Self-energy divergence
The infinite electrostatic field energy U = (q²/8πε₀)·∫ dr/r² stored in the field of a point charge, diverging as 1/r at small radii. Root cause of the pathologies of classical radiation-reaction theory and the target of QED's mass-renormalisation procedure.
Definition
The self-energy divergence is the fundamental sickness of classical electrodynamics of point charges: the electrostatic field energy U = (ε₀/2)∫|E|² d³r stored in the Coulomb field of a point charge q is infinite. The integrand diverges as 1/r² at small radii and the volume integral therefore diverges logarithmically (for a line charge) or as 1/r (for a point). For a spherical charge of radius a the stored field energy is U = q²/(8πε₀ a), which goes to infinity as a → 0.
The divergence is the root of the radiation-reaction pathologies. The classical electron radius r_e = q²/(4πε₀ mc²) ≈ 2.82×10⁻¹⁵ m is the radius at which the stored field energy equals the electron's rest energy mc²; below r_e the field energy exceeds rest energy, so the electron's total energy in classical electrodynamics is dominated by electromagnetic self-energy before radiation reaction even enters the picture. The Abraham-Lorentz derivation computes the self-force of the charged sphere on itself and takes a → 0, at which point the inertia attributed to the electromagnetic field itself diverges — mass-renormalisation absorbs this into the observable (finite) mass m, but the finite residue is the third-order jerk term that then generates the runaway solutions and pre-acceleration problem.
Quantum electrodynamics resolves the classical divergences via systematic renormalisation: the formally infinite bare mass and bare charge are defined in terms of measured (finite) values at a chosen energy scale, and every observable physical quantity comes out finite. The renormalisation programme works because QED is renormalisable as a field theory — a non-trivial property that was not established until 1948 (Tomonaga, Schwinger, Feynman, Dyson). At the classical level, though, the self-energy divergence is simply a marker that classical electrodynamics is not a self-consistent theory of point charges, and that an ultimate honest treatment must pass through either extended-particle models (obsolete) or quantum field theory (the current standard).