 Two-component spinor

A two-component two-component spinor is an element of a two-dimensional complex spin space with invariant antisymmetric form ${ϵ}_{AB}$. In complexified four-dimensional geometry there are independent unprimed and primed spin spaces, and vectors are represented by $x^{A{A}^{\prime }}$. Raising an index uses the antisymmetric form, so ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$. This identity makes the incidence-plane directions ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$ totally null.