# 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

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

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. This explains the name of the “chiral basis” Dirac matrices from the previous section, since they diagonalize the chirality operator. The specific chiral bases we listed allow us write the Dirac spinor $${\psi}$$ as stacked Weyl spinors, since

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

where $${P_{-}}$$ similarly projects to the $${\psi_{L}}$$ half-spinor. $${\psi_{R}}$$ along with the associated rep and projection are called “right-handed” due to the fact that in physics they correspond to particles “spinning” around an axis aligned with the particle’s momentum, using the right-hand rule.

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: $${P_{\pm}\left(C_{0}\left(r,s\right)\right)\rightarrow P_{\mp}\left(C_{0}\left(r,s\right)\right)}$$.