D. Noja, A. Posilicano
THE WAVE EQUATION WITH ONE POINT INTERACTION AND THE ( LINEARIZED )
CLASSICAL ELECTRODYNAMICS OF A POINT PARTICLE
(258K, postscript)
ABSTRACT. We study the point limit of the linearized Maxwell--Lorentz equations
describing the interaction, in the dipole approximation, of an extended
charged particle with the electromagnetic field. We find that this
problem perfectly fits into the framework of singular perturbations
of the Laplacian; indeed we prove that the solutions of the
Maxwell--Lorentz equations converge -- after an infinite mass
renormalization which is necessary in order to obtain a non trivial
limit dynamics -- to the solutions of the abstract wave
equation defined by the self--adjoint operator describing the
Laplacian with a singular perturbation at one point. The elements
in the corresponding form domain have a natural decomposition into
a regular part and a singular one, the singular subspace
being three--dimensional. We obtain that this three--dimensional
subspace is nothing but the velocity particle space, the particle
dynamics being therefore completely determined -- in an explicit
way -- by the behaviour of the singular component of the field.
Moreover we show that the vector coefficient giving the singular
part of the field evolves according to the Abraham--Lorentz--Dirac
equation.