Field equation of motion for this Lagrangian

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?

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

1 Answer

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