Variational principle for the Wheeler–Feynman electrodynamics

Abstract

We adapt the formally-defined Fokker action into a variational principle for the electromagnetic two-body problem. We introduce properly defined boundary conditions to construct a Poincarèinvariant-action-functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an endpoint along each trajectory plus the respective segment of trajectory for the other particle inside the lightcone of each endpoint. We show that the conditions for an extremum of our functional are the mixed-type-neutral-equations with implicit state-dependentdelay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechét-differentiable. We develop a method to calculate the second variation for C 2 orbital perturbations in general and in particular about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some O(1) radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space H 2 0 of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the statedependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits.

Double-Slit and Electromagnetic Models to Complete Quantum Mechanics

“…The electromagnetic fields are simply the coupling terms to the other particle's trajectories as defined by the Euler-Lagrange equations of the variational method. Rigorously speaking, these Euler-Lagrangecoupling-fields are defined only on the trajectories, even though given by the exact same usual formulas of Maxwell's electrodynamics 6 . Our model uses the equations of motion of point charges keeping only the farfield interactions.…”

“…In the Wheeler-Feynman electrodynamics one does not solve differential equations for the electromagnetic fields, but rather the trajectories are the critical points (minimizers) of a variational method with mixed-type boundaries 6 . The electromagnetic fields are simply the coupling terms to the other particle's trajectories as defined by the Euler-Lagrange equations of the variational method.…”

“…(8) and (9)), leaving the bound charges undisturbed. An approximation capturing the essentials of this five-body-dynamics is by orbits with discontinuous velocities 15 of the electromagnetic variational method 6 . In the following we approximate the trajectory of scattered charge 3 by two segments with piecewise-constant-velocity, having a single point where velocity 3 is discontinuous.…”

“…We give a simplified model based on electromagnetic trajectories with vanishing far-fields, a model that explains the qualitative results of interference. In Section III we discuss the equations of motion of point charges of the Wheeler-Feynman electrodynamics and the electromagnetic variational method 6 . In Section III we also work out the details of a model for scattering with each slit represented by a heavy neutral atom, a system with little recoil that is not destroyed upon scattering, unlike a model for bowling.…”

“…As with any modeling, if one starts with a narrow enough setup it is easy to obtain impossibility proofs, and the next step is usually harder, i.e., to define a wider minimum setup. We discuss our models in the light of available experimental work 5 and a finite variational method for the Wheeler-Feynman electrodynamics 6 . No single realistic dynamical model based on electrodynamics has been analyzed down to microscopic detail up to now.…”