March 2026 - The 4-3 Problem

The discrepancy between Maxwellian electrodynamics and the classical interpretation of the elementary character of the electron is known as the "4/3 problem". In it, a sharp distinction is made between the electron dynamics, and the particle/field interaction. Special relativity sees particle mass increasing with velocity. Its formalism treats the 4-momentum of a field distribution as Lorentz-covariant, though this formalism can't describe the process by which electrons move in a stationary three-dimensional Euclidean reference system through the effects of electrical fields alone, due to the instability of classical electron, to the point where constant motion of the electron begins posing a problem. The reference system is chosen arbitrarily, and this is in fact a characteristic of the particle, when describing its dynamics. In the field-description of the electron, its energy-momentum distribution transforms along with the Lorentz-boost as it moves at some speed v = cβ. At rest, each of the electric field components contributes to the energy density with one third of the rest energy density. The energy of a moving charge then is atypical for particles

When further applied to the momentum expression, there are momentum density contributions perpendicular to the velocity, which is also in contradiction to the interpretation of a particle with a rest mass. Unfortunately, the soliton-solution would similarly expect E(β) = γE(0), which would be in line with the pure particle-interpretation. However, when comparing the different energy expressions, the addition of a term equal third of the rest energy with a γ^-1 behavior would provide a close estimate of the expectation. Due to its constancy under Lorentz-indices, it would need to be a potential energy contribution. This could provide an addendum to the 4/3-Problem. At a high-level inspection, whereas "regular" description sees (isolated) electrons as localized in a small volume with quantized charge and a field extending to infinity, this approach would remove the particle character of the electron, instead describing them as purely electromagnetic, through which their dynamics are describable by up to 3 field degrees of freedom.