Majorana–Weyl spinor

Definition

Let

be an even-dimensional complex spinor module, and let $J$ be a Majorana real structure that preserves each half-spin module:

A Majorana–Weyl spinor of positive or negative chirality is a spinor $\psi\in\Delta^\pm$ such that

It is therefore simultaneously a Weyl spinor and a Majorana spinor.

The definition is available only when the dimension and signature admit a real structure compatible with the chiral decomposition. The existence of a Majorana condition and the existence of Weyl spinors separately do not imply that the two conditions are compatible.