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).

**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]; > s.a. spinning particles.

@ __Gravity__: Brody & Hughston AIP(05)ht [quantum spacetime]; Speziale EPJWC(14)-a1404 [loop quantum gravity, and time]; Herfray a1610 [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-a1311-conf [quantization, and non-commutative spacetime].

**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.

@ __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 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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