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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 Cartan formalism (physics) ： ウィキペディア英語版 Cartan formalism (physics) : ''This page covers applications of the Cartan formalism. For the general concept see Cartan connection.'' The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. (See metric tensor.) This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, zweibein, fünfbein, elfbein etc. have been used. Vielbein covers all dimensions. (In German, ''vier'' stands for four and ''viel'' stands for many.) For a basis-dependent index notation, see tetrad (index notation). ==The basic ingredients== Suppose we are working on a differential manifold ''M'' of dimension ''n'', and have fixed natural numbers ''p'' and ''q'' with :''p'' + ''q'' = ''n''. Furthermore, we assume that we are given a SO(''p'', ''q'') principal bundle ''B'' over ''M'' and a SO(''p'', ''q'')-vector bundle ''V'' associated to ''B'' by means of the natural ''n''-dimensional representation of SO(''p'', ''q''). Equivalently, ''V'' is a rank ''n'' real vector bundle over ''M'', equipped with a metric η with signature (''p'', ''q'') (aka non degenerate quadratic form).〔A variant of the construction uses reduction to a Spin(''p'', ''q'') principal spin bundle. In that case, the principal bundle contains more information than the bundle ''V'' together with the metric η, which is needed to construct spinorial fields.〕 The basic ingredient of the Cartan formalism is an invertible linear map $e\backslash colon\backslash to\; M$, between vector bundles over ''M'' where T''M'' is the tangent bundle of ''M''. The invertibility condition on ''e'' is sometimes dropped. In particular if ''B'' is the trivial bundle, as we can always assume locally, ''V'' has a basis of orthogonal sections $f\_a\; =\; f\_1\; \backslash ldots\; f\_n$. With respect to this basis $\backslash eta\_\; =\; \backslash eta(f\_a,\; f\_b)\; =\; (1,\backslash ldots\; 1,\; -1,\; \backslash ldots,\; -1)$ is a constant matrix. For a choice of local coordinates $x^\backslash mu\; =\; x^,\; \backslash ldots,\; x^$ on ''M'' (the negative indices are only to distinguish them from the indices labeling the $f\_a$) and a corresponding local frame $\backslash partial\_\backslash mu\; =\; \backslash frac$ of the tangent bundle, the map ''e'' is determined by the images of the basis sections :$e\_a\; :=\; e(f\_a)\; :=\; e^\backslash mu\_a\; \backslash partial\_\backslash mu.$ They determine a (non coordinate) basis of the tangent bundle (provided ''e'' is invertible and only locally if B is only locally trivialised). The matrix $e^\backslash mu\_a\; ,\; \backslash mu\; =\; -1,\; \backslash dots,\; -n,\; a\; =\; 1,\; \backslash dots,\; n$ is called the tetrad, vierbein, vielbein etc.. Its interpretation as a local frame crucially depends on the implicit choice of local bases. Note that an isomorphism $V\; \backslash cong\; M$ gives a reduction $B\; \backslash to\; (M)$ of the frame bundle, the principal bundle of the tangent bundle. In general, such a reduction is impossible for topological reasons. Thus, in general for continuous maps ''e'', one cannot avoid that ''e'' becomes degenerate at some points of ''M''. 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「Cartan formalism (physics)」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.