Spinor Understanding: QFT vs pure Representation Theory

Spinors are vectors in the representation vector space, not matrices in the image of the representation map.

1. A Dirac spinor or bispinor transforms in the (only) irreducible representation of the Clifford algebra $\mathrm{Cl}(1,3)$. This representation is four-dimensional.

2. A Weyl spinor transforms in an irreducible complex representation of the Lorentz algebra $\mathfrak{so}(1,3)$ (and hence of $\mathrm{Spin}(1,3)$), of which there are two that are denoted by $(1/2,0)$ and $(0,1/2)$, the "left-handed" and "right-handed" representations. These representations are two-dimensional.

3. $\mathfrak{so}(1,3)$ is isomorphic as a Lie algebra to the degree 2 subalgebra of $\mathrm{Cl}(1,3)$, so the Dirac representation - irreducible as a representation of $\mathrm{Cl}(1,3)$ - is also a not necessarily irreducible representation of $\mathfrak{so}(1,3)$.

4. In fact, as a representation of $\mathfrak{so}(1,3)$ the Dirac representation is reducible and isomorphic to $(1/2,0)\oplus (0,1/2)$. This is what the physicist means when they write $\psi = \begin{pmatrix} \psi_L \\ \psi_R\end{pmatrix}$.