Twistor theory

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The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2). The coordinates in such a space are called "twistors."

The twistor theory was stimulated by a rationale indicating its particular usefulness in emergent theories of quantum gravity.

The twistor approach appears to be especially natural for solving the equations of motion of massless fields of arbitrary spin.

In 2003 Edward Witten used twistor theory to understand certain Yang-Mills amplitudes, by relating them to a certain string theory, the topological B model, embedded in twistor space. This field has come to be known as twistor string theory.

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Twistor theory is unique to 4D Minkowski space and does not generalize to other dimensions or metric signatures. At the heart of twistor theory lies the isomorphism between the conformal group Spin(4,2) and SU(2,2), which is the group of linear transformations of determinant 1 over a four dimensional complex vector space leaving a Hermitian norm of signature (2,2) invariant.

$\mathbb{R}^6$ is the real 6D vector space corresponding to the vector representation of Spin(4,2).
$\mathbf{R}\mathbb{P}^5$ is the real 5D projective representation corresponding to the equivalence class of nonzero points in $\mathbb{R}^6$ under scalar multiplication.
$\mathbb{M}^c$ corresponds to the subspace of $\mathbf{R}\mathbb{P}^5$ corresponding to vectors of zero norm. This is conformally compactified Minkowski space.
$\mathbb{T}$ is the 4D complex Weyl spinor representation and is called twistor space. It has an invariant Hermitian sesquilinear norm of signature (2,2).
$\mathbb{PT}$ is a 3D complex manifold corresponding to projective twistor space.
$\mathbb{PT}^+$ is the subspace of $\mathbb{PT}$ corresponding to projective twistors with positive norm (the sign of the norm, but not its absolute value is projectively invariant). This is a 3D complex manifold.
$\mathbb{PN}$ is the subspace of $\mathbb{PT}$ consisting of null projective twistors (zero norm). This is a real-complex manifold (i.e. it has 5 real dimensions, with four of the real dimensions having a complex structure making them two complex dimensions).
$\mathbb{PT}^-$

is the subspace of $\mathbb{PT}$ of projective twistors with negative norm.

$\mathbb{M}^c$ , $\mathbb{PT}^+$ , $\mathbb{PN}$ and $\mathbb{PT}^-$ are all homogeneous spaces of the conformal group.

$\mathbb{M}^c$ admits a conformal metric (i.e. an equivalence class of metric tensors under Weyl rescalings) with signature (+++-). Straight null rays map to straight null rays under a conformal transformation and there is a unique canonical isomorphism between null rays in $\mathbb{M}^c$ and points in $\mathbb{PN}$ respecting the conformal group.

One thing about $\mathbb{M}^c$ is that it is not possible to separate positive and negative frequency solutions locally. However, this is possible in twistor space.

$\mathbb{PT}^+ \simeq SU(2,2)/\left[ SU(2,1) \times U(1) \right]$