Convergence to Bohmian mechanics in a de Broglie-like pilot-wave system

Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian particle does not affect its guiding wave, so the wave field must instead be prescribed by the system geometry. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems, including the "walking droplet" system of Couder and Fort (2006) and its various abstractions, allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush (2024) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie's early "double-solution" theory. We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Finally, we present an application of the present work in developing an analogue for position measurement in a de Broglie-like setting.