How should the singularity of the Coulomb's law be understood? [duplicate]

Question

The electric field at the point $\vec r$ due to a point charge $q$ at the origin $$\vec E=\frac{q}{4\pi\epsilon_0}\frac{1}{r^2}\hat{r}$$ blows up at the origin. In other words, the force between two point charges given by Coulomb's law $$\vec F=\frac{qq'}{4\pi\epsilon_0}\frac{1}{|\vec r-\vec r'|^3}(\vec{r}-\vec{r}')$$ fails at $r\to r'$.

I want to know why this unphysical divergence of Coulomb's law does not occur in real life. Is it because point charges don't exist (making it impossible to bring two charges arbitrarily close) or due to some other physics interfering with Coulomb's law?

In somewhat technical terms, this theory seems to need a UV regulator or a short-distance cut-off. I want to know what is the origin of this cut-off and in particular, whether it has anything to do with the nonexistence of point charges.

Answer 1

I want to know why this unphysical divergence of Coulomb's law does not occur in real life.

The brusque answer would be, it does not occur because it is a defective mathematical model, an invention of human mind motivated by experiments, but world is always richer and more complicated than any set of experiments or model can reveal.

For example, in microscopic systems, while we still believe the Coulomb law is accurate in a sense (the Coulomb potential in atoms/molecules), the whole theory is different from classical or relativistic mechanics of mass points: particles are described via psi functions,and are allowed to be anywhere with some probability. Including being very close to each other (and close to singularity), having arbitrarily big negative potential energy. That is a theoretically possible configuration of the system. There is no substantial problem with this, similarly to classical mechanics of two-body systems. When going further in developing the quantum theory, we may find that the Coulomb law could be inaccurate for very small distances comparable to electron size, but as far as we know, experiments are consistent with electron size being smaller than 1e-18 m, including being zero. (Protons are different, they have size around 1e-15 m),

But on the other hand, it is not so easy to convince ourselves that such singularity can't exist. What is so bad about singularity in electric force?

Sometimes it makes our equations fail to predict what exactly happens when the singularity is reached, i.e. loss of determinism. Or, creation of such singularity means release of infinite amount of EM energy, and infinite energy seems to be a noticeable phenomenon that we never observe.These are unpopular scenarios so we tend to not like them and seek to "fix" the model.

Determinism is a huge problem for some, but not at all for others, who point to quantum theory or probability theory being theories that do not need determinism. And one does not need to seek singularities to find examples of loss of determinism; there are things such as multi-body collisions which are indeterminate, without any infinite forces or energies.

Infinite EM energy seems hard to hide, but maybe it just escapes to infinity really fast without making a big fuss for us to notice. Or it just isn't very visible because of special properties like to short a wavelength.

So, it is thinkable that some singularities from our models (like point particles) may really exist and we just do not observe them for some other reasons, such as: they may be unlikely to occur due to other factors (additional short-range repulsive forces, or background noise breaking the tightly bound state) or may occur occasionally, but their effects are hard to detect (such as the infinite energy quickly escaping to infinity, or somehow not interacting with anything).

So, my message is, do not absolutely reject reality of singularities just because they are ugly; that is not enough. Seek some prediction or property of them that disqualifies them as viable in the real world.