UPDATE October 12, 2022: We erroneously assumed that the electric field does not couple to the spin in the Dirac equation. It does couple. In the nonrelativistic Pauli equation, the spin does not couple to the magnetic field. But the Pauli equation is just an approximation
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Let us put a negative and positive charge as in the diagram, and let them move at a uniform speed out from the screen.
●+
\_______/
-------------- magnetic field lines
______
/ \
● -
The magnetic field lines are the densest between the charges and grow less dense when we move away from the line connecting the charges.
The magnetic field is nonuniform like in the Stern-Gerlach experiment. The magnetic field exerts a force on the magnetic moment of the electron. It is the familiar phenomenon that opposite poles of magnets attract each other.
Let us use the Pauli equation, which should work for non-relativistic electrons.
If we put an electron in the magnetic field, its path will depend on the state of its spin.
The magnetic field couples to the spin of the electron.
Let us next change to a frame where the charges are static. There is no magnetic field in that frame, just an electric field. In that frame, the path of the electron does not depend on the state of its spin.
This contradicts Galilean invariance.
The problem obviously is that the spin and the magnetic moment are encoded in the Pauli equationin a way which is too simple.
Some four years ago we showed that one can avoid the Klein paradox in the Dirac equation by adding the potential energy to the mass of the electron and not making the potential an independent term. That was another example where the minimal coupling does not work.
It has been acknowledged in the past that adding potentials to the Dirac equation is "problematic".
What if we require that the frame is always such that the electron is static in it? This frame is used in calculating the spin-orbit interaction in the hydrogen atom. The electric field of the proton creates a magnetic field in the comoving frame of the orbiting electron.
Conclusions
We have to check if the full relativistic Dirac equation suffers from this problem. If we have an electric field, and a solution where the spin-z is up, can we modify it to a solution where the spin-z is down?
