As we have seen, the Dirac and Majorana reps are in fact isomorphisms from \({C\mathbb{^{C}}\left(r,s\right)}\) and \({C\left(r,s\right)}\), and so are faithful and irreducible; however, as we have also seen they are sometimes reducible when restricting their action to the even subalgebra, decomposing into two chiral irreps. These chiral spinor reps of \({C_{0}\left(r,s\right)}\) can be obtained by projection, revealing some important attributes.

For \({A\in C_{0}\left(r,s\right)}\) and even \({n}\), one can verify that the operators

\(\displaystyle P_{\pm}\left(A\right)\equiv\frac{1}{2}\left(A\pm\sqrt{\Omega^{2}}\,\Omega A\right) \)

are **orthogonal projections**, i.e. they are idempotent, \({P_{\pm}^{2}=P_{\pm},}\) with \({P_{\pm}P_{\mp}=0}\) and \({P_{+}+P_{-}=1}\). Since the unit \({n}\)-vector \({\Omega}\) of \({C\left(r,s\right)}\) commutes with any \({A\in C_{0}\left(r,s\right)}\), we then have the decomposition

\(\displaystyle C_{0}\left(r,s\right)\cong P_{+}\left(C_{0}\left(r,s\right)\right)\oplus P_{-}\left(C_{0}\left(r,s\right)\right), \)

which is also maintained under Clifford multiplication since the unit \({n}\)-vector \({\Omega}\) of \({C\left(r,s\right)}\) commutes with any \({A\in C_{0}\left(r,s\right)}\). Note that this decomposition is not possible if \({n}\) is odd, since then \({\Omega A\notin C_{0}\left(r,s\right)}\) and so \({P_{\pm}\left(C_{0}\left(r,s\right)\right)\notin C_{0}\left(r,s\right)}\). For even \({n}\), the quantity \({\Omega^{2}=\left(-1\right)^{n\left(n-1\right)/2+s}}\) must be positive in order to obtain a real square root. This is only true if \({r-s=0}\) or 4 mod 8, and as we saw above only in the first case is the resulting algebra isomorphic to a real matrix algebra and thus a Majorana-Weyl rep. This restriction is avoided if we apply the decomposition to \({C_{0}\mathbb{^{C}}(n)}\), which is why a Weyl rep exists for any even \({n}\).

In the present context \({\gamma_{5}\equiv\sqrt{\Omega^{2}}\,\Omega}\) is sometimes called the **chirality operator**, and is the generalization of \({\gamma_{5}}\) to arbitrary signature and dimension. We can see that the chiral basis Dirac matrices from the previous section block diagonalize \({C_{0}(3,1)}\) due to the form of \({\gamma_{5}}\); this also allows us write a Dirac spinor \({\psi\in\mathbb{C}^{4}}\) as stacked Weyl spinors

\(\displaystyle P_{+}\left(\psi\right)=\frac{1}{2}\left(I+\gamma_{5}\right)\psi=\begin{pmatrix}0 & 0\\ 0 & I \end{pmatrix}\begin{pmatrix}\psi_{\mathrm{L}}\\ \psi_{\mathrm{R}} \end{pmatrix}=\begin{pmatrix}0\\ \psi_{\mathrm{R}} \end{pmatrix}, \)

where \({P_{-}}\) similarly projects to the \({\psi_{\mathrm{L}}}\), explaining why these are called half-spinors. This stacked decomposition remains invariant under the transformation \({\psi\rightarrow A\psi}\) by any \({A\in C_{0}\left(r,s\right)}\).

In the case of a Lorentzian signature, we can also consider **time reversal** and **parity** operators, which reverse the sign of either the negative signature basis vector or the \({\left(n-1\right)}\) positive signature basis vectors. In either case, for even \({n}\) this consists of reversing an odd number of basis vectors, so that \({\Omega\rightarrow-\Omega}\), and thus under either operation the chiral spinor reps are swapped: \({P_{\pm}\rightarrow P_{\mp}\Rightarrow\psi_{\mathrm{L}}\leftrightarrow\psi_{\mathrm{R}}}\). The parity operation reverses spatial orientation as a mirror image does, which is why the half-spinors are called chiral: they are swapped under parity, just as the right and left hands are swapped in a mirror image.