ON CLASSICAL ELECTRODYNAMICS OF POINT PARTICLES AND MASS RENORMALIZATION Some preliminary results

We consider the problem of nding rigorous results for the dynamics of a classical charged point particle interacting with the electromagnetic eld, as described by the standard Maxwell{Lorentz equations. Some results are given for the corresponding linearized system, i.e. the so called dipole approximation, in the presence of a mechanical linear restoring force. We regularize the system by taking a form factor for the particle (Pauli{Fierz model) and study the limit of the particle's motion as the regularization is removed. We prove that (i) if the regularization is removed but mass is not renormalized the motion is trivial (i.e. the particle does not move at all); (ii) if the regularization is removed and mass is renormalized, the particle's motion corresponding to smooth initial data for the eld has a well deened nontrivial limit; (iii) in the case of vanishing initial eld the limit motion satisses exactly the Abraham{Lorentz{Dirac equation; (iv) for generic initial data the limit motion is runaway.