This is a question on mathematics rather than the physics. It is based on QFT Q3.4, part b on Peskin and Schroeder.
My confusion stems from the fact that we are consider $\chi(x)$ to be a classical anticommuting field,that takes Grassmann numbers as their values and satisfy the following relation:
$$\alpha \beta = -\alpha \beta \quad \text{ for any }\quad \alpha, \beta.$$
So I am given the action which is: $$S = \int d^4x [\chi^\dagger i\bar{\sigma} \cdot\partial\chi + \frac{im}{2}(\chi^T\sigma^2\chi^*- \chi^\dagger \sigma^2 \chi^*)]$$ And then I am trying to show that varying this w.r.t $\chi^*$ yields Majorana equation.
My attempt:
I rewrote $S$ as:
$\large S = \int d^4x [\chi^*_a i\bar{\sigma}^\mu_{ab}\partial_\mu\chi_b + \frac{im}{2}(\chi_a\sigma^2_{ab}\chi_b- \chi^*_a \sigma^2_{ab}\chi^*_b]$
So then I found $\delta S$ and I am at the following step:
$\large \delta S = i \int d^4x [\bar{\sigma}^\mu_{ab} \delta\chi_a^*\partial_\mu\chi_b+\frac{im}{2}\sigma^2_{ab}(-\chi_a^* \delta\chi_b^*-\chi_b^* \delta \chi_a^*)] -----(i)$
Okay, so then this is how I proceeded:
$\large \delta(\chi_a^*\chi_b^*) = \chi_b^* \delta \chi_a^* + \chi_a^* \delta \chi_b^*$
But, $\large \chi_a^* \chi_b^* = -\chi_b^* \chi_a^*$ as given to us in the question.
$\large \rightarrow \delta(-\chi_b^*\chi_a^*) = \chi_b^* \delta \chi_a^* + \chi_a^* \delta \chi_b^* $
$\large \rightarrow -\chi_a^*\delta\chi_b^*-\chi_b^*\delta\chi_a^* =\chi_b^* \delta \chi_a^* + \chi_a^* \delta \chi_b^* $
$\large \rightarrow \chi_a^*\delta\chi_b^* = -\chi_b^*\delta \chi_a^*$
But if I did the last bit correctly, then it means the terms inside the parenthesis in $\delta S$ in eqn(i) will cancel out, and it is not going to give me the Majorana equation.
Could someone help me see where I made the mistake? For this to work out, I feel like I need to get,
$\large \chi_a^*\delta\chi_b^* = \chi_b^*\delta \chi_a^*$
and I am off by a minus sign.
Edit 1: Upon thinking more, I don’t think what I am doing makes sense to me cause I feel like I am not correctly understanding what it means for $\large \chi_b^* \chi_a^* = -\chi_a^* \chi_b^*$. But i also do not know of a way out of this. So any pointer towards right direction would be very helpful.