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Electrons as geometry

by Michael Nielsen on April 4, 2005

(Attention conservation notice (hat tip to Cosma for the term): This post requires some familiarity with Maxwell’s equations to make much sense.)

A recent post by Dave Bacon reminds me of a beautiful old idea by John Wheeler for explaining the electron (and other charged particles) as a combination of non-trivial geometry, and the free electromagnetic field.

Suppose we start with ordinary flat spacetime. Now insert a pair of tiny little punctures, and a little tube connecting those punctures. The tube and the punctures will be used to represent an electron / positron pair.

Now suppose that we have a divergence-free electric field going into one puncture, passing through the tube, and out the other end of the tube.

From the point of view of an observer in the bulk, who is unaware of the puncture, it will look like the electric field has non-zero divergence around each puncture, i.e., it the punctures look like charges.

There are several beautiful things about this picture:

  • The charges of the two punctures are equal and opposite.
  • It puts the electric and magnetic fields on an equal footing – there is no charge for either.

Unfortunately, the idea leaves us wondering (a) why we don’t see magnetic charge in the bulk; (b) why charge is quantized ; and (c) what the dynamics of the tubes is supposed to be.

Still, I really like the idea as a simple example of how non-trivial geometries can give rise to interesting physical phenomena.

From → General

  1. This is very cute, but wouldn’t another problem be that the number of electrons is always equal to the number of positrons? I guess we could treat this as implying that there really are large regions of space-time where anti-matter is predominant…

    (I stole “attention conservation notices” from Bruce Sterling, BTW.)

  2. Good point. Oddly, I saw this idea presented in Schutz’s “Geometrical Methods for Mathematical Physics” and, from memory, he treated the fact that # electrons = # positrons as a feature, not a bug. I gather that the existing evidence all points to the Universe being preponderantly ordinary matter, so this is a big problem.

  3. There is also the little problem that no one knows how to create the spacetime topology you are talking about without there being closed timelike curves in the system (or at least such curves almost always seem to show their face when you slightly perturb most such solutions.)

    Supposedly when Wheeler thought up this idea, he got very excited and told Feynman, then his student, about the idea. Feynman asked Wheeler where all the anti-particles were. And Wheeler sad that he thought that they must be hiding around in the hadrons somehwere!

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