Completion of the Classical Self-Force Problem: Well-Posed Discrete Dynamics and Its Continuum Limit

For over a century, the motion of a charged particle in its own electromagnetic field has resisted complete mathematical treatment. The Abraham-Lorentz-Dirac equation, derived from classical electrodynamics, successfully predicts radiation reaction—the Larmor formula, synchrotron radiation, and energy loss in accelerators. Yet it carries a flaw that has bothered physicists since Lorentz and Abraham first wrote it down in the early 1900s. The equation contains third-order derivatives. Those derivatives allow unphysical solutions where particles accelerate exponentially without any external force. To eliminate these runaway solutions, one must impose a boundary condition that reaches into the future, causing the particle to begin accelerating before the force that pushes it arrives.

These pathologies are not failures of classical electrodynamics. They are artifacts of treating particles as mathematical points. A point has infinite energy density at its location. The self-force calculation subtracts one infinity from another and keeps the finite remainder. The remainder is correct—the ALD self-force matches experiment—but the procedure that produced it cannot be the fundamental theory.

This paper provides the completion that the point-particle idealization lacks. We prove that on a discrete spacetime lattice with a fundamental minimum length, the coupled dynamics of particles and fields is globally well-posed. Solutions exist for all time, are unique, and depend continuously on initial data. The maximum acceleration of any particle is bounded by the lattice spacing. All interactions are causal. There are no runaway solutions and no pre-acceleration.

The continuum limit of this discrete system recovers the ALD equation at macroscopic distances where the point-particle approximation is valid. At distances comparable to the minimum length, the discrete structure provides a well-defined short-distance completion—elastic scattering at a minimum separation—that replaces the singular behavior of the continuum theory.

The result is a completion, not a replacement. The ALD equation remains correct in its domain. The discrete short-distance physics extends classical electrodynamics to all scales without pathologies. This closes a century-old open problem in mathematical physics by identifying the regularization that the point-particle continuum theory could never supply from within.

This work is extracted from the Emergence model, a unified framework in which spacetime itself is a discrete lattice formed from wave intersections on a pre-geometric canvas.