R. Penrose, F. Hadrovich

Twistor Theory

The motivation and one of the initial aims of twistor theory is to provide an adequate formalism for the union of quantum theory and general relativity. Twistors are essentially complex objects, like wavefunctions in quantum mechanics, as well as endowed with holomorphic and algebraic structure sufficient to encode space-time points. In this sense twistor space can be considered more primitive than the space-time itself and indeed provides a background against which space-time could be meaningfully quantised.

In quantum mechanics a state of an object is described by a vector in Hilbert space. This is a linear superposition of some basis vectors, for example, in the case of an electron (or any spin-1/2 particle) it is some linear superposition of the spin "up" and "down" states. The two component vector (z,w) is called a spinor. The norm of this spinor caries no information - indeed, it is usually set to 1. Only the direction is physically significant which forces us to consider the projective (spin) space. In the electron example, spin state is therefore described by a point in CP1, conveniently represented with Riemann sphere.

Stereographic projection is a shape (i.e. angle) preserving map. Hence circles on the sphere will be mapped to circles on the plane (except those containing the south pole, which are mapped to straight lines on the plane). This geometry is preserved by Möbius (fractional linear) transformation. In terms of spinors, this is the action of SL(2,C).

In relativity, an observer sees a t=const. section of its past null cone. This is a celestial (Riemann) 2-sphere that can be stereographically projected onto a complex plane. Möbius transformations of the plane correspond to Lorentz transformations of the celestial sphere.

This "spinorial" point of view has an interesting application in finding the apparent shape of a rapidly moving sphere.

Naively we would expect to see Lorentz contraction. However, no such contraction will be observed: every observer will see a circular outline (moving in some direction; of corse, they do not agree on the direction or the velocity of the sphere).

The spherical outline on the t=const. section of observer's past light cone will remain spherical under Lorentz transformation since we know that Lorentz transformation on the celestial sphere is just Möbius transformation on the image of the sphere under stereographic projection. All these are shape preserving maps which proves out assertion.

Synge argument is another demonstration of the Lorentz-invariance of the circular outline on the celestial sphere. Here the blue timelike hyperplane intersects the observer's past lightcone and t=const. hyperplanes in a circle. It is clear that Lorentz transformation cannot change the shape of the circle.

Now it is time to introduce twistors into the story. We have seen that a space-time point can be represented as a Riemann sphere in terms of some section of its light cone. This is precisely how space-time points are represented in the projective twistor space.

The full twistor space is just a 4-dimensional complex vector space {(Z0,Z1,Z2,Z3)} = C4.

For more technical details see Twistor Primer

Another way of defining twistors is as spinors for O(2,4) group (actually, the subgroup that preserves "time" and "space" orientations). Minkowski space M can be represented in R6 with metric signature (+ + - - - -) as a parabolic intersection of a hyperplane with the null cone of the origin. All generators of the cone intersect M except for the U = W generator. Adding this generator gives the compactified Minkowski space M#. Projectively, M# corresponds to a quadric.

Twistors are essentially complex objects and in order to proceed we shall have to consider the complexification of the compactified Minkowski space. This is a quadric in the 5-dimensional complex projective space. On this quadric, there are two 3-parameter families of totally null 2-planes, one self-dual (SD, alpha-planes), the other anti-self-dual (ASD, beta-planes). Projective twistor space is the space of all alpha-planes. Through every point on the quadric there is a one-parameter family of alpha-planes which means that space-time points are represented as projective lines in the projective twistor space.

This correspondence between the lines in CP3 and points of a quadric in CP5 is known as the Klein correspondence.

In terms of the coordinates (r0,r1,r2,r3) on CM, the complexified Minkowski space, and homogeneous coordinates (Z0,Z1,Z2,Z3) on CP3, the Klein correspondence can be espressed explicitly in terms of the incidence relation.

If we restrict ourselves to real space-time points, the corresponding lines in CP3 lie in a 5-real-dimensional hypersurface PN.

This transparency shows the same matrix equation of incidence relation written in terms of spinor components of a twistor in abstract-index notation. Indices A, A' take values 0 and 1.

In complexified Minkowski space the incidence relation associates an alpha-plane with each projective twistor.

The spinor components of a twistor have a natural interpretation in terms of the momentum and angular momentum of a zero rest mass particle. Technical details can be found in the Twistor Primer.

This identification also yields the spin in the form of Pauli-Lubanski vector.

The quantum mechanical commutation relations for momentum and angular momentum give simple commutation relation for twistors and dual twistors. Hence the quantisation rule for twistor theory: dual twistors are represented with derivative operators. Substituting into the expression for the spin, we observe that Euler homogeneity operator features in the formula. States with well defined spin s are therefore described by functions on twistor space which are homogeneous of degree -2s-2.

Massless field of spin s is a spinor field with 2s indices, primed for s>0 and unprimed for s<0. We expect to be able to encode a spin s massless field with a homgeneity degree -2s-2 twistor function.

Indeed, this is achieved with a contour integral formula shown.

An important observation can be made here that two different twistor functions encode the same field if their difference is a holomorphic function. Thus it is the equivalence classes of functions (strictly speaking, Cech cohomology representatives) that correspond to massless fields. For more information on Cech cohomology see Applications of Sheaf Cohomology in Twistor Theory.

These are the massless fields equations and their solutions in the linearised case of gravitational interaction. As expected, homogeneity degree +2 and -6 functions encode the anti-self-dual and self-dual parts of the gravitational field.