Electrostatic Field Energy of Electron

Doing my physics homework a few weeks ago led me to some startling conclusions (this doesn't really happen often, don't worry too much.) We were learning about electrostatic field energy, and I was doing a problem involving the finding of the electrostatic field energy of a sphere, but later in the problem simply changed the limits of the integral to model an electron, and came out with a funny result:

Energy density in an electric field strength [tex]\vec{\|E\|}[/tex] is:

[tex] \mu_e = \frac{1}{2} \epsilon_0 E^2 [/tex]


And then I set up my integral to sum up all the energy. (dVol represents
the spherical shells of thickness dr surrounding the electron):


[tex]\mu_{tot} = \int_{0}^{\infty} \mu_e (dVol) [/tex]



[tex] = \int_{0}^{\infty} \frac{1}{2} \epsilon_0 (\frac{kq}{r^2})^2 4\pi r^2 dr [/tex]



[tex] = 2\pi \epsilon_0 k^2 q^2 \int_{0}^{\infty} \frac{1}{r^2} dr [/tex]

the rationale for the integral's limit from zero to infinity was that I assumed the electron to be a zero dimensional point particle, which i believe to be the accepted model of what an electron is. If this is so, why is the electrostatic field energy infinite? One of the definitions for electrostatic field energy was the energy it took to create the current electric field. If this is true, it took an infinite amount of energy to create the electric field for an electron??

I would be interested in that too. My ED professor mentioned that, he said that this problem (diverging self-energy) was actually not solved at the time. Maybe it can be done with quantum field theory or whatever.

Does anyone here know if there are current theories that deal with this problem?

QED is the theory that helps to explain the self-energy of the electron.

Anyone know of any good resources on the net that gives a good layman treatment(for now) of QED?

In a large technical/science encyclopedia at the library I saw a chart that gave the wavelengths of EM radiation at various commonly encountered frequencies. The first entry was 0 hz. The wavelength of EM radiation at 0 hz was listed as "infinity". EM radiation at 0 hz, is, of course, a plain electric field.

The wavelength of EM radiation at 0 hz was listed as "infinity". EM radiation at 0 hz, is, of course, a plain electric field.

I wonder if that bothered anyone at the time of writing that. Infinities in physics are usually the indication that a theory has overstepped its bounds.

Layman treatment of QED, the book QED by Feynman, don't believe it has what you are looking for in it though been awhile since I read it and I am away from home right now so I can't check as easily.

Also I don't believe you could actually say they have a size or radius to go out to but I think classically they were something like 3femtometers

There's a lot to say about this before "go learn QED." You've stumbled upon one of the reasons that point particles are inconsistent with classical physics.

If you try to calculate the motion of an accelerated point charge exactly (its not given by the Lorentz force law because the charge's own field back-reacts on itself), you get nonsensical results. This can be interpreted as energy from the field being converted into kinetic energy (and there's an infinite amount of it!). Of course, real charges don't accelerate without bound, and there are ways of reinterpreting the equations so that the problems go away. In the end, though, the only truly rigorous thing to do is to assume the particle has some finite extent. Of course all real objects are finite, so this is only a calculational problem.

This is not inconsistent with the common statement that electrons are point particles. A point particle has a particular meaning in the context of quantum field theory. It also means something in classical physics, although the two notions are not the same. There are bound to be problems when trying to mix ideas from two very different world views (its very common to do it anyway, but that doesn't mean its right).

In the end, you can calculate things in QED and get a reasonable answer. You can also calculate things classicaly on an extended particle, and get a reasonable answer. These agree in the domain where its meaningful to compare the two.

Now you might wonder if the same thing can be done for gravity. In that case, there is no quantum theory you can fall back on, and there are cases where one would like to talk about "point particles" without really talking about quantum objects (stars falling into supermassive black holes, etc.). There are still some open issues here (and in EM as well, depending on who you ask).

Gza -- The sad fact is that nobody has a clue how to calculate the self energy of an electron in a rigorous way. For a long time, early in the 20th century, the emphasis was on calculating this self energy as you did, from standard E&M. The fact that most of the attempts are not currently discussed says it all. In reality, the situation in QED is no better. The only mode of computation is via perturbation theory, which amounts to calculating the coefficients of the electric charge in an infinite series --1, then e, then e**2,...e**n,... Does the series converge? Unknown. Are the e**2 terms finite? No. However,the work of Feynman, Schwinger, and Tomonaga and Dyson, showed how to finesse the infinities by means of renormalization. effectively a scheme to work with cutoffs in the integrals. Nobody understands why, but this adhoc renormalization scheme for QED computations gives results that agree with experiment to 13 decimal places (the magnetic moment of the electron) Over the last 20 years or so in connection with the Standard Model, physicists have studied renormalization in great detail and with great sophistication. But the fact remains: nobody know how to calculate the self energy of the electron.

I would think this weakness in Quantum Field Theory would be highly fertile ground for those that want to poke big holes in standard physics -- relativity is the usual target. But, perhaps fortunately, QED, QFT and renormalization require serious mathematical chops, way beyond those required to "challenge" the Lorentz transfomation. Who knows, perhaps relativity in QED is the problem. It would be tough sledding to tackle such a problem.

You have done a very nice piece of work, and asked a great question.

Regards,
Reilly Atkinson

reilly said:

However,the work of Feynman, Schwinger, and Tomonaga and Dyson, showed how to finesse the infinities by means of renormalization. effectively a scheme to work with cutoffs in the integrals. Nobody understands why, but this adhoc renormalization scheme for QED computations gives results that agree with experiment to 13 decimal places (the magnetic moment of the electron) Over the last 20 years or so in connection with the Standard Model, physicists have studied renormalization in great detail and with great sophistication. But the fact remains: nobody know how to calculate the self energy of the electron.

If someone has come up with a system that acurately predicts the self energy of an electron to 13 decimal places, it seems safe to say that they know how to calculate the self energy of the electron.

Qed And Magnetic Moments

Locrian -- Finding the value of the electron's magnetic moment is a substantially different problem than computing the self energy. Julian Schwinger did the first true relativistic computation of the magnetic moment in 1948. The standard relativistic theory (based on the Dirac Eq.) of the interaction of an electron with electromagnetic fields predicts the magnetic moment of the electron. The 13 decimal places are corrections to that basic result for the magnetic moment -- the corrections come from interactions with virtual photons, done with Feynman diagrams. It's all rather technical, but is a standard topic in any text on QED.

Basic QED gives no info on the electron's mass. The standard approach uses the assumption that the electron's mass is due to electromagnetic self energy. The next steps are unclear.

Regards,
Reilly Atkinson

Whoops, you are right, I misread your post and therefore my rather odd response. Appreciate the correction.

I can ask you a question which may give you more insight than any answer anyone can give you. How does Nature, expressed in the form of the equation you gave, know what dimensional units you are using for the rest energy of the electron? The integral is carried over all values of radius from zero to infinity, and is the same for any system of units you choose. The answer, of course, is that Nature doesn't really "know" what system of units we are using, the electrons mass/energy is only meaningfully expressed as the ratio of the mass of another particle, say a proton. Any such integral, in either classical eletromagnetism or QED, can only yield the value of zero or infinity for the electron's self-energy.

Nor is it possible to establish any relation between the electron's charge and mass, the charge can be expressed in dimensionless (pure number) form, whereas the mass has dimension −1. So the charge doesn't "know" or need to know what units are being used for the mass/energy.

By asking that question you reinvented the wheel, but it's a pretty big wheel!