Cause of radiation resistance I was reading Radiation Damping and light scattering from Feynman's lectures on physics. The following excerpts are from chapter 32, Vol I  

What is this radiation resistance due to? Let us take a simple
  example: let us say that currents are driven up and down in an
  antenna. We find that we have to put work in, if the antenna is to
  radiate energy.....Against what force are we doing this work?  

Then he goes on to say that this problem is resolved in case of such antennas(field of charges in one part of the antenna produce a force on charges in other part), but not for "point" charges or electrons. He provides an obsolete explanation based on the finite size and internal structure of an electron but goes on to say that it is false, and hence the question is unanswered.   
Now knowing that this excellent book was published in 1964, has the modern quantum mechanical ideas solved this problem for an electron?
 A: No, quantum theory only made the divergences easier to ignore. The order of divergences was found to be lower in quantum theory than in classical theory, but the mass and other things still diverge.
There are at least two ways to deal with this. One, which is prevalent today, is to accept this as a sign of the fact that the quantum theory of EM field and electrons is inconsistent without additional fields or structure of the electron, but still very useful and successful in description of low energy situations.
The explanation based on finite size and internal structure is not necessarily false, but it is hard to work out mathematically and is lacking uniqueness in the choice of the structure.
The other way is to hold to the point character of the electron and rectify the rest of the EM theory; mainly the assumption that the EM energy is always distributed with energy density $\frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2 $ (or, which is similar, the assumption that the Lagrangian for EM field is given by $j_\mu A^\mu - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$.) For example, see the paper 
J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct
Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.
http://dx.doi.org/10.1103/RevModPhys.21.425
and the references therein (their special choice of fields as half-retarded, half-avanced is not the only one that solves the divergences, one may work also with purely retarded fields with common boundary conditions, if he does not require validity of the Lorentz-Abraham-Dirac equation).
