Instantaneous Action-at-a-Distance Formulation of Classical Electrodynamics

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Abstract
The possibility of formally translating the interaction of charges from charge ↔ field ↔ charge to charge ↔ charge, where the orbits satisfy Newtonian (second order in t), yet covariant, equations of motion, is exploited for the Wheeler‐Feynman interaction. A method for computing the forces on the charges correct to second order in the coupling constant e2 is presented, and ten constants of the motion correct to e2 are found. The integration is effected via the Noether theorem with the inhomogeneous Lorentz group as symmetry transformations. An important result is that a well‐known correction to the Coulomb interaction which accounts for the uniform motion of charges is revealed to be, to first order in e2, a frame‐invariant expression. The consequent corrected Coulomb dynamics admits first‐order integrals identical to those of the Wheeler‐Feynman dynamics.