Chirality operator

Definition

Let $V$ be an oriented complex quadratic space of even dimension, and let $\Delta$ be a complex spinor module. The oriented Clifford volume element has a scalar normalization $\omega_{\mathbb C}$ satisfying

The chirality operator on $\Delta$ is

The normalizing scalar depends on the dimension, signature, and convention $c(v)^2=\pm g(v,v)$; the defining requirements are $\Gamma^2=1$ and the chosen orientation.

The chirality operator commutes with the even Clifford algebra and anticommutes with Clifford multiplication by every vector:

Its eigenspaces give the half-spin decomposition

Elements of these eigenspaces are Weyl spinors.

In a matrix realization, $\Gamma$ is an appropriately normalized product of all gamma matrices. It is often denoted $\gamma_5$ in four-dimensional physics, but the invariant construction is not tied to that dimension or notation.