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It is straightforward to show from relativistic kinematics that a free electron cannot absorb a photon, as shown in this previous thread.

However, It is also known that using the Dirac equation, you can derive a magnetic moment associated with including an electromagnetic field (for example, in the derivation on page 11 in these lecture notes or page 2 of these notes).

Are the above derivations consistent/coherent only because the electromagnetic field is not quantized there, or is there some other feature (such as spin) that allows photon absorption to take place?

As a separate but related question, why is the Feynman diagram corresponding to the single loop correction of the anomalous magnetic moment even allowed? It seems that energy-momentum conservation as shown in the first link above forbids this process. What am I missing?

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  • $\begingroup$ Remember that virtual particles are off the mass shell, i.e. do not have to satisfy $E^2=p^2+m^2$. $\endgroup$
    – jacob1729
    Commented Nov 26, 2018 at 17:37
  • $\begingroup$ @jacob1729 Are you suggesting that internal loops can cause a breakdown in conservation rules dictated by the kinematics of the real particles? $\endgroup$
    – Peter James Beierle
    Commented Nov 26, 2018 at 17:59

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It's true that a free electron can't absorb a single photon because of kinematics, but that certainly doesn't mean electrons can't interact with the electromagnetic field! It is kinematically allowed to, e.g. absorb multiple photons, or absorb one photon and reemit it.

We measure the magnetic moment from precession experiments in a classical field, and a classical electromagnetic field arises from the coherent superposition of macroscopically many photons. Interaction with such a field has very little in common with absorption or emission of single photons. If you wanted to describe it with a Feynman diagram, it roughly involves the electron absorbing and emitting very many photons, going somewhat off-shell in the intermediate stages.

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