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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 Gauge group (mathematics) ： ウィキペディア英語版 Gauge group (mathematics) A gauge group is a group of gauge symmetries of the Yang – Mills gauge theory of principal connections on a principal bundle. Given a principal bundle $P\backslash to\; X$ with a structure Lie group $G$, a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group $G(X)$ of global sections of the associated group bundle $\backslash widetilde\; P\backslash to\; X$ whose typical fiber is a group $G$ which acts on itself by the adjoint representation. The unit element of $G(X)$ is a constant unit-valued section $g(x)=1$ of $\backslash widetilde\; P\backslash to\; X$. At the same time, gauge gravitation theory exemplifies field theory on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group. It should be emphasized that, in the physical literature on gauge theory, a structure group of a principal bundle often is called the gauge group. In quantum gauge theory, one considers a normal subgroup $G^0(X)$ of a gauge group $G(X)$ which is the stabilizer : $G^0(X)=\backslash$ of some point $1\backslash in\; \backslash widetilde\; P\_$ of a group bundle $\backslash widetilde\; P\backslash to\; X$. It is called the ''pointed gauge group''. This group acts freely on a space of principal connections. Obviously, $G(X)/G^0(X)=G$. One also introduces the ''effective gauge group'' $\backslash overline\; G(X)=G(X)/Z$ where $Z$ is the center of a gauge group $G(X)$. This group $\backslash overline\; G(X)$ acts freely on a space of irreducible principal connections. If a structure group $G$ is a complex semisimple matrix group, the Sobolev completion $\backslash overline\; G\_k(X)$ of a gauge group $G(X)$ can be introduced. It is a Lie group. A key point is that the action of $\backslash overline\; G\_k(X)$ on a Sobolev completion $A\_k$ of a space of principal connections is smooth, and that an orbit space $A\_k/\backslash overline\; G\_k(X)$ is a Hilbert space. It is a configuration space of quantum gauge theory. == References == * Mitter, P., Viallet, C., On the bundle of connections and the gauge orbit manifold in Yang – Mills theory, ''Commun. Math. Phys.'' 79 (1981) 457. * Marathe, K., Martucci, G., ''The Mathematical Foundarions of Gauge Theory'' (North Holland, 1992) ISBN 0-444-89708-9. * Mangiarotti, L., Sardanashvily, G., ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000) ISBN 981-02-2013-8 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「Gauge group (mathematics)」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.