Why isn't it obvious that a particle doesn't interact with its own field, classically? The Wheeler-Feynman absorber theory or any other theory that tries to avoid the notion of field as an independent degree of freedom has always been concerned about infinite self energy of a charged particle. I don't feel confident that I appreciate the gravity of this problem. 
According to the theory of fields every charged particle has a field of its own whose strength varies with distance from the particle and this field is what which the particle uses to exert force on other charged particles. So, if the field is supposed to be an inalienable part attached to the particle that it uses to interact with other particles why would the field be used to exert a force on the particle itself and as such why an infinite self energy at the points very close to it? Am I ignoring any implication of this theory that leads to a logical debacle? I hope with your elaborate answers I can see why this isn't something to be deemed obvious and can appreciate the seriousness of the Abraham force.
 A: The main problem is radiation reaction. It's a fact that if I take a charge and jiggle it about, it will somehow "inject" energy into the electromagnetic field around it; we know this because we can detect this energy in the form of electromagnetic radiation. This means that when I grabbed the charge and jiggled it, I must have performed extra work on it (above the work needed to accelerate its mass) which then ends up as energy in the far field. This extra work can only have been performed against a force, which is electrical in nature since it depends on the charge.  Since the only electric field present is that of the charge, we conclude that the charge must have interacted with its own field in some way.
However, it's also obvious that a point charge cannot interact with its own electric field in the same way that it does with the field of other charges. At $\mathbf r=0$, the electric field $$\mathbf E=q\frac{\mathbf r}{r^3}$$ is singular and ill-defined; it doesn't even have a direction. So the self-interaction needs to be something different.
Everything that follows is an attempt to patch together these two disparate facts. How do you account exactly for that radiated energy, and how can you formulate consistent laws that have energy conservation built in intrinsically? In the far field, it is easy to account for the radiated energy via an electromagnetic energy density proportional to $|\mathbf E|^2$, but if you take this at face value then it blows up if you have point charges. If you try to take this directly but remove the point charge self energies by hand, the resulting theory is clunky, hard to use, and its results are not always Lorentz invariant. If you don't ascribe an energy content to the field (which means you cannot be considering it as a dynamical variable, as done by e.g. Wheeler-Feynman), how do you account for the fact that one can transfer energy via electromagnetic radiation?
More fundamentally, though: do the electromagnetic fields at a given point depend on how they were made? Is it enough to say "the electric field at $\mathbf r$ is 5V/m pointing in the $\hat{\mathbf z}$ direction"? Or do we need to specify which point charges created it? The latter stands very much against the concept of field, and how we use it in practice to calculate and prove things, so we'd need to have a very close look at all of electrodynamics (i.e. where is the 'which-charge' information in Maxwell's equations?). If the former is true, however, with what justification can we just yank out certain specific charge-dependent components from the EM field energy?
So you see, these are tough questions, and they are nowhere near solved, so it's an interesting area (but also it's not clear whether it's even possible to solve these questions, so keep that in mind).
A: I don't really know any high energy QFT, so I don't think I can explain a QFT concept in terms of classical physics and relate the two, but if I've understood the question I can give a pretty classical guess. Coulomb interaction is mediated by the electromagnetic field, and this interaction travels at the speed of light in vaccuum. So the electron "broadcasts" this signal, but it can't interact with the signal because the signal is flying away at the speed of light. As a naive side note: I think that if you tried to construct a theory of superluminal charged particles, you would need to consider the interaction of the particle with itself.
A: It does interact with its own field! This is the only way to get an accelerating charge to radiate away energy -- it has to lose energy itself, so it has to be repulsed by its own field.
