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I am studying Special Relativity. When calculating the Electrostatic Energy of Point Charges, there is self energy which is infinite due to the interaction between the charge and the Coulomb potential produced by itself. The author said this should be ignored as it has no physical interest. It also says it means there is a fundamental limitation for classical electrodynamics (dimension needs to be >1e-25m).

Can you tell me if this is solved in more advanced physics (e.g. quantum electrodynamics) and if this is well understood in the community or it is still a research topic? Thanks a lot!

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2 Answers 2

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In the framework of Quantum Field Theory (a class of theories, to which Quantum Electrodynamics belongs), the concept of elementary particle is modified dramatically.

In Quantum Mechanics, you don't have point particles anymore. Elementary particles are in a way extended. They are represented by complex-valued functions over space, called the wavefunctions. The value of this function at a given spatial point is connected with the probability of observing the particle at that point.

Quantum Field Theory takes this even further. Elementary particles are modeled as quanta of relativistic fields. Thus, there is no place for a point particle model in QFT. And the infinite energy problem does not arise. Well, except that it does :)

Actually, you have to deal with a whole lot more of these nasty infinity problems in QFT. These have been classified into infrared and ultraviolet divergences, based on their origin (infrared divergences originate from large-scale fluctuations, whilst ultraviolet divergences are present in almost all QFTs with interactions and originate from short-scale fluctuations). A systematic approach for dealing with these divergences in the framework of perturbative QFT has been developed. For more details on this, see this answer of mine.

I hope this helps.

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  • $\begingroup$ Thank you very much for showing me the bigger picture! I hope I can understand everything you are talking about one day :-) I am an engineer and trying to self-study physics. I am reading Classical Theory of Field by Landau. And I hope I can finish it this year and start Quantum Electrodynamics next year. I am not smart but looks like I might be able to get a lot of help in this forum. $\endgroup$
    – HYW
    Sep 24, 2016 at 20:46
  • $\begingroup$ @HYW of course you are smart. Stupid people don't bother studying fundamental physics :) I am glad that I was able to help, and I wish you the best of luck in your studies. $\endgroup$ Sep 25, 2016 at 15:21
  • $\begingroup$ In a book, "Actually the root of this difficulty lies in the earlier remarks concerning the infinite electromagnetic "intrinsic mass" of the elementary particles. When in the equation of motion we write a finite mass for the charge, then in doing this we essentially assign to it formally and infinite negative "intrinsic mass" of nonelectromagnetic origin, which together with the electromagnetic mass should result in a finite mass for a particle". This is from an old book. But I wonder if concept such as ""intrinsic mass" of nonelectromagnetic origin" is something people talk about nowadays? $\endgroup$
    – HYW
    Oct 9, 2016 at 14:42
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When calculating the Electrostatic Energy of Point Charges, there is self energy which is infinite due to the interaction between the charge and the Coulomb potential produced by itself.

This is true if we begin with the formula that was introduced for continuous charge distributions $$ E_p = \frac{1}{2}\int d^3\mathbf x \int d^3\mathbf y ~ K\frac{\rho(\mathbf x)\rho(\mathbf y)}{|\mathbf x-\mathbf y|} $$ based on the formula for potential energy of point charges:

$$ E_p = \frac{1}{2}\sum_i \sum_{j}{}^{'} K \frac{q_iq_j}{|\mathbf r_i-\mathbf r_j|}. $$

The prime next to the second sum over $j$ means the sum over $j$ is to be done for $j$ that are not equal to $i$.

For point particles, it makes no sense to use the first formula, point charge has singular density that makes the integral nonsense. The appropriate formula is the original one, where no divergence occurs.

The author said this should be ignored as it has no physical interest. It also says it means there is a fundamental limitation for classical electrodynamics (dimension needs to be >1e-25m).

The author seems to believe the infinite result is "correct" within classical electrodynamics and because the result is unacceptable, he suggests the solution is to limit application of classical electrodynamics to bodies that have finite charge density.

This would be correct if by "classical electrodynamics" we mean a theory where we have to use the first integral formula. This formula does not work for point charged particles.

But if by "classical electrodynamics" we mean Maxwell's eqautions and the Lorentz force formula, there is no necessity to use that formula. For point particles at rest, the electrostatic formula with sums is appropriate and works well for point charges. There is also generalization to general situation where particles move and accelerate, so the field is not static. See for example the papers by Frenkel, Feynman and Wheeler, and Stabler:

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. http://dx.doi.org/10.1007/BF01331692

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433. http://dx.doi.org/10.1103/RevModPhys.21.425

R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Letters, 8, 3, (1964), p. 185-187. http://dx.doi.org/10.1016/S0031-9163(64)91989-4

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  • $\begingroup$ Thank you! I agree with you and thanks for the papers! But to be clear, the author actually derived an equation (your 2nd equation but without prime) from the T00 (energy density) of the Energy Momentum Tensor. And the it says "But we know that in the theory of relativity every elementary particle must be considered as pointlike", for i=j, it will give infinity value for self-energy. So the electron will have infinite mass. $\endgroup$
    – HYW
    Sep 24, 2016 at 21:05
  • $\begingroup$ Therefore, "The physical absurdity of this result shows that the basic principles of electrodynamics itself lead to the result that its application must be restricted to definite limits". I think based on this, one will then come up with your 2nd equation. $\endgroup$
    – HYW
    Sep 24, 2016 at 21:05
  • $\begingroup$ @HYW, could you post a link to the paper or book you refer to? $\endgroup$ Sep 25, 2016 at 10:24
  • $\begingroup$ Hi Ján, here you are: $\endgroup$
    – HYW
    Sep 25, 2016 at 14:37
  • $\begingroup$ books.google.com/… $\endgroup$
    – HYW
    Sep 25, 2016 at 14:38

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