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In the Feynman Lectures Vol 1, Chapter 28 (at the end of section "28–1 Electromagnetism"), it is mentioned:

For those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it there; we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can.

As a curious novice I am interested to know, at a very introductory level, as to how a modern "Unified Theory" could possibly explain such a phenomenon as quoted above, so as to gain insight of how physicists work towards explaining things?

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This problem is nowadays referred to under the names 'self-force' and 'radiation reaction'. In classical electromagnetism it can be solved by noticing that the standard concepts (Maxwell's equations plus the Lorentz force equation) make sense when applied to continuous distributions of charge where there is no infinite charge density (such as a point charge). So charged 'particles' have to be modeled as charged spheres of some very small but non-zero radius. Classical electromagnetism is not able to describe how such a charged sphere could be held together, but it can describe the force exerted by such a body on itself when it is made to accelerate by some externally applied force.

This brings us to quantum mechanics and quantum field theory. The problem of self-force and radiation reaction is closely related to the issue called renormalization. This word refers to the following property of a set of interacting quantum fields, such as the Dirac field (describing electrons and positrons) and the electromagnetic field. When the fields interact, their joint ground state is hard to calculate. One way to approach the problem is to imagine a fiction---a set of fields which do not interact with one another---and then introduce the interactions by a perturbative style of calculation, a kind of Taylor series expansion (a sum of Feynman diagrams). The trouble is that the change introduced by such a 'perturbation' is infinite! So it turns out that the fiction that one started from (the non-interacting fields) is infinitely wrong, in the sense that it differs from the true situation (interacting fields) by an infinite amount. Nonetheless, one can get around this problem by a clever mathematical trick, and this procedure is called renormalization.

I won't go into the details of this mathematical method. The answer to your question is that this procedure is sufficiently robust that we can now calculate things like the interactions between electrons very precisely and with confidence. However, it would be fair to say that the method called renormalization feels as if it is a stop-gap, a method we are using now in the absence of some more general theory which perhaps won't need that method. Such a more general theory could be string theory for example. Thus the puzzle of self-force does still lead one to open questions in fundamental physics.

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    $\begingroup$ I believe Rohrlich in his 3rd edition of "Classical Charged Particles" does provide a solution to self-force problem of point charges in classical framework (Supplement "The Physically Correct Dynamics", also ch. 9-4, 9-5). Basically, it postulates Landau-Lifshitz approximation on the grounds that offensive terms in Lorentz-Abraham-Dirac equation are "small" — i.e., belong to the quantum theory and therefore must be ignored. $\endgroup$ – Joker_vD Oct 28 '19 at 1:35
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    $\begingroup$ @Joker_vD The terms ignored by Landau-Lifshitz approach are not absolutely small, but small compared to a certain combination of charge, mass, size and acceleration. So they cannot be ignored when the acceleration is high enough. Having said that, one usually finds that for electrons the imprecision of ignoring quantum theory comes in sooner than the imprecision of ignoring these further terms in the classical formulae. $\endgroup$ – Andrew Steane Nov 7 '19 at 14:46

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