Some questions about Dirac spinor transformation law I have perhaps meaningless question about Dirac spinors, but I'm at a loss.
The transformation laws for for left-handed and right-handed 2-spinors are
$$
\tag 1 \psi_{a} \to \psi_{a}' = N_{a}^{\quad b} \psi_{b} = \left(e^{\frac{1}{2}\omega^{\mu \nu}\sigma_{\mu \nu}}\right)_{a}^{\quad b}\psi_{b}, \quad \psi^{b}{'} = \psi^{a}(N^{-1})_{a}^{\quad b}, 
$$
$$
\tag 2 \psi_{\dot {a}} \to \psi_{\dot {a}}' = (N^{*})_{\dot {a}}^{\quad \dot {b}} \psi_{\dot {b}} = \left(e^{\frac{1}{2}\omega^{\mu \nu}\tilde {\sigma}_{\mu \nu}}\right)_{\dot {a}}^{\quad \dot {b}}\psi_{\dot {b}}, \quad \psi^{\dot {b}}{'} = \psi^{\dot {a}}(N^{*^{-1}})_{\dot {a}}^{\quad \dot {b}},
$$
where
$$
(\sigma_{\mu \nu})_{a}^{\quad b} = -\frac{1}{4}\left(\sigma_{\mu}\tilde {\sigma}_{\nu}-\sigma_{\nu}\tilde {\sigma}_{\mu}\right), \quad (\tilde {\sigma}_{\mu \nu})_{\quad \dot {a}}^{\dot {b}} = -\frac{1}{4}\left(\tilde {\sigma}_{\mu} \sigma_{\nu}- \tilde {\sigma}_{\nu}\sigma_{\mu}\right),
$$
$$
(\sigma_{\mu})_{b\dot {b}} = (\hat {E}, \sigma_{i}),  \quad (\tilde {\sigma}_{\nu})^{\dot {a} a} = -\varepsilon^{\dot {a}\dot {b}}\varepsilon^{b a} \sigma_{\dot {b} b} = (\hat {E}, -\sigma_{i}).
$$
Why do we always take the Dirac spinor as
$$
\Psi = \begin{pmatrix} \varphi_{a} \\ \kappa^{\dot {b}} \end{pmatrix},
$$
not as
$$
\Psi = \begin{pmatrix} \varphi_{a} \\ \kappa_{\dot {b}} \end{pmatrix}?
$$
According to $(1), (2)$ first one transforms as
$$
\delta \Psi ' = \frac{1}{2}\omega^{\mu \nu}\begin{pmatrix}\sigma_{\mu \nu} & 0 \\ 0 & -\tilde {\sigma}_{\mu \nu} \end{pmatrix}\Psi ,
$$
while the second one - as
$$
\delta \Psi ' = \frac{1}{2}\omega^{\mu \nu}\begin{pmatrix}\sigma_{\mu \nu} & 0 \\ 0 & \tilde {\sigma}_{\mu \nu} \end{pmatrix}\Psi ,
$$
so it is more natural than first, because the first one has both covariant and contravariant components, while the second has only covariant (contravariant components).
 A: I think it is convention to write the conjugate Weyl fermion in,
\begin{equation} 
\left( \begin{array}{c} 
\phi _\alpha   \\  
\bar{\kappa} ^{\dot{\beta }}  
\end{array} \right) 
\end{equation} 
(it is common to put a bar over the conjugate representation), with a raised index in order to comply with the ${} _{ \dot{\alpha} } ^{ \,\, \dot{\alpha} } $ contraction of spinor indicies. Recall that we write,
\begin{equation} 
\phi  \chi \equiv \phi  ^\alpha \chi _\alpha , \quad \psi \bar{\chi} \equiv \phi  _{\dot{\alpha}} \bar{\chi} ^{\dot{\alpha}} 
\end{equation} 
Thus having the particular index structure for the Dirac spinor gives,
\begin{align} 
\bar{ \Psi } \gamma ^\mu \Psi & = \left( \begin{array}{cc}  \kappa ^{\beta }    &\bar{ \phi} _{\dot{\alpha}}\end{array} \right) \left( \begin{array}{cc} 
0  & ( \sigma ^\mu ) _{ \beta  \dot{\beta} } \\  
( \bar{\sigma} ^\mu ) ^{ \dot{\alpha } \alpha } & 0
\end{array} \right) 
\left( \begin{array}{c} 
\phi _\alpha  \\  
\bar{ \kappa} ^{\dot{\beta}}  
\end{array} \right) \\ 
& = \kappa \sigma ^\mu \bar{\kappa} + \bar{\phi} \bar{\sigma} ^\mu \phi 
\end{align} 
where all the dotted indices contract with an "upwards staircase", ${}_{ \dot{\alpha} } ^{ \,\, \dot{\alpha} } $, and undotted with a "downwards staircase", $ {} ^\alpha _{ \,\, \alpha } $.
A: I suspect the origin of this might have to do with the bi-spinor notation. Given a four-vector $b_\mu$, one defines the corresponding bi-spinor, $b\!\!/_{\alpha\dot{\beta}}=b_\mu (\sigma^\mu)_{\alpha\dot{\beta}}$. In this convention, bi-spinors have both lower indices (or upper indices if one uses $(\bar{\sigma}^\mu)^{\beta\dot{\alpha}})$. Once such a choice is made, the index structure of $4\times 4$ gamma matrices is fixed leading to what seems a strange choice for the 
index structure for a Dirac spinor. In order to avoid such details, I usually use a single meta index $A=(\alpha,\dot{\alpha})$ (capital letters) to denote the combination leaving the finer detail only when I need to work explicitly with gamma matrices. I recommend  appendix A of the article by M. Sohnius titled "Introducing Supersymmetry" (Physics Reports 128 (1985) 39-204).
