Majorana spinor

Definition

Let $\Delta$ be a complex spinor module for $\operatorname{Spin}(p,q)$. A real structure on $\Delta$ is an antilinear $\operatorname{Spin}(p,q)$-equivariant map

Relative to such a structure, a Majorana spinor is a spinor $\psi\in\Delta$ satisfying

Equivalently, it is an element of the real form $\Delta^J\subset\Delta$.

The existence and type of an equivariant antilinear structure depend on the dimension, signature, and Clifford sign convention. An antilinear equivariant map with $J^2=-1$ is a quaternionic structure, not a Majorana real structure in the sense above.

The Majorana condition is invariantly a reality condition on the representation. It is not the basis-dependent assertion that the coordinates of a spinor, or all of its gamma matrices, are real.