In General > s.a. complex
$ Twistor space: The space of pairs (ωA, πA ') 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 \(\mathbb C\)P3, the space of all lines through the origin in \(\mathbb C\)4.
$ Null twistors: The ones that satisfy ωA π*A + ω*A' πA ' = 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 \(\mathbb H\)P1 ≅ S7/SU(2) ≅ S4; In fact, \(\mathbb C\)P3 is a bundle over S4, with fiber \(\mathbb C\)P1 ≅ S2.
* And Minkowski space: To construct PT one can complexify \(\mathbb C\) and then take lines through the origin, or compactify M to S4, and then consider the S2-bundle over S4.
@ 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' ωB = –i εAB πA' .
* 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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