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I've read about how people have literally measured the gyromagnetic ratio of a single electron in a Penning trap. Naturally, I am frankly blown away by the exquisite precision of such an experiment.

But now I am wondering: to what degree, if such a thing is possible, has the full EM field of the electron (in its rest frame, let's say) been measured? Or if not a direct measurement, then at least, how confident are we of the shape of that field, based on the indirect constraints imposed by present theories?

For example, if we modeled the electron as a classical spinning surface of uniform charge density, its electric field would be spherically symmetric, and it would have a near dipole magnetic field, which would become a perfect dipole if we let the radius shrink to zero as it is alleged to be.

But to probe a single electron and find out if it truly has that field geometry sounds, to my untrained ear, probably orders of magnitude harder than the measurement of the one-electron $g$-factor. Especially because the probe would have to be another charged fermion, hence the complications of spin interaction...

I'm interested in both components, but especially the electric part, since that's an assumption that feels so ingrained ever since E&M 101: do we really have good reason to believe that each electron has a spherically symmetric electric field? Or could it just be, for example, axi-symmetric -- and maybe not even that close to spherical symmetry! -- such that the net electric field of any measurable object (ie. containing many electrons with randomly oriented spins) would be almost perfectly spherically symmetric?

Adam Herbst
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1 Answers1

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Whether you distinguish between “an electron” and “its EM field” is a philosophical question. The consensus among people actually doing QFT seems to be that you can’t distinguish them, and any question about the shape of an electron’s electromagnetic field is just as much a question about the shape of the electron itself.

With that connection made: one consequence of the Wigner-Eckart theorem is that a spin-half system cannot have any permanent multipole moment larger than a dipole. (Hand-waving explanation and links in this related answer.) Just like for the neutron, the relevant $\mathit{CP}$-odd moments are the magnetic monopole and the electric dipole. Both of these forbidden EM moments are consistent with zero for the electron, to high precision; the Particle Data Group is a good place to look for thorough but accessible experimental reviews.

rob
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