Twistor theory

In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3+1 space-time (Minkowski space) into geometric objects in a 4-dimensional space with a Hermitian form of signature (2,2). This space is called twistor space, and its complex valued coordinates are called "twistors."

Twistor theory was first proposed by Roger Penrose in 1967,^[1] as a possible path to a theory of quantum gravity. The twistor approach is especially natural for solving the equations of motion of massless fields of arbitrary spin.

In 2003, Edward Witten^[2] proposed uniting twistor and string theory by embedding the topological B model of string theory in twistor space. His objective was to model certain Yang–Mills amplitudes. The resulting model has come to be known as twistor string theory (read below). Simone Speziale and collaborators have also applied it to loop quantum gravity.^[3]

Details

Twistor theory is unique to 4D Minkowski space and the (2,2) signature, and does not generalize to other dimensions or 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 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.

$\mathbb{R}^6$ 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, 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.

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

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

Twistor string theory

Main article: Twistor string theory

For many years after Penrose's foundational 1967 paper, twistor theory progressed slowly, in part because of mathematical challenges. Twistor theory also seemed unrelated to ideas in mainstream physics. While twistor theory appeared to say something about quantum gravity, its potential contributions to understanding the other fundamental interactions and particle physics were less obvious.

Witten (2003) proposed a connection between string theory and twistor geometry, called twistor string theory. Witten (2004)^[2] built on this insight to propose a way to do string theory in twistor space, whose dimensionality is necessarily the same as that of 3+1 Minkowski spacetime. 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.

Supertwistors

Witten's twistor string theory is defined on the supertwistor space $\mathbb{CP}^{3|4}$ . Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978.^[4] 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.

Notes

↑ Penrose, R. (1967). "Twistor Algebra". Journal of Mathematical Physics 8 (2): 345. doi:10.1063/1.1705200.
1 2 Witten, Edward (7 October 2004). "Perturbative Gauge Theory as a String Theory in Twistor Space". Communications in Mathematical Physics 252 (1-3): 189–258. arXiv:hep-th/0312171. doi:10.1007/s00220-004-1187-3.
↑ Freidel, Laurent; Speziale, Simone (25 October 2010). "Twistors to twisted geometries". Physical Review D 82 (8). arXiv:1006.0199. doi:10.1103/PhysRevD.82.084041.
↑ 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.

External links

Penrose, Roger (1999) "Einstein's Equation and Twistor Theory: Recent Developments"
Penrose, Roger; Hadrovich, Fedja. "Twistor Theory."
Dunajski, Maciej, "Twistor Theory and Differential Equations."
Hadrovich, Fedja, "Twistor primer."
Andrew Hodges, "Twistor Theory and the Twistor Programme." Includes many links.
Huggett, Stephen (2005) "The Elements of Twistor Theory."
Richard Jozsa (1976) "Applications of Sheaf Cohomology in Twistor Theory."
Mason, L. J., "The twistor programme and twistor strings:From twistor strings to quantum gravity?"
Sämann, Christian (2006) "Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory."
Sparling, George (1999) "On Time Asymmetry."
Spradlin, Marcus (2006), "Progress And Prospects In Twistor String Theory."
MathWorld - Twistors.
Universe Review "Twistor Theory."
Twistor newsletter archives

Theories of gravitation

Standard

Newtonian gravity (NG)	Newton's law of universal gravitation History of gravitational theory

General relativity (GR)	Introduction History Mathematics Resources Tests Post-Newtonian formalism Linearized gravity ADM formalism

Alternatives to
general relativity

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything

Classical	Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Massive gravity Modified Newtonian dynamics (MOND) Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Tensor–vector–scalar Chameleon Pressuron

Quantisation	Euclidean quantum gravity Canonical quantum gravity Wheeler–DeWitt equation Loop quantum gravity Spin foam Causal dynamical triangulation Causal sets DGP model

Unification	Kaluza–Klein theory Dilaton Supergravity

Unification and quantisation	Noncommutative geometry Self-creation cosmology Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string theory Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor string theory

Generalisations / Extensions of GR	Scale relativity Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian theories
and Toy models

Quantum gravity

Central concepts

Toy models

Quantum field theory
in curved spacetime

Black holes

Approaches

String theory	Bosonic string theory M-theory Supergravity Superstring theory

Canonical quantum gravity	Loop quantum gravity Wheeler–DeWitt equation

Euclidean quantum gravity	Hartle–Hawking state

Others	Causal dynamical triangulation Causal sets Noncommutative geometry Spin foam Group field theory Superfluid vacuum theory Twistor theory

Applications

Quantum cosmology

Works by Roger Penrose

Books	The Emperor's New Mind (1989) Shadows of the Mind (1994) The Road to Reality (2004) Cycles of Time (2010)

Coauthored books	The Nature of Space and Time (with Stephen Hawking) (1996) The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright and Stephen Hawking) (1997) White Mars or, The Mind Set Free (with Brian W. Aldiss) (1999)

Academic works	Techniques of Differential Topology in Relativity (1972) Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler) (1987) Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler) (1988)

Topics of twistor theory

Objectives

Principles	Invariance mechanics Background independence

Final objective	Quantum Gravity Theory of Everything

Mathematical concepts

Twistors	Penrose transform Twistor space

Physical concepts

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