Weyl spinor

Definition

Let $V$ be an even-dimensional oriented quadratic space and let

be a complex spinor module decomposed into the $+1$ and $-1$ eigenspaces of its chirality operator. A Weyl spinor of positive or negative chirality is an element of $\Delta^+$ or $\Delta^-$, respectively.

On an even-dimensional spin manifold, a Weyl spinor field is a section of one of the half-spinor bundles $S^+$ or $S^-$. Clifford multiplication by a vector reverses chirality:

The definition requires even dimension and a complex chiral spin representation. A compatible reality condition is separate data; when one exists and preserves a half-spin representation, it defines Majorana–Weyl spinors.