# Chiral decomposition

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.