Radiation Reaction of the Classical Off-Shell Relativistic Charged Particle

It has been shown by Gupta and Padmanabhan that the radiation reaction force of the Abraham–Lorentz–Dirac equation can be obtained by a coordinate transformation from the inertial frame of an accelerating charged particle to that of the laboratory. We show that the problem may be formulated in a flat space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg–Feynman–Schwinger covariant mechanics (the zero mode fields of the 0, 1, 2, 3 components correspond to the Maxwell fields). Without additional constraints, the particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five dimensional field equations provides the classical Abraham–Lorentz–Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution as well as an autonomous non-linear equation of third order for the off-shell mass. The theory does not admit radiation if the particle does not move off-shell. The structure of the equations implies that mass-shell deviation is bounded when the external field is removed.