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If an unbound electron (or indeed any charged particle) is moving through free space, is there a probability that it can spontaneously change energy by emitting a photon, or does this require the presence of an external potential $A_4$.

The important point is that the process is spontaneous, like jumping energy levels in an atom.

I'm pretty sure this violates some conservation law but I just want to check.

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Short answer: No. You can't make the system respect both energy and momentum conservation.

Longer answer: In the presence of another charged particle it is possible, but that's arguably not "free space".

Aside: In free space there are no discrete "levels", it's a continuum.

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Ta! This helps me with my problem. – metzgeer Mar 1 '12 at 4:58
The quick way to see that there is no way to emit a photon is to go to the rest frame of the electron. Then it should be pretty clear that you can't emit a photon make the energy/momentum conservation work. – BebopButUnsteady Mar 1 '12 at 6:37

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