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The (un)detectability of trajectories in pilot-wave theory

This paper argues that despite claims from weak velocity measurements and "surrealistic" trajectory experiments, individual Bohmian particle trajectories remain fundamentally undetectable, as these phenomena are fully consistent with standard quantum mechanics and do not provide evidence for or against pilot-wave theory.

Original authors: Johannes Fankhauser

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Johannes Fankhauser

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 or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Roadmap: Why We Can't See the Path

Imagine you are watching a magic show. The magician pulls a rabbit out of a hat, and you see the rabbit. That's the "manifest" world—the stuff you can touch, see, and record. But what if the rabbit didn't just appear? What if it was actually walking along a secret, invisible tunnel under the stage the whole time? In the world of quantum mechanics, the "rabbit" is a tiny particle like an electron, and the "tunnel" is its path, or trajectory.

For nearly a century, physicists have been arguing about whether these invisible tunnels even exist. The standard rulebook of quantum mechanics (the one used to build lasers and computers) is incredibly good at predicting where the rabbit might show up, but it refuses to say where the rabbit was before it popped out. It treats the rabbit as a fuzzy cloud of possibilities rather than a solid object on a specific road. Some scientists, however, love the idea of a secret tunnel. They propose a theory called "pilot-wave theory" (or de Broglie–Bohm theory), which suggests that particles do have definite paths, guided by an invisible wave. The problem? We can't see these paths. They are "underdetermined," meaning the math allows for infinite different tunnels, all of which lead to the exact same rabbit appearing in the same spot.

Recently, a new trick was proposed to solve this mystery: "weak measurements." Think of this as peeking at the rabbit without actually opening the hat. By gently tapping the system, scientists hoped to trace the invisible tunnel without disturbing the rabbit's journey. If successful, this would prove that the secret tunnels are real and that the pilot-wave theory is the correct description of our universe. But does this trick actually work?

The Paper's Verdict: The Map is Not the Territory

In this chapter, author Johannes Fankhauser investigates whether these "weak measurements" can truly reveal the secret paths of particles. He dives into two famous scenarios: one where scientists tried to map the paths of particles in a double-slit experiment, and another where they tried to prove the paths were "surreal" (meaning they didn't match the tracks left behind).

Fankhauser's main finding is a bit of a buzzkill for those hoping for a quick proof: Weak measurements do not actually show us the real, physical paths of particles.

Here is the breakdown of the adventure:

1. The "Weak" Peek That Wasn't a Peek
Scientists tried to use "weak measurements" to track particles. Imagine trying to follow a ghost by looking at the ripples it leaves in a pond. You can measure the ripples (the weak value), and you can see where the ghost ends up (the strong measurement). By combining these, you can calculate a "velocity" that looks exactly like the path predicted by the pilot-wave theory.

However, Fankhauser argues that this is a trap. The "velocity" you calculate isn't the speed of the ghost; it's just a mathematical feature of the ripples (specifically, the gradient of the wave's phase). It's like measuring the wind speed on a map and assuming that tells you exactly where a specific leaf fell. The paper shows that you can only get the "standard" pilot-wave path if you already assume that the pilot-wave theory is true. If you assume a different, equally valid version of the theory, the math still works, but the "ghost" would have taken a completely different, invisible path. The weak measurement doesn't tell you which path was real; it just tells you about the wave.

2. The "Surreal" Trajectory Trap
The second part of the paper looks at a famous thought experiment involving a "surrealistic trajectory." In this setup, a particle travels through a maze of mirrors and leaves a trail of "spin flips" (like dominoes falling) that seem to mark its path. But according to pilot-wave theory, the particle actually took a different route, one that didn't match the dominoes. Critics said, "Aha! The theory is wrong because the particle didn't go where the dominoes said it did!"

Fankhauser explains that this is a misunderstanding of what the dominoes actually represent. The dominoes (the spin flips) are a record of the wave interacting with the maze, not a direct record of the particle's location. In the pilot-wave world, the particle can be influenced by the wave in a way that makes it move differently than the "trail" suggests. The paper concludes that calling these paths "surreal" is a mistake caused by confusing the map (the spin flips) with the territory (the actual particle). The particle didn't take a weird path; the "trail" just wasn't a perfect map of the path in the first place.

The Bottom Line
The paper concludes that we cannot use these clever experiments to "see" the invisible tunnels. The "weak measurements" are a fascinating tool that helps us understand the behavior of waves, but they don't give us a direct line of sight to the particle's actual journey. Whether the particle is taking the standard path, a wiggly alternative path, or a random walk, the experimental results look exactly the same.

So, the mystery of the invisible road remains unsolved. We can see the rabbit, and we can see the ripples it makes, but the paper argues that we still can't definitively prove which secret tunnel the rabbit walked through. The "weak values" are a beautiful mathematical reflection, but they are not a window into the hidden reality of the particle's motion.

Technical Summary: The (Un)detectability of Trajectories in Pilot-Wave Theory

Problem Statement
The chapter addresses the persistent ambiguity in quantum mechanics (QM) regarding the ontological status of physical systems and the relationship between empirical data (the "manifest domain") and underlying physical reality (the "non-manifest domain"). Specifically, it investigates whether the individual particle trajectories posited by de Broglie–Bohm (pilot-wave) theory are empirically detectable.

A central tension exists within pilot-wave theory: while it postulates deterministic particle trajectories, these trajectories are constructed to be empirically underdetermined. Standard pilot-wave theory is empirically equivalent to standard QM because infinitely many alternative guidance equations (differing by divergence-free vector fields or stochastic terms) yield identical statistical distributions of measurement outcomes. Consequently, the theory is "vastly underdetermined by empirical data." The chapter scrutinizes two specific proposals that claim to resolve this underdetermination or falsify the theory:

Weak Measurements: The claim that "weak velocity measurements" can directly observe Bohmian trajectories, thereby vindicating the standard guidance equation.
Surrealistic Trajectories: The claim that certain experiments (involving spin grids in interferometers) reveal "surreal" trajectories where the Bohmian path contradicts the "actual" path indicated by measurement records, purportedly falsifying the theory.

Methodology
The author employs a rigorous reconstruction of the physical arguments and mathematical formalism underlying both weak measurements and the surrealistic trajectory experiments. The analysis relies on:

Formal Analysis of Weak Values: Examining the operational definition of weak velocity (Wiseman's protocol), which involves a weak interaction followed by a strong post-selection. The author analyzes the conditional probabilities and the role of entanglement between the system and the pointer.
Counterfactual Reasoning: Testing the "Correspondence Assumption" (COR), which posits that the expectation value of a weak pointer, post-selected on a final strong measurement, corresponds to the particle's actual position at the time of the weak interaction.
Bohmian Dynamics in Interferometry: Re-evaluating the "surrealistic" trajectory experiment (Englert et al.) by explicitly calculating the Bohmian velocity field for a particle interacting with a grid of spins, distinguishing between spin flips and spatial displacements of the spin particles.
Ontological Demarcation: Distinguishing between "manifest" variables (observable records, such as pointer positions or spin states) and "non-manifest" variables (actual particle configurations) to identify where naive identifications lead to logical inconsistencies.

Key Contributions and Results

1. Weak Velocity Measurements Do Not Detect Actual Trajectories
The author demonstrates that weak velocity measurements do not provide empirical evidence for the specific trajectories of standard de Broglie–Bohm theory.

Operational Nature: Weak velocity measurements are shown to be measurements of the gradient of the wave function's phase ( $\nabla S$ ), a quantity derivable entirely within standard QM without invoking particle trajectories.
Failure of the Correspondence Assumption (COR): The identification of the weak value with the particle's actual position at the time of interaction relies on the assumption that the conditional probability distribution of the pointer at $t=0$ is well-defined given a post-selection at $t=\tau$ . The author argues this is illicit in a quantum context because, prior to the final strong measurement, the system and pointer are entangled. No "effective collapse" occurs at $t=0$ to define a unique probability measure for the particle's position.
Underdetermination Persists: Because the weak value coincides with the standard guidance equation only if one assumes the standard guidance equation (or that COR holds), the measurement is circular. Any alternative guidance equation (including stochastic variants) that preserves the Born rule yields the same weak velocity field. Thus, weak measurements cannot distinguish between the standard theory and its empirically equivalent alternatives.

2. The "Surrealistic" Trajectory Paradox is a Category Error
The analysis of the surrealistic trajectory experiment reveals that the apparent contradiction arises from a naive conflation of manifest records with non-manifest trajectories.

Mechanism of the Paradox: In the experiment, a particle traverses an interferometer with a grid of spins. The spins flip to indicate a path, but the Bohmian particle is calculated to travel a different path.
Resolution: The author argues that the spin flips constitute a "manifest" record only if the spatial configuration of the spin particles changes (decoherence). In the specific setup described, the spins flip without spatial displacement; their spatial wave functions remain identical and overlap in configuration space. Consequently, the spin chain does not constitute a genuine measurement of the particle's position during the interference region.
Implication: The claim that the spin track represents the "actual" path is unwarranted within standard QM (which lacks trajectories) and inconsistent within pilot-wave theory (where the trajectory is determined by the total wave function, including the overlapping spin states). The paradox dissolves when one recognizes that not all "position" measurements (or spin flips) correspond to the actual position of a particle.

Significance and Claims
The paper concludes that the debate over the detectability of Bohmian trajectories hinges on the insufficient demarcation between accessible measurement records and the ontological entities they are supposed to represent.

On Weak Measurements: The author asserts that weak velocity measurements offer no new empirical or non-empirical arguments in favor of standard de Broglie–Bohm theory over its alternatives. They are best understood as operational manifestations of the wave function's phase gradient within standard QM. The claim that they "vindicate" Bohmian trajectories is a result of naively identifying weak values with actual positions.
On Surrealistic Trajectories: The paper argues that the "surrealistic" trajectory argument fails because it assumes a direct correspondence between a spin-flip record and a particle's trajectory that is not supported by the theory's dynamics. The experiment does not falsify pilot-wave theory; rather, it highlights that in quantum theory, a measurement record does not necessarily reveal the "actual" trajectory of a particle, even if such trajectories are assumed to exist.
Overall Conclusion: The chapter reinforces the view that pilot-wave theory's individual trajectories remain observationally inaccessible by construction. Attempts to detect them via weak measurements or to falsify them via surrealistic trajectories fail because they rely on unjustified assumptions about the relationship between manifest data and non-manifest ontology. The analysis suggests that a coherent treatment of these phenomena requires careful adherence to the distinction between the manifest and non-manifest domains, without which "naive realism" about measurement outcomes leads to paradoxes.

The Invisible Roadmap: Why We Can't See the Path

The Paper's Verdict: The Map is Not the Territory

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