An immediate difficulty presents itself: The vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.
Difficult but not impossible. To find a way around this problem I note first that the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the time-reversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose , the retarded solution, as the physically-relevant solution. Choosing instead the advanced solution would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition
My second key observation is that while the potential of Equation (2) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the fact that , , and all satisfy Equation (1), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line – all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: It stands for ‘singular’ as well as ‘symmetric’.
Because is just as singular as , removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because is known not to affect the motion of the charged particle, this new potential must be entirely responsible for the radiation reaction. We therefore introduce the new potential
The self-action of the charge’s own field is now clarified: A singular potential can be removed from the retarded potential and shown not to affect the motion of the particle. (Establishing this last statement requires a careful analysis that is presented in the bulk of the paper; what really happens is that the singular field contributes to the particle’s inertia and renormalizes its mass.) What remains is a well-behaved potential that must be solely responsible for the radiation reaction. From the radiative potential we form an electromagnetic field tensor , and we take the particle’s equations of motion to be1.
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