= Two-component spinor calculus
{title2=$v^a=v^{AA'},\quad g_{ab}=\epsilon_{AB}\epsilon_{A'B'}$}
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 $\epsilon_{AB}$ and $\epsilon_{A'B'}$ 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, $F_{AA'BB'}=\phi_{AB}\epsilon_{A'B'}+\widetilde\phi_{A'B'}\epsilon_{AB}$. This splits the six-dimensional space of <differential two-forms> into the two three-dimensional eigenspaces of the complexified <Hodge star operator>.
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