Revisiting de Broglie's Double-Solution Pilot-Wave Theory with a Lorentz-Covariant Lagrangian Framework

TL;DR

This paper develops a Lorentz-covariant, local pilot-wave model based on de Broglie's double-solution theory, demonstrating how particle vibrations, wave-particle interactions, and energy relations emerge naturally within a classical field framework.

Contribution

It introduces a variational, Lorentz-invariant pilot-wave system that reproduces de Broglie's ideas with novel local dynamics and particle-wave coupling, advancing the classical field formulation of quantum phenomena.

Findings

Particles exhibit Compton-scale vibrations matching de Broglie's frequency.

Wave patterns dynamically adjust to satisfy the de Broglie relation at the particle's position.

The model implies a classical analogue of the Heisenberg uncertainty principle.

Abstract

The relation between de Broglie's double-solution approach to quantum dynamics and the hydrodynamic pilot-wave system has motivated a number of recent revisitations and extensions of de Broglie's theory. Building upon these recent developments, we here introduce a rich family of pilot-wave systems, with a view to reformulating and studying de Broglie's double-solution program in the modern language of classical field theory. Notably, the entire family is local and Lorentz-invariant, follows from a variational principle, and exhibits time-invariant, two-way coupling between particle and pilot-wave field. We first introduce a variational framework for generic pilot-wave systems, including a derivation of particle-wave exchange of Noether currents. We then focus on a particular limit of our system, in which the particle is propelled by the local gradient of its pilot wave. In this case, we see that the Compton-scale oscillations proposed by de Broglie emerge naturally in the form of particle vibrations, and that the vibration modes dynamically adjust to match the Compton frequency in the rest frame of the particle. The underlying field dynamically changes its radiation patterns in order to satisfy the de Broglie relation $p=\hbar k$ at the particle's position, even as the particle momentum $p$ changes. The wave form and frequency thus evolve so as to conform to de Broglie's "harmony of phases", even for unsteady particle motion. We show that the particle is always dressed with a Compton-scale Yukawa wavepacket, independent of its trajectory, and that the associated energy imparts a constant increase to the particle's inertial mass. Finally, we see that the particle's wave-induced Compton-scale oscillation gives rise to a classical version of the Heisenberg uncertainty principle.