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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 Lichnerowicz formula ： ウィキペディア英語版 Lichnerowicz formula The Lichnerowicz formula (also known as the Lichnerowicz–Weitzenböck formula) is a fundamental equation in the analysis of spinors on pseudo-Riemannian manifolds. In dimension 4, it forms a piece of Seiberg–Witten theory and other aspects of gauge theory. It is named after noted mathematicians André Lichnerowicz who proved it in 1963, and Roland Weitzenböck. The formula gives a relationship between the Dirac operator and the Laplace–Beltrami operator acting on spinors, in which the scalar curvature appears in a natural way. The result is significant because it provides an interface between results from the study of elliptic partial differential equations, results concerning the scalar curvature, and results on spinors and spin structures. Given a spin structure on a pseudo-Riemannian manifold ''M'' and a spinor bundle ''S'', the Lichnerowicz formula states that on a section ψ of ''S'', :$D^2\backslash psi\; =\; \backslash nabla^$ *\nabla\psi + \frac\operatorname\psi where Sc denotes the scalar curvature and $\backslash nabla^$ *\nabla is the connection Laplacian. More generally, given a complex spin structure on a pseudo-Riemannian manifold ''M'', a spinor bundle ''W''± with section $\backslash phi$, and a connection ''A'' on its determinant line bundle ''L'', the Lichnerowicz formula is :$D\_^D\_\backslash phi=\backslash nabla\; \_A^\backslash nabla\_\backslash phi+\backslash fracR\backslash phi+\backslash frac\backslash langle\; F\_^,\backslash phi\backslash rangle.$ Here, $D\_A$ is the Dirac operator $D\_A:\backslash Gamma\; (W^+)\backslash to\; \backslash Gamma\; (W^-),$ and $\backslash nabla\; \_A$ is the covariant derivative associated with the connection A, $\backslash nabla\; \_A:\; \backslash Gamma\; (W^+)\backslash to\; \backslash Gamma(W^+\backslash otimes\; T\_M^$ *). $R$ is the usual scalar curvature (a contraction of the Ricci tensor) and $F\_A^+$ is the self-dual part of the curvature of A. The asterisks denote the adjoint of the quantity and the brackets $\backslash langle\; ,\; \backslash rangle$ denote the Clifford action. ==See also== * Weitzenböck formula 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「Lichnerowicz formula」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.