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| In [[theoretical physics|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 [[metric signature]] (2,2). This space is called [[twistor space]], and its [[complex number|complex]] valued coordinates are called "twistors."
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| Twistor theory was first proposed by [[Roger Penrose]] in 1967,<ref>Penrose, R. (1967) "Twistor algebra," ''J. Math. Phys.'' 8: 345.</ref> as a possible path to a theory of [[quantum gravity]]. The twistor approach is especially natural for solving the equations of motion of [[quantum field theory|massless fields]] of arbitrary [[spin (physics)|spin]].
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| In 2003, [[Edward Witten]]<ref name="Witten2004">Witten, E. (2004) "[http://arxiv.org/abs/hep-th/0312171 Perturbative gauge theory as a string theory in twistor space,]" ''Commun. Math. Phys.'' 252: 189-258.</ref> proposed uniting twistor and [[string theory]] by embedding the [[topological string theory|topological B model]] of string theory in twistor space. His objective was to model certain [[Yang-Mills]] [[S matrix|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]].<ref>http://arxiv.org/abs/1006.0199</ref>
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| ==Details==
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| Twistor theory is unique to 4D Minkowski space and the (2,2) metric signature, and does not generalize to other dimensions or metric signatures. At the heart of twistor theory lies the [[isomorphism]] between the [[conformal group]] [[spin group|Spin(4,2)]] and SU(2,2), which is the group of [[unitary transformation]]s of [[determinant]] 1 over a four dimensional [[complex vector space]]. These transformations leave invariant a Hermitian [[normal matrix|norm]] of signature (2,2).
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| * <math>\mathbb{R}^6</math> is the real 6D [[vector space]] corresponding to the vector representation of Spin(4,2).
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| * <math>\mathbf{R}\mathbb{P}^5</math> is the real 5D [[projective representation]] corresponding to the [[equivalence class]] of nonzero points in <math>\mathbb{R}^6</math> under scalar multiplication.
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| * <math>\mathbb{M}^c</math> corresponds to the subspace of <math>\mathbf{R}\mathbb{P}^5</math> corresponding to vectors of zero norm. This is conformally compactified [[Minkowski space]].
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| * <math>\mathbb{T}</math> is the 4D complex Weyl [[spinor]] representation, called [[twistor space]]. It has an invariant Hermitian sesquilinear norm of signature (2,2).
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| * <math>\mathbb{PT}</math> is a 3D complex manifold corresponding to projective twistor space.
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| * <math>\mathbb{PT}^+</math> is the subspace of <math>\mathbb{PT}</math> 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]].
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| * <math>\mathbb{PN}</math> is the subspace of <math>\mathbb{PT}</math> 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).
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| * <math>\mathbb{PT}^-</math> is the subspace of <math>\mathbb{PT}</math> of projective twistors with negative norm.
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| <math>\mathbb{M}^c</math>, <math>\mathbb{PT}^+</math>, <math>\mathbb{PN}</math> and <math>\mathbb{PT}^-</math> are all [[homogeneous spaces]] of the [[conformal group]].
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| <math>\mathbb{M}^c</math> admits a conformal metric (i.e., an equivalence class of metric tensors under [[Weyl rescaling]]s) with signature (+++−). Straight null rays map to straight null rays under
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| a [[conformal transformation]] and there is a unique canonical [[isomorphism]] between null rays in <math>\mathbb{M}^c</math> and points in <math>\mathbb{PN}</math> respecting the conformal group.
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| In <math>\mathbb{M}^c</math>, it is the case that positive and negative frequency solutions cannot be locally separated. However, this is possible in twistor space. | |
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| <math>\mathbb{PT}^+ \simeq \mathrm{SU}(2,2)/\left[ \mathrm{SU}(2,1) \times \mathrm{U}(1) \right]</math>
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| ==Twistor string theory==
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| {{Main|Twistor string theory}}
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| 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.
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| [[Edward Witten|Witten]] (2003) proposed a connection between [[string theory]] and twistor geometry, called ''twistor string theory''. Witten (2004)<ref name="Witten2004"/> 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 space]]time. 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 amplitude]]s from [[Feynman diagram]]s by using a geometric structure called an [[amplituhedron]].
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| ==Supertwistors==
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| Witten's twistor string theory is defined on the supertwistor space <math>\mathbb{CP}^{3|4}</math>. Supertwistors are a [[supersymmetry|supersymmetric]] extension of twistors introduced by [[Alan Ferber]] in 1978.<ref>{{Citation|bibcode=1978NuPhB.132...55F|doi = 10.1016/0550-3213(78)90257-2|title=Supertwistors and conformal supersymmetry|year=1978|last1=Ferber|first1=A|journal=Nuclear Physics B|volume=132|pages=55–64|postscript=. }}</ref> Along with the standard twistor degrees of freedom, a supertwistor contains N [[fermion]]ic [[Scalar (physics)|scalars]], where N is the [[extended supersymmetry|number of supersymmetries]]. The [[superconformal algebra]] can be realized on supertwistor space.
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| ==See also==
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| *[[Penrose transform]]
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| *[[Twistor space]]
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| *[[Invariance mechanics]]
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| * Baird, Paul "[http://people.math.jussieu.fr/~helein/encyclopaedia/baird-twistors.pdf An Introduction To Twistors]"
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| *{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor algebra | url=http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1 | doi=10.1063/1.1705200 | mr=0216828 | year=1967 | journal=[[Journal of Mathematical Physics]] | volume=8 | pages=345–366|bibcode = 1967JMP.....8..345P | issue=2 }}
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| *{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor quantisation and curved space-time | publisher=Springer Netherlands | doi=10.1007/BF00668831 | year=1968 | journal=International Journal of Theoretical Physics | volume=1 | pages=61–99|bibcode = 1968IJTP....1...61P }}
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| *{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Solutions of the Zero‐Rest‐Mass Equations | url=http://link.aip.org/link/JMAPAQ/v10/i1/p38/s1 | doi=10.1063/1.1664756 | year=1969 | journal=[[Journal of Mathematical Physics]] | volume=10 | issue=1 | pages=38–39|bibcode = 1969JMP....10...38P }}
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| *{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=The twistor programme | doi=10.1016/0034-4877(77)90047-7 | mr=0465032 | year=1977 | journal=Reports on Mathematical Physics | volume=12 | issue=1 | pages=65–76|bibcode = 1977RpMP...12...65P }}
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| *Penrose, Roger (1987) "[http://users.ox.ac.uk/~tweb/00001/index.shtml On the Origins of Twistor Theory]" in ''Gravitation and Geometry'', a volume in honour of I. Robinson. Naples: Bibliopolis.
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| *Penrose, Roger (1999) "[http://users.ox.ac.uk/~tweb/00002/index.shtml The Central Programme of Twistor Theory,]" ''Chaos, Solitons and Fractals'' 10: 581-611.
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| *[[Nima Arkani-Hamed|Arkani-Hamed, Nima]]; Cachazo, Freddy; Cheung, Clifford; Kaplan, Jared (2009) "[http://arxiv.org/abs/0903.2110 The S-Matrix in Twistor Space.]"
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| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Perturbative Gauge Theory As A String Theory In Twistor Space | arxiv=hep-th/0312171 | doi=10.1007/s00220-004-1187-3 | year=2003 |bibcode = 2004CMaPh.252..189W | journal=Communications in Mathematical Physics | volume=252 | pages=189–258 }}
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| ==External links==
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| *[[Roger Penrose|Penrose, Roger]] (1999) "[http://online.itp.ucsb.edu/online/gravity99/penrose/ Einstein's Equation and Twistor Theory: Recent Developments]"
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| * Penrose, Roger; Hadrovich, Fedja. "[http://users.ox.ac.uk/~tweb/00006/index.shtml Twistor Theory.]"
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| * Dunajski, Maciej, "[http://arxiv.org/abs/0902.0274v2 Twistor Theory and Differential Equations.]"
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| * Hadrovich, Fedja, "[http://users.ox.ac.uk/~tweb/00004/ Twistor primer.]"
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| * [[Andrew Hodges]], "[http://www.twistordiagrams.org.uk/general/index.html Twistor Theory and the Twistor Programme.]" Includes many links.
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| * Huggett, Stephen (2005) "[http://people.maths.ox.ac.uk/lmason/Tws/Huggett.pdf The Elements of Twistor Theory.]"
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| * [[Richard Jozsa]] (1976) "[http://users.ox.ac.uk/~tweb/00003/index.shtml Applications of Sheaf Cohomology in Twistor Theory.]"
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| * Mason, L. J., "[http://www.theorie.physik.uni-muenchen.de/Twistor/twistor_files/Talks/Mason.pdf The twistor programme and twistor strings:From twistor strings to quantum gravity?]"
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| * Sämann, Christian (2006) "[http://arxiv.org/abs/hep-th/0603098v1 Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory.]"
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| * Sparling, George (1999) "[http://www.mat.univie.ac.at/~esiprpr/esi731.pdf On Time Asymmetry.]"
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| * Spradlin, Marcus (2006), "[http://conservancy.umn.edu/bitstream/130081/1/spradlin.pdf Progress And Prospects In Twistor String Theory.]"
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| * [http://mathworld.wolfram.com/Twistor.html MathWorld - Twistors.]
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| * Universe Review "[http://universe-review.ca/R15-19-twistor.htm Twistor Theory.]"
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| *[http://people.maths.ox.ac.uk/lmason/Tn/ Twistor newsletter] archives
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| {{Theories of gravitation}}
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| {{quantum gravity}}
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| {{Roger Penrose}}
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| {{Topics of twistor theory}}
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| [[Category:Clifford algebras]]
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| [[Category:Quantum field theory]]
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| [[Category:Theories of gravitation]]
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