My questions should be rather simple. I was trying to get through one of my professor’s papers, and I saw the following notation, first with regards to Dirac and Weyl spinors, but the notation continues throughout $$ \psi=\begin{pmatrix}\lambda_\alpha\\\bar\lambda^{\dot\alpha}\end{pmatrix} $$
In the $\bar\lambda^{\dot\alpha}$ term, what does the dot over the $\alpha$ signify?
Starting from the defining representation $A(\vec{\alpha},\vec{u})$ of ${\rm SL}(2,\mathbf{C})$, also $A^\ast$, $(A^T)^{-1}$ and $(A^\dagger)^{-1}$ furnish representations of ${\rm SL}(2, \mathbf{C})$. The representations $A$ and $A^\ast$ are inequivalent and likewise the representations $(A^T)^{-1}$ and$(A^\dagger)^{-1}$. On the other hand, the representation $A$ and its contragredient representation $(A^T)^{-1}$ are equivalent and, analogously, the complex representation $A^\ast$ and its contragredient $(A^\dagger)^{-1}$.
In index notation (van der Waerden notation), a Weyl spinor $\chi$ transforming with respect to the ${\rm SL}(2, \mathbf{C})$ representation $A$ is written with lower indices, transforming as $\chi^\prime_\alpha = A_\alpha ^{\, \, \beta} \chi_\beta$. A Weyl spinor $\bar{\varphi}$ transforming with respect to the representation $A^\ast$ is written with lower dotted indices, transforming as $\bar{\varphi}_\dot{\alpha}= A^{\ast \, \, \dot{\beta}}_{\, \dot{\alpha}} \bar{\varphi}_\dot{\beta} $. Spinors transforming with respect to the contragredient representation $(A^T)^{-1}$ are denoted by upper indices, $\chi^{\prime \, \alpha}= A^{-1 \, \, \alpha}_{\, \, \, \, \beta} \chi^\beta$. Analogously, spinors transforming with respect to $(A^\dagger)^{-1}$ are written with upper dotted indices, $\bar{\varphi}^{\prime \, \dot{\alpha}}=(A^\ast)^{-1 \, \, \dot{\alpha}}_{\, \, \, \, \dot{\beta}} \bar{\varphi}^\dot{\beta}$.
Employing the fact that $A$ and $(A^T)^{-1}$ are equivalent representations, spinors with upper and lower indices can be related by $\chi^\alpha = \varepsilon^{\alpha \beta} \chi_\beta$ and $\chi_\alpha = \varepsilon_{\alpha \beta} \, \chi^\beta$, where the antisymmetric invariant $\varepsilon$ tensor is defined by $\varepsilon^{1 2} =\varepsilon_{2 1}=1$ (all other elements are determined by antisymmetry). Analogously, one has the relation $\bar{\varphi}^\dot{\alpha}= \varepsilon^{\dot{\alpha} \dot{\beta}} \bar{\varphi}_\dot{\beta} $.
A Dirac spinor consists of two Weyl spinors $\chi_\alpha$ and $\bar{\varphi}^\dot{\alpha}$.
