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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 Polyakov action ： ウィキペディア英語版 Polyakov action In physics, the Polyakov action is the two-dimensional action of a conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L. Brink, P. Di Vecchia and P. S. Howe (in ''Physics Letters'' B65, pages 369 and 471 respectively), and has become associated with Alexander Polyakov after he made use of it in quantizing the string. The action reads :$\backslash mathcal\; =\; \backslash int\; \backslash mathrm^2\; \backslash sigma\; \backslash sqrt\; h^\; g\_\; (X)\; \backslash partial\_a\; X^\backslash mu\; (\backslash sigma)\; \backslash partial\_b\; X^\backslash nu(\backslash sigma)$ where $T$ is the string tension, $g\_$ is the metric of the target manifold, $h\_$ is the worldsheet metric, $h^$ its inverse, and $h$ is the determinant of $h\_$. The metric signature is chosen such that timelike directions are + and the spacelike directions are -. The spacelike worldsheet coordinate is called $\backslash sigma$ whereas the timelike worldsheet coordinate is called $\backslash tau$. This is also known as nonlinear sigma model.〔 〕 The Polyakov action must be supplemented by the Liouville action to describe string fluctuations. == Global symmetries == N.B. : Here, a symmetry is said to be local or global from the two dimensional theory (on the worldsheet) point of view. For example, Lorentz transformations, that are local symmetries of the space-time, are global symmetries of the theory on the worldsheet. The action is invariant under spacetime translations and infinitesimal Lorentz transformations: :(i) $X^\backslash alpha\; \backslash rightarrow\; X^\backslash alpha\; +\; b^\backslash alpha$ :(ii) $X^\backslash alpha\; \backslash rightarrow\; X^\backslash alpha\; +\; \backslash omega^\backslash alpha\_\; X^\backslash beta$ where $\backslash omega\_\; =\; -\; \backslash omega\_$ and $b^\backslash alpha$ is a constant. This forms the Poincaré symmetry of the target manifold. The invariance under (i) follows since the action $\backslash mathcal$ depends only on the first derivative of $X^\backslash alpha$. The proof of the invariance under (ii) is as follows: ::\int \mathrm^2 \sigma \sqrt h^ g_ \partial_a \left( X^\mu + \omega^\mu_ X^\delta \right) \partial_b \left( X^\nu + \omega^\nu_ X^\delta \right) \, |- | |$=\; \backslash mathcal\; +\; \backslash int\; \backslash mathrm^2\; \backslash sigma\; \backslash sqrt\; h^\; \backslash left(\; \backslash omega\_\; \backslash partial\_a\; X^\backslash mu\; \backslash partial\_b\; X^\backslash delta\; +\; \backslash omega\_\; \backslash partial\_a\; X^\backslash delta\; \backslash partial\_b\; X^\backslash nu\; \backslash right)\; +\; O(\backslash omega^2)\; \backslash ,$ |- | |$=\; \backslash mathcal\; +\; \backslash int\; \backslash mathrm^2\; \backslash sigma\; \backslash sqrt\; h^\; \backslash left(\; \backslash omega\_\; +\; \backslash omega\_\; \backslash right)\; \backslash partial\_a\; X^\backslash mu\; \backslash partial\_b\; X^\backslash delta\; +\; O(\backslash omega^2)\; =\; \backslash mathcal\; +\; O(\backslash omega^2)$ |} 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「Polyakov action」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.