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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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  • How are you talking about "acceleration" of the particles here? Either you are in a classical theory, where $\psi$ is just a field and not anything with "particles", or you are in a quantum theory where you need to do the analog of this computation for your theory to claim anything about the classical force law this Lagrangian implies. 2. I don't see how you conclude from the electron being "a mix of left and right" that it "shouldn't be affected at all". A massive fermion is not necessarily an equal mixture.
    • – ACuriousMind Feb 25 '19 at 19:07
  • Well, I'm not sure. I suppose take the classical approximation. On your second point, an electron with average speed 0 would have 50-50 chance of being detected left/right chirality would it not? Maybe I'm wrong. –  Feb 25 '19 at 22:04
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    Well, I've never seen a theory like that, but in pseudoscalar Yukawa theory there is indeed a similar effect. One finds that the force between the fermions is a spin-spin interaction, roughly corresponding with what you say about chirality. – knzhou Feb 25 '19 at 22:19
  • Yes, I'd imagine it would be equivalent to having two massless $Z^0$ bosons acting on left and right chiralities respectively. –  Feb 25 '19 at 23:12
  • @knzhou Indeed, the $h_1$(1170) is an isosinglet axial vector hadron coupling just like that, to nucleons, strongly. Unlike the pion, the coupling links left-handers to left-handers and right-handers to right-handers, like the photon, but with a relative minus sign. In a scattering experiment, this difference should be visible. – Cosmas Zachos Feb 26 '19 at 02:14
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    A fixed mass term breaks the axial symmetry, but you get such a theory for a Weyl superconductor. The (Majorana) mass term is then $me^{i\gamma_5 \theta}$ where $\theta$ is the phase of the superconducting order parameter. – mike stone Dec 31 '21 at 23:18