In mathematics, twistor space is the complex vector space of solutions of the twistor equation . 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, the solutions to the twistor equation are of the form
where and are two constant Weyl spinors and is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by, and with a hermitian form
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
Given a point it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a parametrized by .
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 space
- P := F1(T) = P3(C) = P(C4)
- compactified complexified Minkowski space
- M := F2(T) = G2(C4) = G2,4(C)
- the correspondence space between P and M
- F := F1,2(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.
Famous quotes containing the word space:
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (16421727)