Pilot-Wave Theories as Hidden Markov Models

The original version of the de Broglie-Bohm pilot-wave theory, also called Bohmian mechanics, attempted to treat the wave function or pilot wave as a part of the physical ontology of nature. More recent versions of the de Broglie-Bohm theory appearing in the last few decades have tried to regard the pilot wave instead as an aspect of the theory’s nomology, or dynamical laws. This paper argues that neither of these views is correct, and that the de Broglie-Bohm pilot wave is best understood as a collection of latent variables in the sense of a hidden Markov model, a construct that was not available when de Broglie and Bohm originally formulated what became their pilot-wave theory. This paper also discusses several other challenges for the ontological view of the pilot wave. One such challenge is due to Foldy-Wouthuysen gauge transformations, which connect up with the Deotto-Ghirardi ambiguity in the de Broglie-Bohm theory. Another challenge arises from the freedom to carry out canonical transformations in the wave function’s own notion of phase space, as defined by Strocchi and Heslot.

💡 Research Summary

The paper revisits the de Broglie‑Bohm pilot‑wave theory and argues that the wave function (pilot wave) should not be interpreted either as a physical ontology or as a nomological element of the dynamical laws. Instead, the author proposes that the pilot wave is best understood as the set of latent variables in a hidden Markov model (HMM). After a thorough survey of six dominant views on the quantum state—statistical, epistemic‑over‑measurements, epistemic‑over‑ontology, ontological‑monistic, ontological‑pluralistic, and nomological—the author shows that each faces serious shortcomings: the first two leave the measurement problem untouched, the epistemic‑over‑ontology view runs afoul of the PBR theorem, the ontological‑pluralistic view struggles with the abstract nature of configuration‑space wave functions, and the nomological view is burdened by complex initial conditions and time‑reversal behavior.

The paper then clarifies the notion of Markovianity, distinguishing dynamical Markovianity (future determined solely by the present) from causal Markov conditions used in causal modeling. A non‑Markovian theory can be rendered Markovian by augmenting its state space with unobservable variables, yielding a hidden Markov model. The author lists seven hallmark properties of HMM latent variables—abstraction, non‑uniqueness, unobservability, non‑spatiality, lack of back‑reaction, multivariance, and contingency—and demonstrates that the pilot wave exhibits all of them. The wave function lives in configuration space, is complex‑valued, evolves autonomously under the Schrödinger equation, and guides particle positions via the guiding equation without being directly affected by those positions. Its freedom under gauge and canonical transformations further underscores its non‑unique, abstract character.

Section 3 argues that the pilot‑wave theory for a fixed number of non‑relativistic particles is precisely a hidden Markov model: the observable configuration (particle positions) constitutes the visible state, while the wave function constitutes the hidden state that determines the transition probabilities (in fact, deterministic transitions) to the next configuration. The author points out that the pilot wave’s evolution is independent of the particle configuration (no back‑reaction), satisfying the HMM requirement that latent variables evolve on their own.

Section 4 raises three technical challenges to an ontological reading of the wave function. First, Foldy‑Wouthuysen gauge transformations, originally introduced to separate positive‑ and negative‑energy components in relativistic theory, can be applied to the pilot wave, producing different but physically equivalent representations. This mirrors the Deotto‑Ghirardi ambiguity, showing that the wave function’s representation is highly non‑unique. Second, the Strocchi‑Heslot construction of a phase‑space structure for the wave function (a symplectic manifold with canonical coordinates) allows a wide class of canonical transformations that leave observable predictions unchanged, again emphasizing the wave function’s status as a flexible mathematical device rather than a fixed ontological entity. Third, these freedoms collectively undermine attempts to treat the wave function as a concrete field in three‑dimensional space.

Section 5 disentangles terminology across stochastic processes, causal modeling, and quantum foundations, emphasizing that “hidden variables” in quantum theory (e.g., particle positions) differ from the “latent variables” of an HMM, which are the wave functions themselves in the present analysis.

In conclusion, the author contends that viewing the pilot wave as the latent component of a hidden Markov model resolves the tensions between ontological and nomological interpretations, accommodates the gauge and canonical transformation freedoms, and provides a natural, mathematically precise framework for pilot‑wave dynamics. Future work is suggested on extending the HMM picture to relativistic and field‑theoretic settings, exploring the role of stochastic extensions, and identifying possible experimental signatures of the hidden‑state structure.