Twistor theory

Remarks

For each $x$, the incident twistors form the 2-dimensional subspace $\set{(iX\pi, \pi) : \pi \in \C^2}$, so $L_x$ is a projective line. This gives a bijection between complexified Minkowski space and the lines of $\mathbb{PT}$ that do not meet the line $I = \set{\pi = 0}$. All lines of $\mathbb{PT}$ form the Grassmannian $\mathrm{Gr}(2, 4)$, a compactification of complexified Minkowski space. Example: if $x$ is real, then $X$ is Hermitian, and for $\omega = iX\pi$ one gets $\Sigma(Z) = -i \pi^* X \pi + i \pi^* X \pi = 0$. So real points correspond to lines inside the real hypersurface $\mathbb{PN} = \set{\Sigma = 0}$. Non-example: the line $I$ is not of the form $L_x$ for any point $x$ of complexified Minkowski space, since every $L_x$ contains only twistors with $\pi \ne 0$. Two points $x, y$ are null separated exactly when $\det(X - Y) = 0$, that is, when the lines $L_x$ and $L_y$ meet. Many texts write the incidence relation as $\omega^A = i x^{AA'} \pi_{A'}$ in two-component spinor notation, with an extra factor $1/\sqrt{2}$ in $X$.