Twistors

Twistor theory, introduced by Roger Penrose in 1967, reformulates the geometry of four-dimensional spacetime in terms of twistor space, a complex projective space. A twistor is a basic object encoding information about null geodesics and light rays, turning conformal structure into linear data on twistor space.

For complexified, conformally compactified Minkowski space, a twistor T is a pair of spinor components (ω^A, π_A'). They relate to a spacetime point x via the incidence relation ω^A = x^{AA'} π_{A'}. In this correspondence, a spacetime point corresponds to a Riemann sphere of twistors, while a twistor determines a null line through that point.

Twistor space is projective, denoted CP^3 in the basic case, with dual twistors available as well. The Penrose transform links massless fields on spacetime to holomorphic data on twistor space, making conformal invariance natural in this setting and enabling efficient handling of wave equations and self-dual solutions.

Extensions include supertwistors for supersymmetric theories and ambitwistor theory, which underpins modern approaches to scattering amplitudes. In the 2000s, twistor methods inspired alternative formulations of gauge theory, notably Witten's twistor string theory for perturbative N=4 super Yang–Mills, with ongoing developments in ambitwistor approaches.

Limitations include a focus on conformally invariant or massless aspects; real spacetime requires reality conditions and further structure. Twistor theory is a powerful, specialized tool that complements spacetime methods rather than replacing them.