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Say your dirac equation had the normal vector boson intereaction term:

$i\overline{\psi}\gamma^\mu A_\mu\psi + i\psi \gamma^\mu A_\mu \overline{\psi}$

What would the effect be of another field B with terms:

$\overline{\psi}\gamma^\mu B_\mu\psi - \psi \gamma^\mu B_\mu \overline{\psi}$

But this time without the $i$ factor.

The first one is the normal U(1) vector boson and so would cause the electron to move in the field. What would the effect of the second one be? Would that also cause the electron to move? Or perhaps just change phase?

(I may have got the signs mixed up. Both should be real valued.)

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