Difference between Coulomb law and QED

The use of 'point particles' to describe the results of classical electromagnetism is questionable if it is assumed the particles really are point-like but have a finite charge. Therefore one must avoid invoking such impossible entities. Introductory textbooks usually do not go into this.

One can treat the whole of classical electromagnetism without ever invoking the notion of a point particle with finite charge. In fact one ought to do this, because the notion of a point particle with finite charge is unphysical. Such an entity would have an electric field around it scaling as $1/r^2$, and consequently an energy density scaling as $1/r^4$; upon integrating this energy density over volume one has a value that goes as $1/r$ so is infinite in the limit $r rightarrow 0$. This means the entity would have infinite inertia. There have been attempts to argue that one can simply ignore this infinite inertia, but then other problems arise such as self-acceleration of small dipoles. Overall, the classical point particle with finite charge does not make physical sense and no such thing exists in the natural world.

With this in mind, we now have two issues: how is classical electromagnetism done correctly, and what is the nature of charged 'particles' such as electrons.

Classical electromagnetism is done correctly by insisting that charge density $rho$ (charge per unit volume) cannot be infinite. If one wishes to treat small physical entities, then one can do so by allowing them to have finite charge density, and then their total charge tends to zero when their physical size does. When one refers to a 'point charge' it can then be taken as understood that what one has in mind is something like a small spherical body with a radius $r$ that is small compared to all significant distances in the problem under discussion, but not zero, and not so small that the energy of the particle's own field exceeds the particle's rest energy. The latter distance can be estimated from $$ frac{q^2}{4piepsilon_0 r} le m c^2 $$ where $m$ is the rest mass. If a small entity has mass $m$ and radius $r$ then its charge must be less than the limit set by this equation. If a small entity has mass $m$ and charge $q$ then its radius must be larger than the limit set by this equation.

When a charged body is accelerated by the application of a force, the forces associated with different parts of the body exerting forces on one another via their fields create a net contribution called self-force which complicates the analysis. This is also called radiation reaction. It is negligible at ordinary accelerations, but just noticeable in modern particle accelerators.

Finally, what about electrons and things like that? These entities cannot be fully described by classical physics; they require quantum physics for a good understanding. This is provided by quantum electrodynamics. An electron is an excitation of a set of interacting fields called Dirac field and electromagnetic field. It can be localized in space, but it cannot be perfectly localized at a point; as one attempts to do that one encounters large fields which in turn lead to electron-positron pair creation. On the other hand, the electron does not have spatial structure itself, at least on distance scales down to femtometres which have been probed in collision experiments, but it does possess a fixed charge. Such an entity really cannot be described by classical mechanics at that distance scale. However, the behaviour of electrons at larger distance scales, larger than than atoms for example, can be modeled reasonably well by treating them as small spheres---but not as point particles. This is just a model, but it is better than no model at all. A suitable size to pick for the small sphere is the radius given by the above formula: $$ r_c = frac{e^2}{4piepsilon_0 m c^2} $$ This is called the classical radius of the electron, value about $2.8 times 10^{-15}$ m.

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Refs. I am referring to my own work here because it is the best resolution of the issues that I am aware of. The first paper below begins with a short review of other literature. The second paper is very readable I think and will interest anyone interested in these issues.

A. Steane, Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism, arXiv:1402.1106 doi 10.1119/1.4897951

A. Steane, Tracking the radiation reaction energy when charged bodies accelerate, arXiv:1408.1349, doi 10.1119/1.4914421