Dirac spinor

Definition

Let $\Delta$ be a chosen complex spinor module. A Dirac spinor is an element of $\Delta$. On a spin manifold, a Dirac spinor field is a section of the corresponding complex spinor bundle.

In even dimension the complex spinor module has the chiral decomposition

A Dirac spinor may therefore have components of both chiralities. A spinor lying entirely in one summand is a Weyl spinor.

“Dirac spinor” specifies the complex spin representation in which the spinor takes values; it does not assert that the spinor satisfies the Dirac equation.