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## Main Question or Discussion Point

Using this convention

[tex] A \overleftrightarrow{\partial }_{\mu} B =:A \overrightarrow{\partial}_{\mu} B - A \overleftarrow{\partial}_{\mu} B [/tex]

one can write the QED Lagrangian density simply as

[tex] \mathcal{L}_{QED} =\frac{i}{2} \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \overleftrightarrow{\partial }_{\mu} \Psi^{\beta} -m \bar{\Psi}_{\alpha}\Psi^{\alpha} +g \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \Psi^{\beta} A_{\mu} -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} -\frac{1}{2\xi} \left(\partial^{\mu}A_{\mu}\right)^{2} + \left(\partial^{\mu}\bar{\eta}\right) \left(\partial_{\mu} \eta}\right) [/tex]

,where "g" is the coupling constant (plus/minus the electron's charge depending on the convention), [itex]\eta [/itex] is the ghost field associated to the gauge parameter [itex] \epsilon [/itex] and [itex] \bar{\eta} [/itex] is a ghost field from the nonminimal spectrum.

The question doesn't concern the uniqueness of the gauge-fixing term (one can implement various gauges, chosing a gauge-fixing fermion is, up to a point, arbitrary), but the consistent cross-interaction term, Dirac field - gauge abelian one-form field. The question is

Is [itex] g \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \Psi^{\beta} A_{\mu} [/itex] the only consistent cross-interaction between a massive Dirac field and a gauge abelian one-form field ? If so, how would one go about & prove it...?

Daniel.

[tex] A \overleftrightarrow{\partial }_{\mu} B =:A \overrightarrow{\partial}_{\mu} B - A \overleftarrow{\partial}_{\mu} B [/tex]

one can write the QED Lagrangian density simply as

[tex] \mathcal{L}_{QED} =\frac{i}{2} \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \overleftrightarrow{\partial }_{\mu} \Psi^{\beta} -m \bar{\Psi}_{\alpha}\Psi^{\alpha} +g \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \Psi^{\beta} A_{\mu} -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} -\frac{1}{2\xi} \left(\partial^{\mu}A_{\mu}\right)^{2} + \left(\partial^{\mu}\bar{\eta}\right) \left(\partial_{\mu} \eta}\right) [/tex]

,where "g" is the coupling constant (plus/minus the electron's charge depending on the convention), [itex]\eta [/itex] is the ghost field associated to the gauge parameter [itex] \epsilon [/itex] and [itex] \bar{\eta} [/itex] is a ghost field from the nonminimal spectrum.

The question doesn't concern the uniqueness of the gauge-fixing term (one can implement various gauges, chosing a gauge-fixing fermion is, up to a point, arbitrary), but the consistent cross-interaction term, Dirac field - gauge abelian one-form field. The question is

Is [itex] g \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta} \Psi^{\beta} A_{\mu} [/itex] the only consistent cross-interaction between a massive Dirac field and a gauge abelian one-form field ? If so, how would one go about & prove it...?

Daniel.