Twistor space

In mathematics, twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 . It was described in the 1960s by Roger Penrose and MacCallum. According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.

For Minkowski space, denoted M , the solutions to the twistor equation are of the form

Ω A ( x ) = ω A − i x A A ′ π A ′

where ω A and π A ′ are two constant Weyl spinors and x A A ′ = σ μ A A ′ x μ is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by Z α = ( ω A , π A ′ ) , and with a hermitian form

Σ ( Z ) = ω A π ¯ A + ω ¯ A ′ π A ′

which is invariant under the group SU(2,2) which is 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

ω A = i x A A ′ π A ′ .

This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted PT, which is isomorphic as a complex manifold to C P 3 .

Given a point x ∈ M it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a C P 1 parametrized by π A ′ .

The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is

T := C⁴. It has associated to it the double fibration of flag manifolds P ←_μ F _ν→ M, where

projective twistor spacecompactified complexified Minkowski spacethe correspondence space between P and MF := F_1,2(T)

In the above, P stands for projective space, G a Grassmannian, and F a flag manifold. The double fibration gives rise to two correspondences, c := ν . μ⁻¹ and c⁻¹ := μ . ν⁻¹.

M is embedded in P₅ ~=~ P(Λ²T) by the Plücker embedding and the image is the Klein quadric.