Twistor space
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation formula_1. It was described in the 1960s by Roger Penrose and Malcolm 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. Informal motivation. In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying  four-dimensional space formula_2 it might be valuable to identify it with formula_3 However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space formula_4 parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in formula_2. It turns out that vector bundles with self-dual connections on formula_2(instantons) correspond bijectively to holomorphic bundles on complex projective 3-space formula_7 Formal definition. For Minkowski space, denoted formula_8, the solutions to  the twistor equation are of the form where formula_10 and formula_11 are two constant Weyl spinors and formula_12 is a point in Minkowski space. The formula_13 are the Pauli matrices, with formula_14 the indexes on the matrices. This twistor space is a four-dimensional complex vector space, whose points are denoted by formula_15, 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 This incidence relation is preserved under  an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted formula_18, which is isomorphic as a complex manifold to formula_4. Given a point formula_20 it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a formula_21 parametrized by formula_11. 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 It has associated to it the double fibration of flag manifolds  formula_24 where formula_25 is the projective twistor space and formula_8 is the compactified complexified Minkowski space and the correspondence space between formula_25 and formula_8 is In the above, formula_32 stands for projective space, formula_33 a Grassmannian, and formula_34 a flag manifold. The double fibration gives rise to two (see also Penrose transform), formula_35 and formula_36 The compactified complexified Minkowski space formula_8 is embedded in formula_38 by the Plücker embedding; the image is the Klein quadric.