Wave-Corpuscle Mechanics for Electric Charges

Abstract

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the 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 the 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 and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion 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 the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum.

Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

“…Namely, we derive here point dynamics governed by Newton's law from wave dynamics using concepts and methods of the classical field theory. That continues our study of a neoclassical model of electric charges which is designed to provide an accurate description of charge interaction with the electromagnetic field from macroscopic down to atomic scales, [7]- [11].…”

“…More examples of nonlinearities can be found in [7]- [11], and many facts from the theory of the NLS equations can be found in [20], [51]. The NLS equations with logarithmic nonlinearity are studied in [21], [19], [20].…”

“…Therefore, for given twice continuously differentiable potentials ϕ and A, a trajectory is a trajectory of concentration of asymptotic solutions of the NLS if and only if this trajectory satisfies Newton's law (1.9). The phase function S of the wave-corpuscle can be interpreted as the phase of de Broglie wave (see [7] for details). Hence, the described relation between Newtonian trajectories and concentrating asymptotic solutions can be interpreted as what is known as the wave-particle duality.…”

Explicit mass renormalization and consistent derivation of radiative response of classical electron

“…In the point localization limit the internal structure of the extended charge and the adjacent field disappear, but not without a trace: the value of the mass and the magnitude of the radiation reaction in the LAD equation depend on the parameters which determine dynamics of the extended charge and its adjacent field. Note that the introduction of Poincaré stresses based on a combination of the adjacent field with the nonlinearity is not the only possible way to balance the EM self-interaction for an extended charge in the field framework; it is possible to use only a properly defined nonlinearity without use of an adjacent field as in [4], but the analysis in the point limit meets with technical difficulties. The model involving the adjacent field which we use here allows for a detailed analysis in the point limit, and it is closer to the original Dirac's approach.…”

Relativistic Point Dynamics and Einstein Formula as a Property of Localized Solutions of a Nonlinear Klein-Gordon Equation

“…where m is a positive mass parameter and χ is a constant which in physical applications coincides with (or is close to) the Planck constant . The covariant derivatives in equation (3) are defined by∂…”