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Consider a Dirac action with a pseudo-vector potential:

$$S = \overline{\psi}(\gamma^\mu(\partial_\mu + i\gamma_5 A_\mu) + m_e)\psi$$

i.e. exactly like a Dirac equation with an electromegnetic potential except of the matrix $\gamma_5$.

$\tfrac{1}{2}(1\pm \gamma_5)$ are projetion operators to left/right chiral states of the electron. Basically $\gamma_5$ would give the potential a sign change depending on the chirality of the electron compared to a normal vector potential.

We might imagine that $A_\mu$ is some spherical potential, for the sake of argument.

As far as I can tell, this potential would accelerate left-handed particles towards it and accelerate right-handed particles away from it. And not depend on charge.

But then an electron nearly at rest would be a mix of left and right handed so shouldn't be affected at all (on average). Which is strange.

Is this correct? Precisely, what effect would this have on left/right handed electron/positrons?

(One might assume the electrons are moving at speeds in which a classical approximation is appropriate).

Edit: Might it be like the field of a magnetic monopole? (Just a guess.)

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Here's the Dirac equation for an electron $$S = \overline{\psi}(i\gamma^\mu(\partial_\mu -eA_\mu) - m_e)\psi$$

Note, by replacing

$$-e \rightarrow +\gamma_5$$

you have removed the coupling of the field $A_\mu$ with particle $m_e$.

Hence, there is no charge current.

Also note, the projection operator for the electron only works when the energy of the particle is large compared to its rest mass.

Cinaed Simson
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