# Dirac spinor in the chiral basis

In the chiral basis, the gamma matrices take the form $$\gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix}$$ and therefore one can calculate what the left and right projectors look like: $$P_R=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}, \quad P_L=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}.$$ Given a Dirac spinor with components $$\psi=(\psi_1,\psi_2,\psi_3,\psi_4)^T$$, it is pretty clear that the Weyl spinors should become $$\psi_R:=P_R\psi=\begin{pmatrix}\psi_1 \\ \psi_2 \\ 0 \\ 0\end{pmatrix}, \quad \psi_L:=P_L\psi=\begin{pmatrix}0 \\ 0 \\ \psi_3 \\ \psi_4\end{pmatrix}$$ and one can reconstruct the spinor by summing over both of them, as $$\psi=\psi_R+\psi_L$$. I've been told however that in this basis we can decompose the Dirac spinor in terms of the Weyl spinors as $$\psi=\begin{bmatrix}\psi_R \\ \psi_L \end{bmatrix}.$$ This can't be possinle, if $$\psi_R$$ and $$\psi_L$$ are the objects with four components defined above. So it is probably a notational issue; who are these $$\psi_R,\psi_R$$ and what is their relation with $$P_R\psi, P_L\psi$$?

The projectors $$P_R, P_L$$ project $$\psi \in \mathcal{H}\cong \mathbb{R}^4$$ onto the right- and left-handed sectors of the representation of the Lorentz algebra, which are each a two-dimensional vector space, hence (locally) isomorphic to $$\mathbb{R}^2$$.
What "sum" is meant here is the direct sum $$\oplus$$: $$\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} \oplus \begin{pmatrix} \psi_3 \\ \psi_4 \end{pmatrix} = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix}.$$ You correctly observed that the projector operators drop the two entries not in the projected space, so $$P_L \mathcal{H} \cong \mathbb{R}^2$$, likewise for $$P_R \mathcal{H}$$. So the statement "$$\psi = \psi_1 + \psi_2$$" really means $$\mathcal{H} \cong P_L \mathcal{H} \oplus P_R \mathcal{H}.$$ To answer your last question, then, $$\psi_R$$ and $$\psi_L$$ are the members of $$\mathbb{R}^2$$ which can be identified (via isomorphism) to the images of the projections of $$\mathcal{H}$$.