In quantum-field theories, matter positive frequencies can go forward in time, and antimatter negative frequencies can go backward in time {twistor, quantum mechanics}| (Penrose). In Minkowski space, twistors are spinors and complex-conjugate spinors.
Riemann sphere
Complex numbers graph to planes. Plane can be at Riemann sphere equator. Pole point can be at infinity. Line from pole through plane can intersect Riemann sphere. Real numbers are on equator. Positive frequencies are in upper hemisphere. Riemann sphere is twistor space. Twistor space has two plane dimensions and three space-time-point dimensions. Adding spin makes six real dimensions {projective twistor space}.
space-time and quantum mechanics
Perhaps, general relativity and quantum mechanics unify using twistors. Space-time relates to quantum-mechanics complex amplitudes through Riemann spheres. Riemann-sphere space-time points have light-ray sets. Space-time events are Riemann-sphere directions, showing which past events can affect future event. In twistor space, light rays are points, so twistor space is not local. Photons have right or left circular polarization {helicity}. Half-spin particles have up and down spin superpositions, as observer sees Riemann sphere. Riemann spheres can have inscribed icosahedrons, which define 20 sphere points. Points join three edges, which can be like three space dimensions. Points combine two independent entangled fermion spins, with spin +1/2 or -1/2. Riemann tensor has 20 components in flat space-time. Perhaps, complex numbers can relate general relativistic space-time to spin quantum mechanics [Penrose, 2004]. At different velocities, transformation groups {Möbius transformation, twistor} can find curvature.
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Date Modified: 2022.0224