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I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114).

Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \gamma^{\mu} \partial_{\mu} P_{L} \chi + \bar{F} F + (F + \bar{F})(mZ + gZ^2) - \bar{\chi} P_{L} (m + 2 g Z) \chi. $$

Here $\chi$ is a Majorana spinor, $P_L$ is the projection operator, $Z$ is a complex scalar field, $F$ is a complex scalar auxiliary field.

I don't understand how one derives the equations of motion for the field $\chi$ and $\bar{\chi}$, when projection operators are involved. Do I just calculate $$\partial_{\sigma} \frac{ \partial \mathcal{L}}{\partial (\partial_{\sigma} P_L \chi)} = \frac{\partial \mathcal{L}}{\partial P_L \chi} ~?$$ Also, are $P_L \chi$ and $P_R \chi$ are considered independent fields? I would get $$ \frac{\partial \mathcal{L}}{\partial P_L \chi} = - \bar{\chi} (m + 2g Z) $$ and $$\partial_{\sigma} \frac{ \partial \mathcal{L}}{\partial (\partial_{\sigma} P_L \chi)} = - \partial_{\sigma} \bar{\chi} \gamma^{\sigma}. $$

And if I want to compute the equations of motions for $\bar{\chi}$, how do I do that?

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