Twistors |

**In General** > s.a. complex structures.

$ __Twistor space__: The space
of pairs (*ω*^{A},
*π*_{A '}) of a spinor
and a complex conjugate spinor; It has 8 (or in some versions 6) dimensions.

$ __Projective twistor space__: The
space PT of equivalence classes of twistors (under multiplication by a non-zero
complex number) is \(\mathbb C\)P^{3}, the space of
all lines through the origin in \(\mathbb C\)^{4}.

$ __Null twistors__:
The ones that satisfy *ω*^{A}
*π**_{A}
+ *ω**^{A'}
*π*_{A'} = 0;
They correspond to null lines in Minkowski space.

* __Graphic representation__: A non-null
twistor can be sketched as a series of nested doughnuts of various sizes travelling
at the speed of light along their shared axis, a Robinson congruence.

* __Relationships__: Notice that each
line in PT induces a line in \(\mathbb H\)^{2},
i.e., an element of \(\mathbb H\)P^{1} ≅
S^{7}/SU(2) ≅ S^{4};
In fact, \(\mathbb C\)P^{3} is a bundle over
S^{4}, with fiber \(\mathbb C\)P^{1}
≅ S^{2}.

* __And Minkowski space__:
To construct PT one can complexify \(\mathbb C\) and then take lines through the
origin, or compactify *M* to S^{4}, and then
consider the S^{2}-bundle over S^{4}.

@ __References__: Penrose JMP(67);
Penrose IJTP(68),
in(81) [curved spacetime];
Woodhouse CQG(85);
Bandyopadhyay & Ghosh IJMPA(89);
Penrose in(99),
GRG(06);
Atiyah et al PRS(17)-a1701 [history].

**And Physics**
> s.a. angular momentum [at null infinity]; locality [relative
locality]; loop quantum gravity and spin-foam models.

* __Idea__: One replaces
Minkowski space *M* by PT, and then translates problems on
*M* to problems on PT; The basic objects here are null lines;
A null geodesic in Minkowski is a null projective twistor, points are
intersections of null lines or 2-spheres of null projective twistors.

* __Motivation__: Twistors
incorporate the concepts of energy, momentum and spin, and this allows
them to work as basic building blocks to describe everything; They also
allow quantum fluctuations to set in at the very basic level of definition
of points; Null lines can fluctuate, causal relations are more basic.

* __Twistor equation__:

∂_{AA'} *ω*^{B}
= −i *ε*_{A}^{B}
*π*_{A'} .

* __Applications__: Maxwell's
equations and some components of the Einstein equation come out very naturally,
and twistors are used to find solutions of Yang-Mills and Einstein's equations.

* __Twistor graphs__: The analog
of Feynman graphs; It seems that they should be always finite; In fact each
one could correspond to (infinitely?) many Feynman diagrams.

@ __Particles__: Fedoruk & Zima ht/02-conf [twistorial superparticle],
JKU(03)ht,
ht/04-proc [spinning];
Bars & Picón PRD(06)ht/05,
PRD(06);
Mezincescu et al JPA(16)-a1508 [massive supersymmetric particle];
García & Guillen JHEP(20)-a2006 [10D massless superparticle];
> s.a. spinning particles.

@ __Gravity__: Brody & Hughston AIP(05)ht [quantum spacetime];
Speziale EPJWC(14)-a1404 [loop quantum gravity, and time];
Herfray JMP(17)-a1610,
Sharma a2104 [twistor action].

@ __Twistor strings__:
Cachazo & Svrček PoS-ht/05;
Musser SA(10)jun;
> s.a. spacetime foam; string theory.

@ __Related topics__:
Penrose CQG(97) [and light rays];
Cederwall PLB(00) [in anti-de Sitter spacetime];
Sinkovics & Verlinde PLB(05) [6D *N* = 4 super-Yang-Mills];
Wolf JPA(10)-a1001-ln [and supersymmetric gauge theories];
Dalhuisen & Bouwmeester JPA(12) [and knotted electromagnetic fields];
Livine et al PRD(12) [twistor networks];
Metzner CQG(13),
CQG(13) [higher-dimensional black holes];
Lukierski & Woronowicz IJMPA(14)-a1311-fs [quantization, and non-commutative spacetime];
Popov a2104 [twistor-space action for Yang-Mills theory].

**General References** > s.a. Hyperkähler Structure.

@ __And mathematics__: Atiyah & Ward CMP(77).

@ __Textbooks and reviews__:
Penrose & MacCallum PRP(73);
Penrose in(75);
Sparling in(75);
Hughston 79;
Madore et al PRP(79) [intro];
Penrose in(80);
Penrose & Ward in(80); Ward in(81);
Huggett IJTP(85);
Ward & Wells 90;
Huggett & Tod 94;
Esposito 95;
Penrose CQG(99)A;
Bars ht/06-ln;
Adamo a1712-ln [intro].

@ __Proceedings, collections__: Huggett ed-94.

@ __Symplectic twistor spaces__:
Vaisman JGP(86).

@ __Twistor-spinors__: Lichnerowicz LMP(89);
Hayashi MPLA(01)ht [spin-3/2].

@ __Twistor conformal field theory__:
Hodges, Penrose & Singer PLB(89).

@ __Generalizations__: Hannabuss LMP(01)ht [non-commutative (Moyal) deformation];
da Rocha & Vaz PoS-mp/04;
Baird & Wehbe CMP(11) [on a finite graph];
Lin & Zheng JGP(17)-a1609 [higher-dimensional].

@ __Related topics__: Low JMP(90) [causal geometry];
Field & Low JGP(98) [linking];
Zunger PRD(00) [on coset spaces];
Frauendiener & Sparling JMP(00) [local twistors and conformal field equations];
Ilyenko JMP(02)ht/01 [representation of null 2-surfaces];
Chamblin CQG(04)ht [and holographic bound];
Arcaute et al mp/06 [and Clifford algebra];
Bloch a1302 [twistor integrals];
> s.a. loop space.

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