are signatures of the local kinematic properties predicted by relativistic Bohmian theories, specifically linked to the quantum potential.

The authors note that while the results match theoretical predictions, a fundamental conceptual tension remains regarding the interpretation of these superluminal flows for point-particle mechanics in flat spacetime. They suggest that a full reconciliation may require broader field-theoretic formulations. The work provides a benchmark for future theoretical developments and opens an experimental route to reassessing the role of quantum trajectories in relativistic spacetime, with potential extensions to correlated photons and fermions (Dirac equation).

Technical Summary: Observation of Relativistic Bohmian Dynamics

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