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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク twistor space ： ウィキペディア英語版 twistor space In mathematics, twistor space is the complex vector space of solutions of the twistor equation $\backslash nabla\_^\backslash Omega^=0$. It was described in the 1960s by Roger Penrose and MacCallum.〔R. Penrose and M. A. H. MacCallum, Twistor theory: An approach to the quantisation of fields and space-time. 〕 According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.〔Hodges, Andrew, "One to Nine" 2009〕 For Minkowski space, denoted $\backslash mathbb$, the solutions to the twistor equation are of the form :$$ \Omega^A(x)=\omega^A-ix^\pi_ where $\backslash omega^A$ and $\backslash pi\_$ are two constant Weyl spinors and $x^=\backslash sigma^\_\backslash mu\; x^$ is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by $Z^=(\backslash omega^,\backslash pi\_)$, and with a hermitian form :$$ \Sigma(Z)=\omega^\bar\pi_+\bar\omega^\pi_ 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. Points in Minkowski space are related to subspaces of twistor space through the incidence relation :$$ \omega^=ix^\pi_. 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 $\backslash mathbb^3$. Given a point $x\backslash in\; M$ it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a $\backslash mathbb^1$ parametrized by $\backslash pi\_$. 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 T := C4. It has associated to it the double fibration of flag manifolds P ←μ F ν→ M, where :projective twistor space ::P := F1(T) = P3(C) = P(C4) :compactified complexified Minkowski space ::M := F2(T) = G2(C4) = G2,4(C) :the correspondence space between P and M ::F := F1,2(T) In the above, P stands for projective space, G a Grassmannian, and F a flag manifold. The double fibration gives rise to two correspondences, ''c'' := ν . μ−1 and ''c''−1 := μ . ν−1. M is embedded in P5 ~=~ P(Λ2T) by the Plücker embedding and the image is the Klein quadric. == 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 R4 it might be valuable to identify it with C2. However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space P3(C) parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in R4. It turns out that vector bundles with self-dual connections on R4(instantons) correspond bijectively to holomorphic bundles on complex projective 3-space P3(C). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「twistor space」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.