Pilot-Wave Theories as Hidden Markov Models

This paper argues that the de Broglie-Bohm pilot wave should be interpreted as a collection of latent variables within a hidden Markov model rather than as a physical entity or a dynamical law, while also highlighting ontological challenges posed by gauge and canonical transformations.

The Gist

Imagine you are watching a high-tech, automated dance performance. You see the dancers (the particles) moving across the stage in very specific, intricate patterns. You want to know why they are moving that way.

In physics, there is a famous theory called Bohmian Mechanics (or Pilot-Wave Theory). It says there are two things happening: there are the actual dancers (the particles), and there is an invisible "ghostly" wind or wave (the pilot wave) that pushes the dancers around.

For decades, scientists have argued about what that "ghostly wave" actually is. Is it a real, physical thing like a gust of wind? Is it just a law of nature, like gravity?

This paper, written by Jacob Barandes, argues that both sides are wrong. He suggests a third, much more clever option: The wave isn't a "thing" or a "law"—it’s a mathematical "cheat sheet" used to make a complicated story look simple.

Here is the breakdown of his argument using three simple analogies.

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1. The "Hidden Clock" (The Hidden Markov Model)

Imagine you are watching a movie of a person walking. Suddenly, the person freezes for exactly one minute, then starts walking again. To you, it looks like the universe just "glitched." It looks like the person's movement depends on something that happened a minute ago. This is what physicists call non-Markovian—it means the future depends on the past in a weird, jumpy way.

But what if there is a hidden clock behind the scenes? If you knew the clock was ticking, you’d realize the person isn't "glitching"; they are just following a schedule. The clock is a "latent variable"—it’s not a physical object you can touch, but it makes the "glitchy" movement look like a smooth, predictable process.

Barandes argues that the Quantum Wave is exactly like that hidden clock. The universe might actually be "glitchy" and complicated (non-Markovian), but we use the "Wave" as a mathematical tool to make it look like a smooth, predictable dance (Markovian).

2. The "Ptolemaic Epicycles" (The Fake Circles)

Back in ancient times, astronomers noticed planets moving in weird loops in the sky. To explain this, they invented "epicycles"—tiny little circles moving on top of bigger circles. They treated these tiny circles as if they were real, physical objects in space.

Eventually, we realized the epicycles weren't real; they were just a mathematical way to describe the much simpler reality of gravity and orbits.

Barandes is saying that treating the "Pilot Wave" as a real, physical object is like the ancient astronomers treating epicycles as real. The wave looks like it's "pushing" particles, but it’s actually just a mathematical "loop" we've drawn to explain a more complex reality. The wave is a mathematical shadow, not a physical object.

3. The "Multiple Maps" Problem (The Gauge Ambiguity)

Imagine you are trying to navigate a city using a map. However, you realize that for every street, there are a thousand different maps that all show the same streets but use different colors, different symbols, or different ways of drawing the turns.

If you try to say, "The blue line on this specific map is the actual, physical road," someone else could show you a red map that predicts the exact same driving route and say, "No, the red line is the real road!"

Barandes points out a massive problem for the "Pilot Wave" theory: math shows that you can change the "wave" in infinite ways (using something called Foldy-Wouthuysen transformations) without changing the actual movement of the particles.

If you can change the "wave" completely and the particles still move the same way, how can you claim the wave is a real, physical thing? It’s like claiming the "blue line" on a map is a physical object when the map itself is just a representation.

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The Big Picture Summary

The paper concludes that the "Pilot Wave" is a Hidden Markov Model.

Instead of being a physical "ghost wind" pushing particles, the wave is a collection of latent variables—mathematical placeholders that help us translate a messy, complicated, "glitchy" universe into a clean, predictable set of equations.

In short: The wave isn't the wind blowing the dancers; it's just the sheet music the dancers are following to make sense of a very chaotic orchestra.

Technical Summary

Technical Summary: Pilot-Wave Theories as Hidden Markov Models

Author: Jacob A. Barandes
Date: February 12, 2026

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1. The Problem: The Ontological Status of the Wave Function

The central problem addressed in this paper is the metaphysical ambiguity surrounding the quantum state (the wave function, $\Psi$). In the context of de Broglie-Bohm (pilot-wave) theory, there is a long-standing debate regarding whether the pilot wave is:

• Ontological: A physical, real entity (a "field") existing in configuration space.
• Nomological: A feature of the dynamical laws of nature (like a law of motion).

The author argues that both views are conceptually flawed. The ontological view struggles with the "polydimensional" nature of the wave function (existing in $3N$-dimensional configuration space rather than 3D physical space), while the nomological view struggles with the wave function's contingent initial conditions and complex time-dependence, which are atypical for fundamental laws.

2. Methodology: The Hidden Markov Model (HMM) Framework

The author introduces a novel interpretational framework by applying the mathematical construct of Hidden Markov Models (HMMs)—a concept developed in the 1960s and 70s—to quantum foundations.

The methodology involves:

• Defining Markovianity: A theory is dynamically Markovian if its present state uniquely determines its future state.
• Identifying Latent Variables: If a non-Markovian process (where the future depends on the past) can be made to appear Markovian by adding unobservable variables, those variables are "latent" or "hidden."
• Comparative Analysis: The author identifies seven "smoking-gun" characteristics of latent variables (Abstraction, Non-uniqueness, Unobservability, Non-spatiality, Absence of backreaction, Multivariateness, and Contingency) and tests whether the pilot wave $\Psi$ satisfies them.

3. Key Contributions and Results

A. The Pilot Wave as a Latent Variable
The paper’s primary contribution is the proof that the de Broglie-Bohm pilot wave is not a physical field or a law, but a collection of latent variables in an HMM. The author demonstrates that $\Psi$ satisfies all seven hallmark characteristics of latent variables. Specifically, the "absence of backreaction"—the fact that the particles do not affect the evolution of the wave function—is a definitive indicator of a latent variable. This allows the theory to be viewed as a deterministic HMM of a fundamentally non-Markovian universe.

B. Connection to the Deotto-Ghirardi Ambiguity
The author provides a new mathematical derivation for the Deotto-Ghirardi ambiguity (the ability to add divergence-free terms to the current density without changing empirical predictions). He shows that this ambiguity is not an ad hoc mathematical quirk but a direct consequence of Foldy-Wouthuysen gauge transformations.

C. Challenges to Ontological Realism
The paper presents two major technical challenges to the ontological view:

1. Gauge Non-Invariance: The author shows that under Foldy-Wouthuysen transformations, the pilot wave, the probability current, and even the particle trajectories are not gauge-invariant. In physics, structures that are not gauge-invariant are typically not considered "real" or ontological.
2. Strocchi-Heslot Phase Space Ambiguity: The author argues that the wave function possesses its own infinite-dimensional phase space. Because there are no observables within this space to "fix" a preferred coordinate system, there are infinitely many "canonical frames" for the wave function. This makes the "ontology" of the wave function fundamentally ill-defined.

4. Significance

The significance of this work is twofold:

• Philosophical/Interpretational: It offers a "third way" to resolve the ontological-nomological fork in Bohmian mechanics. By treating the wave function as a latent variable, one avoids the "epicycles" of treating abstract, non-spatial, non-backreacting mathematical constructs as physical objects.
• Foundational: It suggests a radical shift in how we view quantum mechanics. The author posits that quantum theory itself might be the result of attempting to force fundamentally non-Markovian physical processes into a Markovian mathematical paradigm. This positions the "exotic" features of quantum mechanics (like interference and entanglement) as the natural consequences of this mathematical embedding.