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I would like to know what is the theoretical explanation (if it's possible something more than just "The Dirac equation") of the fact that the g-factor for the electron relating the magnetic moment with the spin is double than that relating the magnetic moment with the angular momentum of the electron in a circular orbit.
This is explained in any textbook treating the Dirac equation. Please read first such a standard account and then amend your question by giving some details about what still remains unclear (if anything).
Please explain it then, Arnold. I don't think I've ever seen a satisfactory explanation myself.
@JohnDuffield: Use the minimal prescription for the external field in the Klein Gordon equation and expand. Multiply the Dirac equation in an external magnetic field with $\gamma \partial$ and simplify. Compare the results. I don't see anything unsatisfactory.
@Arnold Neumaier : with respect, that doesn't explain it.
@JohnDuffield did you try what Arnold said, where did you get stuck? The purpose of PhysicsOverflow is generally not to give popular-level explanations ...
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