Brian Bi

How does the unification of electricity and magnetism work when comparing charge and spin?

On one level, I've learned that a relativistic transformed E-field or B field is the other (I think). So I think of E fields as originating from charges or changing magnetic fields, and B fields as originating from moving charges, changing E fields, or quantum spins. How are the spins and charges related?
I interpret the question as follows: if a spinning charged particle generates magnetic fields, then what is responsible for generating the electric field that you see when we boost into a different reference frame? Does the spin turn into a charge somehow?

Well, now you are trying to combine classical and quantum thinking, because in classical electrodynamics we think in terms of charges and currents which generate E and B fields, but in quantum electrodynamics the E and B fields are an afterthought, the "true" field is more closely related to the four-potential, and instead of charges and currents we deal in wave functions.

But what you might not realize is that there is a current density associated with the electron, even though everyone always tells you that you can't think of the electron as a spinning ball of charge! This current density is given by the expression

\(j^\mu = -e\overline{\psi}\gamma^\mu\psi\)

where \(e\) is the absolute value of the charge of the electron, and \(\psi\) is the Dirac spinor describing the electron's wavefunction; a solution to the Dirac equation.

It can be shown that even when the electron is at rest (in a given reference frame), the current density can be nonzero, and circulates around the electron's spin axis. The four-current \(j^\mu\) on the left hand side of the previously given equation is a Lorentz four-vector, so upon transformation to another frame it transforms correctly and results in an electric dipole moment for the electron which is not present at rest! So the spinning electron does have charge and current densities which are related in a natural way just like the charge and current densities of classical systems, and can in fact be seen to give rise to the electron's magnetic dipole moment—even though the picture of the electron as a spinning ball of charge is inapplicable.

I don't have the mathematical sophistication to explain the details but you can find them in this paper: What is spin?