Pilot-Wave Theories as Hidden Markov Models

TL;DR

The paper reframes the quantum wave function and pilot-wave dynamics as a hidden Markov model, proposing that the wave function acts as a latent variable that renders non-Markovian quantum evolution Markovian when augmented with appropriate latent structure. It surveys historical and methodological contexts for pilot-wave theories, introduces precise Bohmian formulations with equivariance and current-derived guiding equations, and then formalizes a continuous latent-field version of the HMM that reproduces pilot-wave behavior while highlighting the seven latent-variable characteristics. The work also argues that traditional ontological and nomological readings face challenges from interference phenomena and Foldy–Wouthuysen gauge non-uniqueness, suggesting the latent-variable interpretation as a natural, falsifiable-parsimonious account. Overall, the approach reframes foundational questions about quantum ontology and dynamics, offering a mathematically explicit framework that connects pilot-wave theory to well-developed concepts in stochastic modeling with potential implications for interpretation and generalization to broader quantum settings.

Abstract

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.