Effective dynamics of a classical point charge

•Extension of the classical action principle for dissipative systems.•New derivation of the Abraham–Lorentz force for a point charge.•Absence of a runaway solution of the Abraham–Lorentz force.•Acausality in classical electrodynamics.•Renormalization of classical electrodynamics of point charges.

The effective Lagrangian of a point charge is derived by eliminating the electromagnetic field within the framework of the classical closed time path formalism. The short distance singularity of the electromagnetic field is regulated by an UV cutoff. The Abraham–Lorentz force is recovered and its similarity to quantum anomalies is underlined. The full cutoff-dependent linearized equation of motion is obtained, no runaway trajectories are found but the effective dynamics shows acausality if the cutoff is beyond the classical charge radius. The strength of the radiation reaction force displays a pole in its cutoff-dependence in a manner reminiscent of the Landau-pole of perturbative QED. Similarity between the dynamical breakdown of the time reversal invariance and dynamical symmetry breaking is pointed out.