In my first QFT exam I was supposed to derive the equations of motion for all fields for this Lagrangian:

$$\mathcal{L} = \bar{\Psi}(i\gamma^\mu\partial_\mu-M)\Psi+g\bar{\Psi}\gamma^\mu\Psi\bar{\Psi}\gamma_\mu\Psi.$$

For $\Psi$ I ended up with the following:

$$\partial_\mu(\bar{\Psi}i\gamma^\mu)+M\bar{\Psi}-g[\bar{\Psi}\gamma^\mu\bar{\Psi}\gamma_\mu\Psi + \bar{\Psi}\gamma^\mu\Psi\bar{\Psi}\gamma_\mu] = 0.$$

In the exam review I was told that one can simplify the term in the square brackets to 2*something but I can't remember what exactly. My question is why can one simplify this term since the gamma matrices do not commute with the spinor fields?

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    $\begingroup$ Put spinor indices on everything and check what is a number and what not. $\endgroup$ – Oбжорoв Sep 17 at 11:45
  • $\begingroup$ I ended up with $2\bar{\Psi}\gamma^\mu\bar{\Psi}\gamma_\mu\Psi$, is this correct? $\endgroup$ – Peter Hidor Sep 18 at 17:24

$\bar{\Psi} \gamma^\mu \Psi$ is indeed a number. It multiplies $\bar{\Psi}\gamma^\mu$ in both terms.


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