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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 $P_R\psi, P_L\psi$?

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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 $\mathcal{H}$.

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  • $\begingroup$ Thank you for your answer, I did suspect it was something like that. Unfortunately the exposure I had on this topic was quite mathematically sloppy and I feel too many details were glossed over. $\endgroup$ Apr 16 at 15:23
  • $\begingroup$ Glad it helps. My favorite exposition of this topic is chapter 10 in Schwartz's Quantum Field Theory and the Standard Model. He starts with representations of the Lorentz group, which motivates this decomposition very naturally (chapter 10.2). He's also not afraid to flash mathematical terminology, which makes it easy to google for extra context. $\endgroup$
    – jsborne
    Apr 16 at 15:53

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