Twistors| Twistors |
In General [> s.a. complex
structures.]
$ Twistor space: The
space of pairs (
A,
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 nonzero complex number) is CP3,
the space
of all lines through the origin in C4.
$ Null twistors:
The ones that satisfy A
*A + *A' 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 H2,
i.e., an element of HP1 
S7/SU(2)
S4;
In fact,
CP3 is a bundle over S4,
with fiber CP1 S2.
* And Minkowski: To construct PT one can complexify 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].
* 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' B
= –i 
AB 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-in
[twistorial superparticle], JKU(03)ht,
ht/04-in
[spinning]; Bars & Picón PRD(06)ht/05,
PRD(06).
@ Twistor strings: Cachazo & Svrcek ht/05-ln; > s.a. spacetime
foam.
@ Related topics: Penrose CQG(97)
[and light rays]; Cederwall PLB(00)
[in AdS]; Brody & Hughston ht/05-in
[and quantum spacetime]; Sinkovics & Verlinde PLB(05)
[6D N = 4 super-Yang-Mills].
General References > s.a. Hyperkähler.
@ 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 [nc
(Moyal) deformation]; da Rocha & Vaz mp/04-in.
@ 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
13 jun 2008