What does a dot over a spinor index signify?

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?

• possibly useful: en.wikipedia.org/wiki/Van_der_Waerden_notation Commented Jan 3, 2023 at 18:38
• That certainly helps a lot, thanks! So, is the only reason you use a dot is to signify it’s a different chirality? Commented Jan 3, 2023 at 18:43

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 $$\bar{\varphi}^\dot{\alpha}$$.