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What follows is from Professor Barton Zwiebach of MIT:

The only problem with such a large field is an infinite self-energy of the point particle limit. This problem is not really there in QED, as any infinite self energy is renormalized away. There is actually an effective version of Maxwell’s equations, called nonlinear electrodynamics in which point charges have finite self energy, and electric fields cannot exceed a limit (just like the speed of objects is $< c$). This is also true in classical string theory. It is explained in Chapter 20 of my book on string theory."

What does it mean that the energy is renormalized?

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Without a great understanding of QED, I will not attempt to go into renormalization.

The short answer, which I hope is satisfactory, is that this problem arises from us humans using a simplified model to make sense of reality.

It is simple for us to use $\vec{E} = \frac{q}{\epsilon_0r^2}\hat{r}$ to describe the field from a point charge. And it works for most cases, but you notice there's a big problem when you try to evaluate E at $r = 0$! Since you don't really need to do that for practical purposes, you stick with the point charge model.

Maxwell's Electrodynamics ignores the $r=0$ problem, and does just fine. QED zooms in on $r=0$ to try and figure out what's going on.

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  • $\begingroup$ As the quote correctly points out, the only problem with such a large field is an infinite self-energy of the point particle limit. This answer does not consider that self-energy at all. Maybe that's enough to sketch just how far from complete it is? $\endgroup$ – Emilio Pisanty Jan 2 '18 at 18:15

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