Editing Twistor theory (section)

==The twistor correspondence==
Denote [[Minkowski space]] by <math>M</math>, with coordinates <math>x^a = (t, x, y, z)</math> and Lorentzian metric <math>\eta_{ab}</math> signature <math>(1, 3)</math>. Introduce 2-component spinor indices <math>A = 0, 1;\; A' = 0', 1',</math> and set

:<math>x^{AA'} = \frac{1}{\sqrt{2}}\begin{pmatrix} t - z & x + iy \\ x - iy & t + z \end{pmatrix}.</math>

Non-projective twistor space <math>\mathbb{T}</math> is a four-dimensional complex vector space with coordinates denoted by <math>Z^{\alpha} = \left(\omega^{A},\, \pi_{A'}\right)</math> where <math>\omega^A</math> and <math>\pi_{A'}</math> are two constant [[Weyl spinor]]s. The hermitian form can be expressed by defining a complex conjugation from <math>\mathbb{T}</math> to its dual <math>\mathbb{T}^*</math> by <math>\bar Z_\alpha = \left(\bar\pi_A,\, \bar \omega^{A'}\right)</math> so that the Hermitian form can be expressed as

:<math>Z^\alpha \bar Z_\alpha = \omega^{A}\bar\pi_{A} + \bar\omega^{A'}\pi_{A'}.</math>

This together with the holomorphic volume form, <math>\varepsilon_{\alpha\beta\gamma\delta} Z^\alpha dZ^\beta \wedge dZ^\gamma \wedge dZ^\delta</math> is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation

:<math>\omega^{A} = ix^{AA'}\pi_{A'}.</math>

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space <math>\mathbb{PT},</math> which is isomorphic as a complex manifold to <math>\mathbb{CP}^3</math>. A point <math>x\in M</math> thereby determines a line <math>\mathbb{CP}^1</math> in <math>\mathbb{PT}</math> parametrised by <math>\pi_{A'}.</math> A twistor <math>Z^\alpha</math> is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take <math>x</math> to be real, then if <math>Z^\alpha \bar Z_\alpha</math> vanishes, then <math>x</math> lies on a light ray, whereas if <math>Z^\alpha \bar Z_\alpha</math> is non-vanishing, there are no solutions, and indeed then <math>Z^{\alpha}</math> corresponds to a massless particle with spin that are not localised in real space-time.