Future-null spinor factorization ($X=\chi {\chi }^{†}$)

A nonzero Hermitian rank-one positive semidefinite matrix has zero determinant and positive trace, representing a future-directed null vector. It factors as $\chi {\chi }^{†}$, with an irrelevant spinor phase. A past-directed null vector instead requires a minus sign; a positive outer product alone does not cover both time directions.

With $X_{12}=x+iy$, a spinor whose components have phases $e^{-i\varphi /2}$ and $e^{i\varphi /2}$ has $x+iy\propto e^{-i\varphi }$, so its physical azimuth is $-\varphi$. The usual positive azimuth for those phases uses $X_{12}=x-iy$. Both conventions are valid if used consistently.