 Two-component spinor calculus

The complexified tangent space of a four-dimensional Lorentzian manifold factors locally into an unprimed and a primed two-dimensional spinor representation. Their alternating forms ${ϵ}_{AB}$ and ${ϵ}_{A^{\prime }{B}^{\prime }}$ encode the metric tensor and raise or lower spinor indices. A convention must specify the order of contraction with these alternating forms. An antisymmetric spacetime tensor decomposes into symmetric unprimed and primed spinors, $F_{A{A}^{\prime }B{B}^{\prime }}={\varphi }_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}+{\tilde{\varphi }}_{A^{\prime }{B}^{\prime }}{ϵ}_{AB}$. This splits the six-dimensional space of differential two-forms into the two three-dimensional eigenspaces of the complexified Hodge star operator.