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?


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.