Renormalization of the Lorentz-Abraham-Dirac equation for radiation reaction force in classical electrodynamics

Abstract

While he derived the equation for the radiation force, Dirac (1938) mentioned a possibility to use different choices for the 4-momentum of an emitting electron. Particularly, the 4-momentum could be non-colinear to the electron 4-velocity. This ambiguity in the electron 4-momentum allows us to assume that the mass of emitting electron may be an operator, or, at least, a 4-tensor instead of being the usually assumed scalar, which relates the 4-velocity of a bare charge to the total momentum of a dressed point electron, the latter being a total of the momentum of the bare electron and that of the own electromagnetic field. On applying the re-normalization procedure to the mass operator, we arrive at an interesting dichotomy. The first choice (more close to traditional one) ensures the radiation force to be orthogonal to the 4-velocity. In this way the re-normalization results in the Lorentz-Abraham-Dirac equation or in the Eliezer equation. However, the 4-momentum of electron in this case is not well defined: the equality in the relativistic entity (E/c)^2=m^2c^2+p^2 appears to be broken and even the energy is not definite positive. The latter is an underlying reason for the 'run-away' solution. The other choice is to require the radiation force to be orthogonal to the 4-momentum (not to the 4-velocity). In this case the energy and momentum are well-defined and obey the relationship (E/c)^2=m^2c^2+p^2. Remarkably, the equations of a particle's motion in this case differ significantly from all the known versions. They appear to be well founded. They are simple, easy to solve, and can be applied to simulate the particle motion in the focus of an ultra-bright laser.