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, 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 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.
Penrose's original version of twistor theory is uniquely adapted to four-dimensional Minkowski space, with its signature (3,1) metric (although some of the most profound work in pure mathematics growing out of these ideas has actually occurred in other signatures, and even in other dimensions). 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.
- is the real 6D vector space corresponding to the vector representation of Spin(4,2).
- is the real 5D projective representation corresponding to the equivalence class of nonzero points in under scalar multiplication.
- corresponds to the subspace of corresponding to vectors of zero norm. This is conformally compactified Minkowski space.
- is the 4D complex Weyl spinor representation, called twistor space. It has an invariant Hermitian sesquilinear norm of signature (2,2).
- is a 3D complex manifold corresponding to projective twistor space.
- is the subspace of 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.
- is the subspace of 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).
- is the subspace of of projective twistors with negative norm.
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 and points in respecting the conformal group.
In , it is the case that positive and negative frequency solutions cannot be locally separated. However, this is possible in twistor space.
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) 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.
Witten's twistor string theory is defined on the supertwistor space . Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978. 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.
- Penrose, R. (1967). "Twistor Algebra". Journal of Mathematical Physics. 8 (2): 345. Bibcode:1967JMP.....8..345P. doi:10.1063/1.1705200.
- 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:. Bibcode:2004CMaPh.252..189W. doi:10.1007/s00220-004-1187-3.
- Freidel, Laurent; Speziale, Simone (25 October 2010). "Twistors to twisted geometries". Physical Review D. 82 (8). arXiv:. Bibcode:2010PhRvD..82h4041F. 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.
- Baird, Paul "An Introduction To Twistors"
- Penrose, Roger (1967), "Twistor algebra", Journal of Mathematical Physics, 8 (2): 345–366, Bibcode:1967JMP.....8..345P, doi:10.1063/1.1705200, MR 0216828
- Penrose, Roger (1968), "Twistor quantisation and curved space-time", International Journal of Theoretical Physics, Springer Netherlands, 1: 61–99, Bibcode:1968IJTP....1...61P, doi:10.1007/BF00668831
- Penrose, Roger (1969), "Solutions of the Zero‐Rest‐Mass Equations", Journal of Mathematical Physics, 10 (1): 38–39, Bibcode:1969JMP....10...38P, doi:10.1063/1.1664756
- Penrose, Roger (1977), "The twistor programme", Reports on Mathematical Physics, 12 (1): 65–76, Bibcode:1977RpMP...12...65P, doi:10.1016/0034-4877(77)90047-7, MR 0465032
- Penrose, Roger (1987) "On the Origins of Twistor Theory" in Gravitation and Geometry, a volume in honour of I. Robinson. Naples: Bibliopolis.
- Penrose, Roger (1999) "The Central Programme of Twistor Theory," Chaos, Solitons and Fractals 10: 581-611.
- Arkani-Hamed, Nima; Cachazo, Freddy; Cheung, Clifford; Kaplan, Jared (2009) "The S-Matrix in Twistor Space."
- Witten, Edward (2003), "Perturbative Gauge Theory As A String Theory In Twistor Space", Communications in Mathematical Physics, 252: 189–258, arXiv:, Bibcode:2004CMaPh.252..189W, doi:10.1007/s00220-004-1187-3
- 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