# 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 $$\psi = \begin{pmatrix} \psi_L \\ \psi_R\end{pmatrix}$$.