I've been referring to Pg.36-Pg.38 in Introduction to Supersymmetry by Wiedamann. For understanding the precise origin of dotted, undotted indices on Spinors. He starts off my saying that $M$ acts on $\Psi$ as a representation on the spinor space.He says, that a spinor $\Psi_A$ transforms as $$\Psi_A = M_A^B\psi_B$$ under a rep $M$ of $SL(2,\mathbb{C})$. Then he proves that $M^{-1T}$ is an equivalent representation to $M$. He introduces the standard lowering and raising $\epsilon$ through this. And shows that, $\psi^A \equiv \epsilon^{AB}\psi_B$ is an object that transforms in the $M^{-1T}$ and therefore, we give it a new notation of contra variant index, $$\Psi^A = (M^{-1T})^{A}_B\Psi^A$$ So far everything is fine. But on Pg.39, he writes $$\psi^A \equiv \epsilon^{AB}\psi_B = -\psi_B\epsilon^{BA}$$ I'm unclear with how the second part come that is, $-\psi_B\epsilon^{BA}$. I know $\epsilon$ is anti-symmetric but, what is the need for changing the order? Why not $-\epsilon^{BA}\psi_B$?
Understanding the Spinors as Mathematical Objects
Now, I know that there is a very uncanny resemblance of a metric and $\epsilon$ here. I fail to see the exact connection. It does the same role as raising and lower indices. Namely, from a $\mathbb{R}^4$ manifold point of view, I know that the metric gives a notion of distance for vectors in the tangent space $T_p(\mathcal{M})$ of a manifold $\mathcal{M}$. Therefore, it makes sense to write $g_{\mu\nu} x^\nu x^\nu$, for vector which are labeled as $x_\nu$ and $x_\mu$ in a basis. FOR CONVIENCE (I hope I'm right with this), we choose to write $g_{\mu\nu}x^\nu \equiv x_\mu$ as column vector to have an easy way of having the action of duals on its space.
Similarly. I'm guessing the whole point of $\epsilon$ is a notion of distance on the spinor space with a spinor vector whose components labeled by $\psi_A$. And $\epsilon^{AB}\psi_A\psi_B$ gives a notion of distance on this space. In which case, I dont' see why not $-\epsilon^{BA}\psi_B =- \psi_B\epsilon^{BA}$. As they are just some numbers. I would like to know if this is what we do. Probably, someone with the whole picture of a supermanifold and metrics on it can explain me the relation between these.Because, the fact that we consider these labels $\Psi_A$ and $\Psi_B$ to be anti-commuting $\{\Psi_A,\Psi_B\} =0$ seems to suggest to be that these are Grassmann numbers on superspace if I'm not wrong.