[Paper Review] Wave-corpuscle mechanics for elementary charges

Anatoli Babin, Alexander Figotin|arXiv (Cornell University)|Dec 14, 2008

Geophysics and Sensor Technology50 references3 citations

TL;DR

This paper introduces wave-corpuscle mechanics as a classical field theory framework to resolve the self-energy divergence problem of point charges in electromagnetism. By modeling elementary charges as localized, complex-valued wave functions with a nonlinear self-interaction term, the theory yields exact solutions that reproduce Newtonian dynamics for nonrelativistic motion and de Broglie wave behavior, while maintaining Lorentz and gauge invariance.

ABSTRACT

It is well known that a concept of point charge interacting with electromagnetic (EM) field has a problem. To address that problem we introduce a concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient - a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free or accelerated motion. It carries explicitly features of a point charge and the de Broglie wave. A system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton motion equations for point charges interacting with the Lorentz forces. In this regime the nonlinearities are "stealthy" and don't show explicitly anywhere, but they provide for binding forces that keep localized every individual charge.

Motivation & Objective

• To resolve the classical self-energy divergence problem of point charges interacting with the electromagnetic field.
• To develop a Lorentz- and gauge-invariant classical field theory that describes elementary charges as localized wave-particles.
• To reproduce the dynamics of point charges under Lorentz forces in the nonrelativistic limit using wave-corpuscle solutions.
• To unify features of classical electrodynamics, de Broglie’s matter waves, and quantum-like behavior within a deterministic classical framework.

Proposed method

• Introduces a complex-valued wave function over spacetime to represent each charge, replacing the point-particle model.
• Constructs a local, gauge-invariant, and Lorentz-invariant Lagrangian with a nonlinear self-interaction term to bind the wave function into a localized structure.
• Derives Euler-Lagrange equations from the Lagrangian to describe free and accelerated motion of wave-corpuscles.
• Uses the energy-momentum tensor and averaged quantities to recover point-particle dynamics in the nonrelativistic regime.
• Employs Green’s functions and Fourier transforms to solve field equations and analyze wave function structure.
• Demonstrates that the nonlinear term acts as a 'stealthy' binding force, invisible in classical equations but essential for localization.

Experimental results

Research questions

• RQ1 Can a classical field theory describe localized, point-like charges without the self-energy divergence problem of classical electrodynamics?
• RQ2 How can a wave-corpuscle model reproduce the Lorentz force law for nonrelativistic motion while preserving gauge and Lorentz invariance?
• RQ3 What role does the nonlinear self-interaction term play in maintaining charge localization and enabling de Broglie wave behavior?
• RQ4 How do wave-corpuscle solutions for accelerated charges relate to classical point-particle dynamics and the de Broglie hypothesis?
• RQ5 To what extent does this framework reproduce features of quantum mechanics, such as the Schrödinger equation and continuity equations?

Key findings

• The wave-corpuscle model provides exact solutions to the Euler-Lagrange equations that describe both free and accelerated motion of charges with point-like features.
• In the nonrelativistic limit, the wave-corpuscle dynamics reproduces Newton’s second law with Lorentz forces, even though the nonlinearities are 'stealthy' and do not explicitly appear in the equations of motion.
• The nonlinear self-interaction term ensures charge localization and acts as a cohesive force, preventing wave function dispersion.
• For a single free charge at rest, the wave-corpuscle solution exhibits a spherically symmetric form factor whose potential asymptotically approaches the Coulomb potential.
• The theory reproduces the Schrödinger equation in the nonrelativistic limit when the wave function is expressed in terms of probability density and current, with the same continuity equation.
• The energy-momentum tensor derived from the wave-corpuscle model allows recovery of point-particle mechanics via spatial averaging, validating the classical limit.