Twistors| Twistors |
In General > s.a. complex
structures.
$ 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); 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' 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 JMP-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; 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 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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