In Dirac Eq in presence of EM field we normally include the minimal coupling based on $U(1)$ symmetry argument, as $$\gamma_\mu (\partial_\mu +\frac{ie}{\hbar c} A_\mu) + \frac{mc}{\hbar}.$$ Now I am wondering why we don't include the derivative of field like, $\gamma_\mu \gamma_\nu \partial_\nu A_\mu$? I understand that multiplying Dirac eq with $\gamma_\mu (\partial_\mu +\frac{ie}{\hbar c} A_\mu) - \frac{mc}{\hbar}$ give rise to $F_{\mu \nu} \sigma_{\mu\nu}$ term (which is derivative term), but I am not sure why we don't include that to Dirac Eq in the first place. This term is permitted by symmetry.



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