View source for Twistor theory

{{Short description|Possible path to quantum gravity proposed by Roger Penrose}}
In [[theoretical physics]], '''twistor theory''' was proposed by [[Roger Penrose]] in 1967<ref name ="Pen1">{{cite journal|last1=Penrose|first1=R.|date=1967|title=Twistor Algebra|journal=[[Journal of Mathematical Physics]]|volume=8|issue=2|pages=345–366|bibcode=1967JMP.....8..345P|doi=10.1063/1.1705200}}</ref> as a possible path<ref name ="PenMac">{{Cite journal|last1=Penrose|first1=R.|last2=MacCallum|first2=M.A.H.|title=Twistor theory: An approach to the quantisation of fields and space-time|journal=Physics Reports|volume=6|issue=4|pages=241–315|doi=10.1016/0370-1573(73)90008-2|year=1973|bibcode=1973PhR.....6..241P}}</ref> to [[quantum gravity]] and has evolved into a widely studied branch of [[Theoretical physics|theoretical]] and [[mathematical physics]]. Penrose's idea was that [[twistor space]] should be the basic arena for physics from which [[space-time]] itself should emerge. It has led to powerful mathematical tools that have applications to [[Differential geometry|differential]] and [[integral geometry]], [[nonlinear differential equation]]s and [[representation theory]], and in physics to [[general relativity]], [[quantum field theory]], and the theory of [[scattering amplitude]]s.

Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of [[general relativity]] in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, [[Roger Penrose]] has credited [[Ivor Robinson (physicist)|Ivor Robinson]] as an important early influence in the development of twistor theory, through his construction of so-called ''Robinson congruences''.<ref>{{cite book |first=Roger |last=Penrose |chapter=On the Origins of Twistor Theory |title=Gravitation and Geometry, a Volume in Honour of Ivor Robinson |editor-first=Wolfgang |editor-last=Rindler |editor2-first=Andrzej |editor2-last=Trautman |publisher=Bibliopolis |year=1987 |isbn=88-7088-142-3 }}</ref>

==Overview<!--'Ambitwistor' and 'Super-ambitwistor' redirect here-->==
[[Projective space|Projective]] [[twistor space]] <math>\mathbb{PT}</math> is [[Projective space|projective 3-space]] <math>\mathbb{CP}^3</math>, the simplest [[3-dimensional]] [[complex manifold|compact algebraic variety]]. It has a physical interpretation as the space of [[massless particle]]s with [[Spin (physics)|spin]]. It is the [[projectivisation]] of a 4-dimensional [[complex vector space]], non-projective twistor space <math>\mathbb{T}</math>, with a [[Hermitian form]] of [[Metric signature|signature]] (2,&nbsp;2) and a [[holomorphic]] [[volume form]]. This can be most naturally understood as the space of [[Chirality (physics)|chiral]] ([[Weyl spinor|Weyl]]) [[spinor]]s for the [[conformal group]] <math>SO(4,2)/\mathbb{Z}_2</math> of [[Minkowski space]]; it is the [[fundamental representation]] of the [[spin group]] <math>SU(2,2)</math> of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective [[pure spinor]]s<ref name="HS1">{{cite journal | last1=Harnad | first1=J. | last2=Shnider | first2=S. | title=Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions | journal=Journal of Mathematical Physics | volume=33 | issue=9 | year=1992 | doi=10.1063/1.529538 | pages=3197–3208 | bibcode=1992JMP....33.3197H }}</ref><ref name="HS2">{{cite journal | last1=Harnad | first1=J. | last2=Shnider | first2=S. | title=Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces | journal=Journal of Mathematical Physics | volume=36 | issue=9 | year=1995 | doi=10.1063/1.531096 | pages=1945–1970 | doi-access=free | bibcode=1995JMP....36.1945H }} </ref> for the conformal group.<ref name="PenRin">{{Cite book |title=Spinors and Space-Time |last1=Penrose |first1=Roger |last2=Rindler |first2=Wolfgang |publisher=Cambridge University Press |year=1986 |isbn=9780521252676 |pages=Appendix |language=en |doi=10.1017/cbo9780511524486}}</ref><ref>{{Cite journal |last1=Hughston |first1=L. P. |last2=Mason |first2=L. J. |date=1988 |title=A generalised Kerr-Robinson theorem |journal=Classical and Quantum Gravity |language=en |volume=5 |issue=2 |pages=275 |doi=10.1088/0264-9381/5/2/007 |issn=0264-9381 |bibcode=1988CQGra...5..275H |s2cid=250783071 }}</ref>

In its original form, twistor theory encodes [[physical field]]s on Minkowski space in terms of [[complex analytic]] objects on twistor space via the [[Penrose transform]]. This is especially natural for [[Massless particle|massless fields]] of arbitrary [[Spin (physics)|spin]]. In the first instance these are obtained via [[contour integral]] formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations can be more deeply understood as [[Čech cohomology|Čech]] representatives of analytic [[cohomology classes]] on regions in <math>\mathbb{PT}</math>. These correspondences have been extended to certain nonlinear fields, including [[self-dual]] gravity in Penrose's [[Einstein field equations#Nonlinearity|nonlinear]] [[graviton]] construction<ref name="Penrose1976">{{cite journal | last1 = Penrose | first1 = R. | year = 1976 | title = Non-linear gravitons and curved twistor theory | url = | journal = Gen. Rel. Grav. | volume = 7 | issue = 1 | pages = 31–52 | doi = 10.1007/BF00762011 | bibcode = 1976GReGr...7...31P | s2cid = 123258136 }}</ref> and self-dual [[Yang–Mills field]]s in the so-called Ward construction;<ref>{{Cite journal |last=Ward |first=R. S. |author-link=Richard S. Ward |title=On self-dual gauge fields |journal=Physics Letters A |volume=61 |issue=2 |pages=81–82 |doi=10.1016/0375-9601(77)90842-8 |year=1977 |bibcode=1977PhLA...61...81W}}</ref> the former gives rise to [[Deformation (mathematics)|deformations]] of the underlying complex structure of regions in <math>\mathbb{PT}</math>, and the latter to certain holomorphic vector bundles over regions in <math>\mathbb{PT}</math>. These constructions have had wide applications, including inter alia the theory of [[integrable system]]s.<ref>{{Cite book |title=Twistor geometry and field theory |last=Ward |first=R. S. |date=1990 |publisher=Cambridge University Press |others=Wells, R. O. |isbn=978-0521422680 |location=Cambridge [England] |oclc=17260289}}</ref><ref>{{Cite book |title=Integrability, self-duality, and twistor theory |last1=Mason |first1=Lionel J. |last2=Woodhouse |first2=Nicholas M. J. |date=1996 |publisher=Clarendon Press |isbn=9780198534983 |location=Oxford |oclc=34545252}}</ref><ref>{{Cite book |title=Solitons, instantons, and twistors |last=Dunajski |first=Maciej |date=2010 |publisher=Oxford University Press |isbn=9780198570622 |location=Oxford |oclc=507435856}}</ref>

The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for [[Yang–Mills–Higgs equations|Yang–Mills–Higgs]] [[Magnetic monopole|monopoles]] and [[instanton]]s (see [[ADHM construction]]).<ref>{{Cite journal |last1=Atiyah |first1=M. F. |last2=Hitchin |first2=N. J. |last3=Drinfeld |first3=V. G. |last4=Manin |first4=Yu. I. |title=Construction of instantons |journal=Physics Letters A |volume=65 |issue=3 |pages=185–187 |doi=10.1016/0375-9601(78)90141-x |year=1978 |bibcode=1978PhLA...65..185A}}</ref> An early attempt to overcome this restriction was the introduction of '''ambitwistors'''<!--boldface per WP:R#PLA--> by Isenberg, Yasskin and Green,<ref name="IYG">{{Cite journal| last1=Isenberg |first1=James |last2=Yasskin |first2=Philip B. |last3=Green |first3=Paul S. |title=Non-self-dual gauge fields |journal=Physics Letters B |volume=78 |issue=4 |pages=462–464 |doi=10.1016/0370-2693(78)90486-0 |year=1978 |bibcode=1978PhLB...78..462I}}</ref> and their [[superspace]] extension, '''super-ambitwistors'''<!--boldface per WP:R#PLA-->, by [[Edward Witten]].<ref name = "Wi1">{{Cite journal |last=Witten |first=Edward |title=An interpretation of classical Yang–Mills theory |journal=Physics Letters B |volume=77 |issue=4–5 |pages=394–398 |doi=10.1016/0370-2693(78)90585-3 |year=1978 |bibcode=1978PhLB...77..394W}}</ref> Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as a [[complexification]] or [[cotangent bundle]] of the original twistor description. By extending the ambitwistor correspondence to suitably defined formal neighborhoods, Isenberg, Yasskin and Green<ref name="IYG"/> showed the equivalence between the vanishing of the curvature along such extended null lines and the full Yang–Mills field equations.<ref name="IYG"/> Witten<ref name="Wi1"/> showed that a further extension, within the framework of super Yang–Mills theory, including [[Fermionic field|fermionic]] and scalar fields, gave rise, in the case of ''N''&nbsp;=&nbsp;1 or 2 [[supersymmetry]], to the constraint equations, while for ''N''&nbsp;=&nbsp;3 (or 4), the vanishing condition for supercurvature along super null lines (super ambitwistors) implied the full set of [[N = 4 supersymmetric Yang–Mills theory|field equations]], including those for the fermionic fields. This was subsequently shown to give a {{clarify span|1-1|date=May 2024}} equivalence between the null curvature constraint equations and the supersymmetric Yang-Mills field equations.<ref name = "HLHS">{{Cite journal | last1=Harnad | first1=J. | last2=Légaré | first2=M. | last3=Hurtubise | first3=J. | last4=Shnider | first4=S. | title=Constraint equations and field equations in supersymmetric N = 3 Yang-Mills theory | journal=Nuclear Physics B | volume=256 | pages=609–620 | doi=10.1016/0550-3213(85)90410-9 | year=1985 | bibcode=1985NuPhB.256..609H }}</ref><ref name = "HHS">{{Cite journal | last1=Harnad | first1=J. | last2=Hurtubise | first2=J. | last3=Shnider | first3=S. | title=Supersymmetric Yang-Mills equations and supertwistors | journal=Annals of Physics | volume=193 | issue=1 | pages=40–79 | doi=10.1016/0003-4916(89)90351-5 | year=1989 | bibcode=1989AnPhy.193...40H }}</ref> Through dimensional reduction, it may also be deduced from the analogous super-ambitwistor correspondence for 10-dimensional, ''N''&nbsp;=&nbsp;1 super-Yang–Mills theory.<ref name=W2>{{cite journal | first1=E. | last1=Witten | title=Twistor-like transform in ten dimensions | journal=Nuclear Physics | volume=B266 | pages=245–264 | year=1986 | issue=2 | doi=10.1016/0550-3213(86)90090-8 | bibcode=1986NuPhB.266..245W }}</ref><ref name=HS>{{cite journal | first1=J. | last1=Harnad | first2=S. | last2=Shnider | title=Constraints and Field Equations for Ten Dimensional Super Yang-Mills Theory | journal=Commun. Math. Phys. | volume=106 | pages=183–199 | year=1986 | issue=2 | doi=10.1007/BF01454971 | bibcode=1986CMaPh.106..183H | s2cid=122622189 | url=http://projecteuclid.org/euclid.cmp/1104115696 }}</ref>

Twistorial formulae for [[Fundamental interaction|interactions]] beyond the self-dual sector also arose in Witten's [[twistor string theory]],<ref name="Witten2004">{{cite journal |last1=Witten |first1=Edward |date=2004 |title=Perturbative Gauge Theory as a String Theory in Twistor Space |journal=Communications in Mathematical Physics |volume=252 |issue=1–3 |pages=189–258 |arxiv=hep-th/0312171 |bibcode=2004CMaPh.252..189W |doi=10.1007/s00220-004-1187-3 |s2cid=14300396}}</ref> which is a quantum theory of holomorphic maps of a [[Riemann surface]] into twistor space. This gave rise to the remarkably compact RSV (Roiban, Spradlin and Volovich) formulae for tree-level [[S-matrices]] of Yang–Mills theories,<ref>{{Cite journal |last1=Roiban |first1=Radu |last2=Spradlin |first2=Marcus |last3=Volovich |first3=Anastasia |date=2004-07-30 |title=Tree-level S matrix of Yang–Mills theory |journal=Physical Review D |volume=70 |issue=2 |pages=026009 |doi=10.1103/PhysRevD.70.026009 |bibcode=2004PhRvD..70b6009R |arxiv=hep-th/0403190 |s2cid=10561912}}</ref> but its gravity degrees of freedom gave rise to a version of conformal [[supergravity]] limiting its applicability; [[conformal gravity]] is an unphysical theory containing [[Ghost (physics)|ghosts]], but its interactions are combined with those of Yang–Mills theory in loop amplitudes calculated via twistor string theory.<ref>{{Cite journal |last1=Berkovits |first1=Nathan |last2=Witten |first2=Edward |date=2004 |title=Conformal supergravity in twistor-string theory |journal=Journal of High Energy Physics |language=en |volume=2004 |issue=8 |pages=009 |doi=10.1088/1126-6708/2004/08/009 |issn=1126-6708 |bibcode=2004JHEP...08..009B |arxiv=hep-th/0406051 |s2cid=119073647}}</ref>

Despite its shortcomings, twistor string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism<ref>{{Cite journal |last1=Cachazo |first1=Freddy |last2=Svrcek |first2=Peter |last3=Witten |first3=Edward |date=2004 |title=MHV vertices and tree amplitudes in gauge theory |journal=Journal of High Energy Physics |language=en |volume=2004 |issue=9 |pages=006 |doi=10.1088/1126-6708/2004/09/006 |issn=1126-6708 |bibcode=2004JHEP...09..006C |arxiv=hep-th/0403047 |s2cid=16328643}}</ref> loosely based on disconnected strings, but was given a more basic foundation in terms of a twistor action for full Yang–Mills theory in twistor space.<ref>{{Cite journal |last1=Adamo |first1=Tim |last2=Bullimore |first2=Mathew |last3=Mason |first3=Lionel |last4=Skinner |first4=David |title=Scattering amplitudes and Wilson loops in twistor space |journal=Journal of Physics A: Mathematical and Theoretical |volume=44 |issue=45 |pages=454008 |doi=10.1088/1751-8113/44/45/454008 |year=2011 |bibcode=2011JPhA...44S4008A |arxiv=1104.2890 |s2cid=59150535}}</ref> Another key development was the introduction of [[BCFW recursion]].<ref>{{Cite journal |last1=Britto |first1=Ruth |author1-link= Ruth Britto |last2=Cachazo |first2=Freddy |last3=Feng |first3=Bo |last4=Witten |first4=Edward |date=2005-05-10 |title=Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang–Mills Theory |journal=Physical Review Letters |volume=94 |issue=18 |pages=181602 |doi=10.1103/PhysRevLett.94.181602 |pmid=15904356 |bibcode=2005PhRvL..94r1602B |arxiv=hep-th/0501052 |s2cid=10180346}}</ref> This has a natural formulation in twistor space<ref>{{Cite journal |last1=Mason |first1=Lionel |last2=Skinner |first2=David |date=2010-01-01 |title=Scattering amplitudes and BCFW recursion in twistor space |journal=Journal of High Energy Physics |language=en |volume=2010 |issue=1 |pages=64 |doi=10.1007/JHEP01(2010)064 |issn=1029-8479 |bibcode=2010JHEP...01..064M |arxiv=0903.2083 |s2cid=8543696}}</ref><ref>{{Cite journal |last1=Arkani-Hamed |first1=N. |last2=Cachazo |first2=F. |last3=Cheung |first3=C. |last4=Kaplan |first4=J. |date=2010-03-01 |title=The S-matrix in twistor space |journal=Journal of High Energy Physics |language=en |volume=2010 |issue=3 |pages=110 |doi=10.1007/JHEP03(2010)110 |issn=1029-8479 |bibcode=2010JHEP...03..110A |arxiv=0903.2110 |s2cid=15898218}}</ref> that in turn led to remarkable formulations of scattering amplitudes in terms of [[Grassmann integral]] formulae<ref>{{Cite journal |last1=Arkani-Hamed |first1=N. |last2=Cachazo |first2=F. |last3=Cheung |first3=C. |last4=Kaplan |first4=J. |date=2010-03-01 |title=A duality for the S matrix |journal=Journal of High Energy Physics |language=en |volume=2010 |issue=3 |pages=20 |doi=10.1007/JHEP03(2010)020 |issn=1029-8479 |bibcode=2010JHEP...03..020A |arxiv=0907.5418 |s2cid=5771375}}</ref><ref>{{Cite journal |last1=Mason |first1=Lionel |last2=Skinner |first2=David |date=2009 |title=Dual superconformal invariance, momentum twistors and Grassmannians |journal=Journal of High Energy Physics |language=en |volume=2009 |issue=11 |pages=045 |doi=10.1088/1126-6708/2009/11/045 |issn=1126-6708 |bibcode=2009JHEP...11..045M |arxiv=0909.0250 |s2cid=8375814}}</ref> and [[polytope]]s.<ref>{{Cite journal |last=Hodges |first=Andrew |date=2013-05-01 |title=Eliminating spurious poles from gauge-theoretic amplitudes |journal=Journal of High Energy Physics |language=en |volume=2013 |issue=5 |pages=135 |doi=10.1007/JHEP05(2013)135 |issn=1029-8479 |bibcode=2013JHEP...05..135H |arxiv=0905.1473 |s2cid=18360641}}</ref> These ideas have evolved more recently into the positive [[Grassmannian]]<ref>{{cite arXiv |last1=Arkani-Hamed |first1=Nima |last2=Bourjaily |first2=Jacob L. |last3=Cachazo |first3=Freddy |last4=Goncharov |first4=Alexander B. |last5=Postnikov |first5=Alexander |last6=Trnka |first6=Jaroslav |date=2012-12-21 |title=Scattering Amplitudes and the Positive Grassmannian |eprint=1212.5605 |class=hep-th}}</ref> and [[amplituhedron]].

Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlying [[string theory]]. The extension to gravity was given by Cachazo & Skinner,<ref>{{Cite journal |last1=Cachazo |first1=Freddy |last2=Skinner |first2=David |date=2013-04-16 |title=Gravity from Rational Curves in Twistor Space |journal=Physical Review Letters |volume=110 |issue=16 |pages=161301 |doi=10.1103/PhysRevLett.110.161301 |pmid=23679592 |bibcode=2013PhRvL.110p1301C |arxiv=1207.0741 |s2cid=7452729}}</ref> and formulated as a twistor string theory for [[maximal supergravity]] by David Skinner.<ref>{{cite arXiv |last=Skinner |first=David |date=2013-01-04 |title=Twistor Strings for ''N''&nbsp;=&nbsp;8 Supergravity |eprint=1301.0868 |class=hep-th}}</ref> Analogous formulae were then found in all dimensions by Cachazo, He and Yuan for Yang–Mills theory and gravity<ref>{{Cite journal |last1=Cachazo |first1=Freddy |last2=He |first2=Song |last3=Yuan |first3=Ellis Ye |date=2014-07-01 |title=Scattering of massless particles: scalars, gluons and gravitons |journal=Journal of High Energy Physics |language=en |volume=2014 |issue=7 |pages=33 |doi=10.1007/JHEP07(2014)033 |issn=1029-8479 |bibcode=2014JHEP...07..033C |arxiv=1309.0885 |s2cid=53685436}}</ref> and subsequently for a variety of other theories.<ref>{{Cite journal |last1=Cachazo |first1=Freddy |last2=He |first2=Song |last3=Yuan |first3=Ellis Ye |date=2015-07-01 |title=Scattering equations and matrices: from Einstein to Yang–Mills, DBI and NLSM |journal=Journal of High Energy Physics |language=en |volume=2015 |issue=7 |pages=149 |doi=10.1007/JHEP07(2015)149 |issn=1029-8479 |bibcode=2015JHEP...07..149C |arxiv=1412.3479 |s2cid=54062406}}</ref> They were then understood as string theories in ambitwistor space by Mason and Skinner<ref>{{Cite journal |last1=Mason |first1=Lionel |last2=Skinner |first2=David |date=2014-07-01 |title=Ambitwistor strings and the scattering equations |journal=Journal of High Energy Physics |language=en |volume=2014 |issue=7 |pages=48 |doi=10.1007/JHEP07(2014)048 |issn=1029-8479 |bibcode=2014JHEP...07..048M |arxiv=1311.2564 |s2cid=53666173}}</ref> in a general framework that includes the original twistor string and extends to give a number of new models and formulae.<ref>{{Cite journal |last=Berkovits |first=Nathan |date=2014-03-01 |title=Infinite tension limit of the pure spinor superstring |journal=Journal of High Energy Physics |language=en |volume=2014 |issue=3 |pages=17 |doi=10.1007/JHEP03(2014)017 |issn=1029-8479 |bibcode=2014JHEP...03..017B |arxiv=1311.4156 |s2cid=28346354}}</ref><ref>{{Cite journal |last1=Geyer |first1=Yvonne |last2=Lipstein |first2=Arthur E. |last3=Mason |first3=Lionel |date=2014-08-19 |title=Ambitwistor Strings in Four Dimensions |journal=Physical Review Letters |volume=113 |issue=8 |pages=081602 |doi=10.1103/PhysRevLett.113.081602 |pmid=25192087 |bibcode=2014PhRvL.113h1602G |arxiv=1404.6219 |s2cid=40855791}}</ref><ref>{{Cite journal |last1=Casali |first1=Eduardo |last2=Geyer |first2=Yvonne |last3=Mason |first3=Lionel |last4=Monteiro |first4=Ricardo |last5=Roehrig |first5=Kai A. |date=2015-11-01 |title=New ambitwistor string theories |journal=Journal of High Energy Physics |language=en |volume=2015 |issue=11 |pages=38 |doi=10.1007/JHEP11(2015)038 |issn=1029-8479 |bibcode=2015JHEP...11..038C |arxiv=1506.08771 |s2cid=118801547}}</ref> As string theories they have the same [[critical dimension]]s as conventional string theory; for example the [[Type II string theory|type&nbsp;II]] supersymmetric versions are critical in ten dimensions and are equivalent to the full field theory of type&nbsp;II supergravities in ten dimensions (this is distinct from conventional string theories that also have a further infinite hierarchy of massive higher spin states that provide an [[ultraviolet completion]]). They extend to give formulae for loop amplitudes<ref>{{Cite journal |last1=Adamo |first1=Tim |last2=Casali |first2=Eduardo |last3=Skinner |first3=David |date=2014-04-01 |title=Ambitwistor strings and the scattering equations at one loop |journal=Journal of High Energy Physics |language=en |volume=2014 |issue=4 |pages=104 |doi=10.1007/JHEP04(2014)104 |issn=1029-8479 |bibcode=2014JHEP...04..104A |arxiv=1312.3828 |s2cid=119194796}}</ref><ref>{{Cite journal |last1=Geyer |first1=Yvonne |last2=Mason |first2=Lionel |last3=Monteiro |first3=Ricardo |last4=Tourkine |first4=Piotr |date=2015-09-16 |title=Loop Integrands for Scattering Amplitudes from the Riemann Sphere |journal=Physical Review Letters |volume=115 |issue=12 |pages=121603 |doi=10.1103/PhysRevLett.115.121603 |pmid=26430983 |bibcode=2015PhRvL.115l1603G |arxiv=1507.00321 |s2cid=36625491}}</ref> and can be defined on curved backgrounds.<ref>{{Cite journal |last1=Adamo |first1=Tim |last2=Casali |first2=Eduardo |last3=Skinner |first3=David |date=2015-02-01 |title=A worldsheet theory for supergravity |journal=Journal of High Energy Physics |language=en |volume=2015 |issue=2 |pages=116 |doi=10.1007/JHEP02(2015)116 |issn=1029-8479 |bibcode=2015JHEP...02..116A |arxiv=1409.5656 |s2cid=119234027}}</ref>

==The twistor correspondence==
Denote [[Minkowski space]] by <math>M</math>, with coordinates <math>x^a = (t, x, y, z)</math> and Lorentzian metric <math>\eta_{ab}</math> signature <math>(1, 3)</math>. Introduce 2-component spinor indices <math>A = 0, 1;\; A' = 0', 1',</math> and set

:<math>x^{AA'} = \frac{1}{\sqrt{2}}\begin{pmatrix} t - z & x + iy \\ x - iy & t + z \end{pmatrix}.</math>

Non-projective twistor space <math>\mathbb{T}</math> is a four-dimensional complex vector space with coordinates denoted by <math>Z^{\alpha} = \left(\omega^{A},\, \pi_{A'}\right)</math> where <math>\omega^A</math> and <math>\pi_{A'}</math> are two constant [[Weyl spinor]]s. The hermitian form can be expressed by defining a complex conjugation from <math>\mathbb{T}</math> to its dual <math>\mathbb{T}^*</math> by <math>\bar Z_\alpha = \left(\bar\pi_A,\, \bar \omega^{A'}\right)</math> so that the Hermitian form can be expressed as

:<math>Z^\alpha \bar Z_\alpha = \omega^{A}\bar\pi_{A} + \bar\omega^{A'}\pi_{A'}.</math>

This together with the holomorphic volume form, <math>\varepsilon_{\alpha\beta\gamma\delta} Z^\alpha dZ^\beta \wedge dZ^\gamma \wedge dZ^\delta</math> is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation

:<math>\omega^{A} = ix^{AA'}\pi_{A'}.</math>

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space <math>\mathbb{PT},</math> which is isomorphic as a complex manifold to <math>\mathbb{CP}^3</math>. A point <math>x\in M</math> thereby determines a line <math>\mathbb{CP}^1</math> in <math>\mathbb{PT}</math> parametrised by <math>\pi_{A'}.</math> A twistor <math>Z^\alpha</math> is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take <math>x</math> to be real, then if <math>Z^\alpha \bar Z_\alpha</math> vanishes, then <math>x</math> lies on a light ray, whereas if <math>Z^\alpha \bar Z_\alpha</math> is non-vanishing, there are no solutions, and indeed then <math>Z^{\alpha}</math> corresponds to a massless particle with spin that are not localised in real space-time.

==Variations<!--'History of twistor theory' redirects here-->==
===Supertwistors===

Supertwistors are a [[supersymmetry|supersymmetric]] extension of twistors introduced by Alan Ferber in 1978.<ref name="Fer">{{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|issue = 1|pages=55–64|postscript=. }}</ref> Non-projective twistor space is extended by [[fermion]]ic coordinates where <math>\mathcal{N}</math> is the [[extended supersymmetry|number of supersymmetries]] so that a twistor is now given by <math>\left(\omega^A,\, \pi_{A'},\, \eta^i\right), i = 1, \ldots, \mathcal{N}</math> with <math>\eta^i</math> anticommuting. The super conformal group <math>SU(2,2|\mathcal{N})</math> naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The <math>\mathcal{N} = 4</math> case provides the target for Penrose's original twistor string and the <math>\mathcal{N} = 8</math> case is that for Skinner's supergravity generalisation.

=== Higher dimensional generalization of the Klein correspondence ===

A higher dimensional generalization of the [[Klein quadric | Klein correspondence]] underlying twistor theory, applicable to isotropic subspaces of conformally compactified (complexified) Minkowski space and its super-space extensions, was developed by [[John Harnad |J. Harnad]] and S. Shnider.<ref name ="HS1"/><ref name = "HS2"/>

=== Hyperkähler manifolds ===
[[Hyperkähler manifold]]s of dimension <math>4k</math> also admit a twistor correspondence with a twistor space of complex dimension <math>2k+1</math>.<ref>{{cite journal|last4=Roček | first4=M. | last3=Lindström | first3=U. | last2=Karlhede | first2=A. | last1=Hitchin | first1=N. J. | title=Hyper-Kähler metrics and supersymmetry | url=https://projecteuclid.org/download/pdf_1/euclid.cmp/1104116624 |mr=877637 | year=1987 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=108 | issue=4 | pages=535–589 | doi=10.1007/BF01214418| bibcode=1987CMaPh.108..535H | s2cid=120041594 }}</ref>

===Palatial twistor theory<!--'Googly problem' and 'Palatial twistor theory' redirect here-->===
The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.<ref name="Penrose1976"/> A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of [[Chirality (physics)|right-handed]] fields. Infinitesimally, these are encoded in twistor functions or [[cohomology]] classes of [[Homogeneous function|homogeneity]] −6. The task of using such twistor functions in a fully nonlinear way so as to obtain a ''[[Helicity (particle physics)|right-handed]]'' nonlinear graviton has been referred to as the ('''gravitational''') '''googly problem'''<!--boldface per WP:R#PLA-->.<ref name="Penrose1000">Penrose 2004, p. 1000.</ref> (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.) The most recent proposal in this direction by Penrose in 2015 was based on [[noncommutative geometry]] on twistor space and referred to as '''palatial twistor theory'''<!--boldface per WP:R#PLA-->.<ref>{{Cite journal|doi=10.1098/rsta.2014.0237|title=Palatial twistor theory and the twistor googly problem|year=2015|last1=Penrose|first1=Roger|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=373|issue=2047|pmid=26124255|s2cid=13038470|page=20140237|bibcode=2015RSPTA.37340237P |doi-access=free}}</ref> The theory is named after [[Buckingham Palace]], where [[Michael Atiyah]]<ref>[https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/ "Michael Atiyah's Imaginative State of Mind"] – ''[[Quanta Magazine]]''</ref> suggested to Penrose the use of a type of "[[noncommutative algebra]]", an important component of the theory. (The underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutative [[holomorphic]] twistor [[Quantum group|quantum algebra]].)

==See also==
* [[Background independence]]
* [[Complex spacetime]]
* [[History of loop quantum gravity]]
* [[Robinson congruences]]
* [[Spin network]]
* [[Twisted geometries]]

==Notes==
{{Reflist|22em}}

==References==
* [[Roger Penrose]] (2004), ''[[The Road to Reality]]'', Alfred A. Knopf, ch. 33, pp.&nbsp;958–1009.
* Roger Penrose and [[Wolfgang Rindler]] (1984), ''Spinors and Space-Time; vol. 1, Two-Spinor Calculus and Relativitic Fields'', Cambridge University Press, Cambridge.
* Roger Penrose and Wolfgang Rindler (1986), ''Spinors and Space-Time; vol. 2, Spinor and Twistor Methods in Space-Time Geometry'', Cambridge University Press, Cambridge.

==Further reading==
* {{cite journal | doi = 10.1098/rspa.2017.0530 | volume=473 | title=Twistor theory at fifty: from contour integrals to twistor strings | year=2017 | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | page=20170530 | last1 = Atiyah | first1 = Michael | last2 = Dunajski | first2 = Maciej | last3 = Mason | first3 = Lionel J.| issue=2206 | pmid=29118667 | pmc=5666237 | arxiv=1704.07464 | bibcode=2017RSPSA.47370530A | s2cid=5735524 | doi-access=free }}
* Baird, P., "[http://people.math.jussieu.fr/~helein/encyclopaedia/baird-twistors.pdf An Introduction to Twistors]."
* Huggett, S. and Tod, K. P. (1994).&nbsp;[https://www.worldcat.org/oclc/831625586 ''An Introduction to Twistor Theory''], second edition. Cambridge University Press.&nbsp;{{ISBN|9780521456890}}.&nbsp;[[OCLC]]&nbsp;[https://www.worldcat.org/oclc/831625586 831625586].
* Hughston, L. P. (1979) ''Twistors and Particles''. Springer Lecture Notes in Physics 97, Springer-Verlag. {{ISBN|978-3-540-09244-5}}. 
* Hughston, L. P. and Ward, R. S., eds (1979) ''Advances in Twistor Theory''. Pitman. {{ISBN|0-273-08448-8}}.
* Mason, L. J. and Hughston, L. P., eds (1990) ''Further Advances in Twistor Theory, Volume I: The Penrose Transform and its Applications''. Pitman Research Notes in Mathematics Series 231, Longman Scientific and Technical. {{ISBN|0-582-00466-7}}.
* Mason, L. J., Hughston, L. P., and Kobak, P. K., eds (1995) ''Further Advances in Twistor Theory, Volume II: Integrable Systems, Conformal Geometry, and Gravitation''. Pitman Research Notes in Mathematics Series 232, Longman Scientific and Technical. {{ISBN|0-582-00465-9}}.
* Mason, L. J., Hughston, L. P., Kobak, P. K., and Pulverer, K., eds (2001) ''Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces''. Research Notes in Mathematics 424, Chapman and Hall/CRC. {{ISBN|1-58488-047-3}}.
* {{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 | url-status=dead | archive-url=https://archive.today/20130112095407/http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1 | archive-date=2013-01-12 }}
* {{Citation | last1=Penrose | first1=Roger | title=Twistor Quantisation and Curved Space-time | doi=10.1007/BF00668831 | year=1968 | journal=International Journal of Theoretical Physics | volume=1 | issue=1 | pages=61–99|bibcode = 1968IJTP....1...61P | s2cid=123628735 }}
* {{Citation | last1=Penrose | first1=Roger | 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 | url-status=dead | archive-url=https://archive.today/20130112125501/http://link.aip.org/link/JMAPAQ/v10/i1/p38/s1 | archive-date=2013-01-12 }}
* {{Citation | last1=Penrose | first1=Roger | 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 }}
* {{cite journal | last1 = Penrose | first1 = Roger | year = 1999 | title = The Central Programme of Twistor Theory | url = http://users.ox.ac.uk/~tweb/00002/index.shtml | journal = Chaos, Solitons and Fractals | volume = 10 | issue = 2–3| pages = 581–611 | doi = 10.1016/S0960-0779(98)00333-6 | bibcode = 1999CSF....10..581P }}
* {{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=2004 |bibcode = 2004CMaPh.252..189W | journal=Communications in Mathematical Physics | volume=252 | issue=1–3 | pages=189–258 | s2cid=14300396 }}

==External links==
* [[Roger Penrose|Penrose, Roger]] (1999), "[http://online.itp.ucsb.edu/online/gravity99/penrose/ Einstein's Equation and Twistor Theory: Recent Developments]"
* Penrose, Roger; Hadrovich, Fedja. "[http://users.ox.ac.uk/~tweb/00006/index.shtml Twistor Theory.]"
* Hadrovich, Fedja, "[http://users.ox.ac.uk/~tweb/00004/ Twistor Primer.]"
* Penrose, Roger. "[http://users.ox.ac.uk/~tweb/00001/index.shtml On the Origins of Twistor Theory.]"
* [[Richard Jozsa|Jozsa, Richard]] (1976), "[http://users.ox.ac.uk/~tweb/00003/index.shtml Applications of Sheaf Cohomology in Twistor Theory.]"
* {{cite journal |last=Dunajski |first=Maciej |title=Twistor Theory and Differential Equations |arxiv=0902.0274 |date=2009 |journal=J. Phys. A: Math. Theor. |volume=42 |issue=40 |page=404004 |doi=10.1088/1751-8113/42/40/404004 |bibcode=2009JPhA...42N4004D |s2cid=62774126 }}
* [[Andrew Hodges]], [https://www.twistordiagrams.org.uk/papers/index.html Summary of recent developments.]
* Huggett, Stephen (2005), "[http://people.maths.ox.ac.uk/lmason/Tws/Huggett.pdf The Elements of Twistor Theory.]"
* 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?]"
* {{cite thesis |type=PhD |last=Sämann |first=Christian |date=2006 |title=Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory |arxiv=hep-th/0603098 |publisher=Universität Hannover}}
* Sparling, George (1999), "[https://www.mat.univie.ac.at/~esiprpr/esi731.pdf On Time Asymmetry.]"
* {{cite web |last=Spradlin |first=Marcus |date=2012 |url=https://conservancy.umn.edu/bitstream/handle/11299/130081/1/spradlin.pdf |title=Progress and Prospects in Twistor String Theory |hdl=11299/130081 }}
* [http://mathworld.wolfram.com/Twistor.html MathWorld: Twistors.]
* Universe Review: "[http://universe-review.ca/R15-19-twistor.htm Twistor Theory.]"
* [http://people.maths.ox.ac.uk/lmason/Tn/ Twistor newsletter] archives.

{{Theories of gravitation}}
{{quantum gravity}}
{{Roger Penrose}}
{{Topics of twistor theory}}
{{Standard model of physics}}

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[[Category:Roger Penrose]]