Radiation reaction in electrodynamics and the elimination of runaway solutions

Abstract

The familiar Abraham-Lorentz theory of radiation reaction in classical non-relativistic electrodynamics exhibits many problems such as “runaway solutions” and violation of causality. As shown by many authors, such problems can be alleviated by dropping the assumption of a point electron. We also drop this assumption (by introducing a form-factor with a large cutoff frequency Ω) but we present a new approach based on the use of the generalized quantum Langevin equation. For an electric dipole interaction, an exact treatment is possible and we obtain a new equation of motion which, in spite of being third order, does not lead to runaway solutions or solutions which violate causality (the sole proviso being that Ω cannot exceed an upper limit of 3Mc 3 2e 2 =1.60×10 23 s -1 ). Furthermore, Ω appears in the third-derivative term but we show that, to a very good approximation, this term may be dropped so that we end up with a simple second-order equation which does not contain Ω and whose solutions are well-behaved.