# Twistor theory

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In theoretical and mathematical physics, twistor theory is a theory proposed by Roger Penrose in 1967,[1] as a possible path to a theory of quantum gravity.

In twistor theory, the Penrose transform maps Minkowski space into twistor space, taking the geometric objects from a 4-dimensional space with a Hermitian form of signature (2,2) to geometric objects in twistor space, specified by complex coordinates, called twistors. The twistor approach is especially natural for solving the equations of motion of massless fields of arbitrary spin.[2]

## Overview

Penrose's twistor theory is unique to four-dimensional Minkowski space,[3] with its signature (3,1) metric.[4] 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 unitary transformations of determinant 1 over a four-dimensional complex vector space that leave invariant a Hermitian form of signature (2,2), see classical group.

• ${\displaystyle \mathbb {R} ^{6}}$ is the real 6D vector space corresponding to the vector representation of Spin(4,2).
• ${\displaystyle \mathbf {R} \mathbb {P} ^{5}}$ is the real 5D projective representation corresponding to the equivalence class of nonzero points in ${\displaystyle \mathbb {R} ^{6}}$ under scalar multiplication.
• ${\displaystyle \mathbb {M} ^{c}}$ corresponds to the subspace of ${\displaystyle \mathbf {R} \mathbb {P} ^{5}}$ corresponding to vectors of zero norm. This is conformally compactified Minkowski space.
• ${\displaystyle \mathbb {T} }$ is the 4D complex Weyl spinor representation, called twistor space. It has an invariant Hermitian sesquilinear norm of signature (2,2).
• ${\displaystyle \mathbb {PT} }$ is a 3D complex manifold corresponding to projective twistor space.
• ${\displaystyle \mathbb {PT} ^{+}}$ is the subspace of ${\displaystyle \mathbb {PT} }$ 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.
• ${\displaystyle \mathbb {PN} }$ is the subspace of ${\displaystyle \mathbb {PT} }$ 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).
• ${\displaystyle \mathbb {PT} ^{-}}$ is the subspace of ${\displaystyle \mathbb {PT} }$ of projective twistors with negative norm.

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

${\displaystyle \mathbb {M} ^{c}}$ 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 ${\displaystyle \mathbb {M} ^{c}}$ and points in ${\displaystyle \mathbb {PN} }$ respecting the conformal group.

In ${\displaystyle \mathbb {M} ^{c}}$, it is the case that positive and negative frequency solutions cannot be locally separated. However, this is possible in twistor space:

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

## Variations

### Supertwistors

Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978.[5] Along with the standard twistor degrees of freedom, a supertwistor contains N fermionic scalars, where N is the number of supersymmetries. The superconformal algebra can be realized on supertwistor space.

### Twistor string theory

Twistor theory progressed slowly, in part because of mathematical challenges (see Palatial twistor theory). Twistor theory also seemed unrelated to ideas in mainstream physics: while twistor theory appeared to say something about quantum gravity (after the original 1976 construction of the left-handed non-linear graviton),[6][7][8] its potential contributions to understanding the other fundamental interactions and particle physics were less obvious.[9]

In 2003, Edward Witten[10] proposed uniting twistor and string theory by embedding the topological B model of string theory in twistor space, whose dimensionality is necessarily the same as that of 3+1 Minkowski spacetime. His objective was to model planar N = 4 Yang–Mills amplitudes.

The resulting model, defined on the supertwistor space ${\displaystyle \mathbb {CP} ^{3|4}}$ (see Supergroup for the vertical bar notation), has come to be known as twistor string theory. Although Witten has said that "I think twistor string theory is something that only partly works," his work has given new life to the twistor research program. For example, twistor string theory may simplify calculating scattering amplitudes from Feynman diagrams by using a geometric structure called an amplituhedron. Simone Speziale and collaborators have also applied twistor string theory to loop quantum gravity.[11]

Penrose himself rejects string theory, and criticizes it in his book, Fashion, Faith, and Fantasy in the New Physics of the Universe.[12][13]

### Palatial twistor theory

The problem of somehow modifying a certain type of twistor functions (i.e. the –6 homogeneity twistor functions) to obtain a right-handed non-linear graviton has been referred to as the (gravitational) googly problem (the word "googly" is a term used in the game of cricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity).[6] It took nearly 40 years (since the publication of Penrose 1976a/1976b) to find a plausible solution and provide an appropriate construction for this. The result was obtained by Penrose in 2015 and was dubbed palatial twistor theory[14] (named after Buckingham Palace, the place in which Michael Atiyah suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory).[15][16]

## Notes

1. ^ Penrose, R. (1967). "Twistor Algebra". Journal of Mathematical Physics. 8 (2): 345. Bibcode:1967JMP.....8..345P. doi:10.1063/1.1705200.
2. ^ This is due to the fact that there is a relationship between the analytic continuation properties needed for the "positive frequency" condition in complex solutions of quantum field equations and a certain generalization, to massless fields of arbitrary spin (see Roger Penrose, "On the Origins of Twistor Theory" in: Gravitation and Geometry, a Volume in Honour of I. Robinson, Biblipolis, Naples 1987).
3. ^ Roger Penrose (1971), "Angular momentum: an approach to combinatorial spacetime," in T. Bastin (ed.), Quantum Theory and Beyond, Cambridge University Press: "The twistor group is the (+,+,−,−) pseudo-unitary group SU(2,2), which is locally isomorphic with O(2,4). In turn, O(2,4) is locally isomorphic with the fifteen-parameter (local) conformal group of space-time."
4. ^ Penrose 2004, p. 891: "it is an essential implication of my own particular "twistorial" perspective that spacetime indeed have the directly observed values of one time and three space dimensions (i.e. '1 + 3 dimensions')."
5. ^ Ferber, A. (1978), "Supertwistors and conformal supersymmetry", Nuclear Physics B, 132: 55–64, Bibcode:1978NuPhB.132...55F, doi:10.1016/0550-3213(78)90257-2.
6. ^ a b Penrose 2004, p. 1000.
7. ^ Penrose, R. (1976a). "The non-linear graviton." Gen. Rel. Grav. 7, 171–6.
8. ^ Penrose, R. (1976b). "Non-linear gravitons and curved twistor theory." Gen. Rel. Grav. 7, 31–52.
9. ^ Penrose 2004, p. 1001.
10. ^ Witten, Edward (6 October 2004). "Perturbative Gauge Theory as a String Theory in Twistor Space". Communications in Mathematical Physics. 252 (1-3): 189–258. Bibcode:2004CMaPh.252..189W. arXiv:. doi:10.1007/s00220-004-1187-3.
11. ^ Freidel, Laurent; Speziale, Simone (25 October 2010). "Twistors to twisted geometries". Physical Review D. 82 (8). Bibcode:2010PhRvD..82h4041F. arXiv:. doi:10.1103/PhysRevD.82.084041.
12. ^ "Fashion, Faith, and Fantasy in the New Physics of the Universe - Roger Penrose". Princeton.edu. 11 January 2017.
13. ^ Graham Farmelo (12 November 2016). "Fashion, Faith and Fantasy in the New Physics of the Universe by Roger Penrose – review". Guardian.com.
14. ^ Penrose R. (2015). "Palatial twistor theory and the twistor googly problem." Phil. Trans. R. Soc. A 373: 20140237.
15. ^ Michael Atiyah's Imaginative State of Mind – Quanta Magazine
16. ^ The underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutative holomorphic twistor quantum algebra.