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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 Spin connection ： ウィキペディア英語版 Spin connection In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations. ==Definition== Let $e\_\backslash mu^$ be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthogonal space time vector fields that diagonalize the metric tensor :$g\_\; =\; e\_\backslash mu^\; e\_\backslash nu^\; \backslash eta\_,$ where $g\_$ is the spacetime metric and $\backslash eta\_$ is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that $g\_$, when written in terms of the basis $e\_\backslash mu^$, is locally flat. The vierbein field indices can be raised or lowered by the metric $g^$ and/or $\backslash eta^$. For example, $e^=g^\; e\_\backslash nu^$. The spin connection is given by :$\backslash omega\_^=e\_\backslash nu^\; \backslash Gamma^\backslash nu\_e^\; +\; e\_\backslash nu^\; \backslash partial\_\backslash mu\; e^\; =\; e\_\backslash nu^\; \backslash Gamma^\backslash nu\_e^\; -\; e^\; \backslash partial\_\backslash mu\; e\_\backslash nu\; ^,$ where $\backslash Gamma^\backslash sigma\_$ is the affine connection. Or purely in terms of the vierbein field as〔M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Vol. 2.〕 :$\backslash omega\_^=\backslash frace^(\backslash partial\_\backslash mu\; e\_\backslash nu^-\backslash partial\_\backslash nu\; e\_\backslash mu^)-\backslash frace^(\backslash partial\_\backslash mu\; e\_\backslash nu^-\backslash partial\_\backslash nu\; e\_\backslash mu^)-\backslash frace^e^(\backslash partial\_\backslash rho\; e\_-\backslash partial\_\backslash sigma\; e\_)e\_\backslash mu^,$ which by definition is anti-symmetric in its internal indices $a,\; b$. The spin connection $\backslash omega\_\backslash mu^$ defines a covariant derivative $D\_\backslash mu$ on generalized tensors. For example its action on $V\_\backslash nu^$ is :$D\_\backslash mu\; V\_\backslash nu^\; =\; \backslash partial\_\backslash mu\; V\_\backslash nu^\; +\; ^a\}\_b\; V\_\backslash nu^\; -\; \backslash Gamma^\backslash sigma\_\; V\_\backslash sigma^$ 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「Spin connection」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.