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| | The name of the author is Jayson. Invoicing is my occupation. Alaska is where I've usually been living. The favorite pastime for him and his children is to play lacross and he would never give it up.<br><br>Feel free to visit my blog post: best psychic readings, [http://checkmates.co.za/index.php?do=/profile-56347/info/ click this link here now], |
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| In [[mathematics]], '''twistor space''' is the [[complex number|complex]] [[vector space]] of solutions of the twistor equation <math>\nabla_{A'}^{(A}\Omega^{B)}=0 </math>. It was described in the 1960s by [[Roger Penrose]] and MacCallum.<ref>R. Penrose and M. A. H. MacCallum, Twistor theory: An approach to the quantisation of fields and space-time. {{doi|10.1016/0370-1573(73)90008-2}}</ref> According to [[Andrew Hodges]], twistor space is useful for conceptualizing the way photons travel through space, using four [[complex number]]s. He also posits that twistor space may aid in understanding the [[asymmetry]] of the [[weak nuclear force]].<ref>Hodges, Andrew, "One to Nine" 2009</ref>
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| For [[Minkowski space]], denoted <math>\mathbb{M}</math>, the solutions to the twistor equation are of the form
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| :<math>
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| \Omega(x)=\omega^A-ix^{AA'}\pi_{A'}
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| </math>
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| where <math>\omega^A</math> and <math>\pi_{A'}</math> are two constant [[Weyl spinor]]s and <math>x^{AA'}=\sigma^{AA'}_\mu x^{\mu}</math> is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by <math>Z^{\alpha}=(\omega^{A},\pi_{A'})</math>, and with a hermitian form
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| :<math>
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| \Sigma(Z)=\omega^{A}\bar\pi_{A}+\bar\omega^{A'}\pi_{A'}
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| </math>
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| which is invariant under the group SU(2,2) which is a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.
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| Points in Minkowski space are related to subspaces of twistor space through the incidence relation
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| :<math>
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| \omega^{A}=ix^{AA'}\pi_A.
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| </math>
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| This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted '''PT''', which is isomorphic as a complex manifold to <math>\mathbb{CP}^3</math>.
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| Given a point <math>x\in M</math> it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of
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| a <math>\mathbb{CP}^1</math> parametrized by <math>\pi_{A'}</math>.
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| The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is | |
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| '''T''' := '''C'''<sup>4</sup>. It has associated to it the [[double fibration]] of [[flag manifold]]s '''P''' ←<sub>μ</sub> '''F''' <sub>ν</sub>→ '''M''', where
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| :projective twistor space | |
| ::'''P''' := '''F'''<sub>1</sub>('''T''') = '''P'''<sub>3</sub>('''C''') = '''P'''('''C'''<sup>4</sup>)
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| :compactified complexified Minkowski space
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| ::'''M''' := '''F'''<sub>2</sub>('''T''') = '''G'''<sub>2</sub>('''C'''<sup>4</sup>) = '''G'''<sub>2,4</sub>('''C''')
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| :the correspondence space between '''P''' and '''M'''
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| ::'''F''' := '''F'''<sub>1,2</sub>('''T''')
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| In the above, '''P''' stands for [[projective space]], '''G''' a [[Grassmannian]], and '''F''' a [[flag manifold]]. The double fibration gives rise to two [[correspondence (mathematics)|correspondences]], ''c'' := ν . μ<sup>−1</sup> and ''c''<sup>−1</sup> := μ . ν<sup>−1</sup>.
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| '''M''' is embedded in '''P'''<sub>5</sub> ~=~ '''P'''(Λ<sup>2</sup>'''T''') by the [[Plücker embedding]] and the image is the [[Klein quadric]].
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| == Rationale == | |
| In the (translated) words of [[Jacques Hadamard]]: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying '''R'''<sup>4</sup> it might be valuable to identify it with '''C'''<sup>2</sup>. However, since there is no [[Archetype|canonical]] way of doing so, instead all [[isomorphism]]s respecting orientation and metric between the two are considered. It turns out that complex projective 3-space '''P'''<sub>3</sub>('''C''') parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in '''R'''<sup>4</sup>. It turns out that vector bundles with self-dual connections on '''R'''<sup>4</sup>([[instanton]]s) [[bijection|correspond bijectively]] to holomorphic bundles on complex projective 3-space '''P'''<sub>3</sub>('''C''').
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| ==See also==
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| *[[Roger Penrose]]
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| *[[Twistor theory]]
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| ==References==
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| {{reflist}}
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| *Ward, R.S. and [[Raymond O. Wells, Jr.|Wells, Raymond O. Jr.]], ''Twistor Geometry and Field Theory'', Cambridge University Press (1991). ISBN 0-521-42268-X.
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| *Huggett, S. A. and Todd, K. P., ''An introduction to twistor theory'', Cambridge University Press (1994). ISBN 978-0-521-45689-0.
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| {{Topics of twistor theory}}
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| [[Category:Complex manifolds]]
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