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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 Affine gauge theory ： ウィキペディア英語版 Affine gauge theory Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold $X$. For instance, these are gauge theory of dislocations in continuous media when $X=\backslash mathbb\; R^3$, the generalization of metric-affine gravitation theory when $X$ is a world manifold and, in particular, gauge theory of the fifth force. ==Affine tangent bundle== Being a vector bundle, the tangent bundle $TX$ of an $n$-dimensional manifold $X$ admits a natural structure of an affine bundle $ATX$, called the ''affine tangent bundle'', possessing bundle atlases with affine transition functions. It is associated to a principal bundle $AFX$ of affine frames in tangent space over $X$, whose structure group is a general affine group $GA(n,\backslash mathbb\; R)$. The tangent bundle $TX$ is associated to a principal linear frame bundle $FX$, whose structure group is a general linear group $GL(n,\backslash mathbb\; R)$. This is a subgroup of $GA(n,\backslash mathbb\; R)$ so that the latter is a semidirect product of $GL(n,\backslash mathbb\; R)$ and a group $T^n$ of translations. There is the canonical imbedding of $FX$ to $AFX$ onto a reduced principal subbundle which corresponds to the canonical structure of a vector bundle $TX$ as the affine one. Given linear bundle coordinates : $(x^\backslash mu,\backslash dot\; x^\backslash mu),\; \backslash qquad\; \backslash dot\; x\text{'}^\backslash mu=\backslash frac\backslash dot\; x^\backslash nu,\; \backslash qquad\backslash qquad\; (1)$ on the tangent bundle $TX$, the affine tangent bundle can be provided with affine bundle coordinates : $(x^\backslash mu,\backslash widetilde\; x^\backslash mu=\backslash dot\; x^\backslash mu\; +a^\backslash mu(x^\backslash alpha)),\; \backslash qquad\; \backslash widetilde\; x\text{'}^\backslash mu=\backslash frac\backslash widetilde\; x^\backslash nu\; +\; b^\backslash mu(x^\backslash alpha).\; \backslash qquad\backslash qquad\; (2)$ and, in particular, with the linear coordinates (1). 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「Affine gauge theory」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.