Why do $\psi_a$ and $\bar{\psi}_{\dot{\alpha}}$ represent two different degrees of freedom?

Question

I am taking a course in QFT and I've been introduced to the concept of left-handed (undotted) and right-handed spinors (dotted).

I know that left-handed spinors are associated with the irreducible representation $(1/2,0)$ of the Lorentz group ($\mathcal{L}$) and the right-handed spinors with the representation $(0,1/2)$ and that these two representations are not equivalent.

My professor told us in class that $\psi_a$ and $\bar{\psi}_{\dot{\alpha}}$ represent two different degrees of freedom but he didn't say why. I've been trying to understand it myself but I haven't been able of convincing myself of why it must be so (I have very little knowledge of Lie algebras).

Why do these spinors represent two different degrees of freedom?

Answer 1

In fact these representations are related to each other by discrete transformations of the Lorentz group. But by the definition, dotted spinors are transformed as complex conjugated undotted spinors, and this provides understanding of the statement that $\psi_{a}$ and $\bar{\psi}_{\dot{a}}$ are different degrees of freedom: the situation is similar (but not equally coincide with) to the statement that complex scalar field $\varphi$ and conjugated one $\varphi^{*}$ are independent degrees of freedom.