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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 Grassmann variable ： ウィキペディア英語版 Grassmann number In mathematical physics, a Grassmann number, named after Hermann Grassmann, (also called an anticommuting number or anticommuting c-number) is a mathematical construction which allows a path integral representation for Fermionic fields. A collection of Grassmann variables $\backslash theta\_i$ are independent elements of an algebra which contains the real numbers that anticommute with each other but commute with ordinary numbers $x$: : $\backslash theta\_i\; \backslash theta\_j\; =\; -\backslash theta\_j\; \backslash theta\_i\backslash qquad\backslash theta\_i\; x\; =\; x\; \backslash theta\_i.$ In particular, the squares of the generators vanish: : $(\backslash theta\_i)^2\; =\; 0,\backslash ,$ since $\backslash theta\_i\; \backslash theta\_i\; =\; -\backslash theta\_i\; \backslash theta\_i.$ In other words, a Grassmann number is a non-zero square-root of zero. In order to reproduce the path integral for a Fermi field, the definition of Grassmann integration needs to have the following properties: * linearity :: $\backslash int\backslash ,(f(\backslash theta)\; +\; b\; g(\backslash theta))\backslash ,\; d\backslash theta\; =\; a\; \backslash int\backslash ,f(\backslash theta)\backslash ,\; d\backslash theta\; +\; b\; \backslash int\backslash ,g(\backslash theta)\backslash ,\; d\backslash theta$ * partial integration formula :: $\backslash int\; \backslash left()\backslash ,\; d\backslash theta\; =\; 0.$ This results in the following rules for the integration of a Grassmann quantity: :: $\backslash int\backslash ,\; 1\backslash ,\; d\backslash theta\; =\; 0$ :: $\backslash int\backslash ,\; \backslash theta\backslash ,\; d\backslash theta\; =\; 1.$ Thus we conclude that the operations of integration and differentiation of a Grassmann number are identical. In the path integral formulation of quantum field theory the following Gaussian integral of Grassmann quantities is needed for fermionic anticommuting fields: : $\backslash int\; \backslash exp\backslash left()\; \backslash ,d\backslash theta\backslash ,d\backslash eta\; =\; \backslash det\; A$ with ''A'' being an ''N'' × ''N'' matrix. The algebra generated by a set of Grassmann numbers is known as a Grassmann algebra. The Grassmann algebra generated by ''n'' linearly independent Grassmann numbers has dimension 2''n''. Grassmann algebras are the prototypical examples of supercommutative algebras. These are algebras with a decomposition into even and odd variables which satisfy a graded version of commutativity (in particular, odd elements anticommute). ==Exterior algebra == The Grassmann algebra is the exterior algebra of the vector space spanned by the generators. The exterior algebra is defined independent of a choice of basis. 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「Grassmann number」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.