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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 Berezin integral ： ウィキペディア英語版 Berezin integral In mathematical physics, a Berezin integral, named after Felix Berezin, (or Grassmann integral, after Hermann Grassmann) is a way to define integration of elements of the exterior algebra (Hermann Grassmann 1844). It is called integral because it is used in physics as a sum over histories for fermions, an extension of the path integral. ==Integration on an exterior algebra== Let $\backslash Lambda^n$ be the exterior algebra of polynomials in anticommuting elements $\backslash theta\_,\backslash dots,\backslash theta\_$ over the field of complex numbers. (The ordering of the generators $\backslash theta\_1,\backslash dots,\backslash theta\_n$ is fixed and defines the orientation of the exterior algebra.) The Berezin integral on $\backslash Lambda^$ is the linear functional $\backslash int\_\backslash cdot\backslash textrm\backslash theta$ with the following properties: :$\backslash int\_\backslash theta\_\backslash cdots\backslash theta\_\backslash ,\backslash mathrm\backslash theta=1,$ :$\backslash int\_\backslash frac\backslash theta=0,\backslash \; i=1,\backslash dots,n$ for any $f\backslash in\backslash Lambda^n,$ where $\backslash partial/\backslash partial\backslash theta\_$ means the left or the right partial derivative. These properties define the integral uniquely. The formula :$\backslash int\_f\backslash left(\; \backslash theta\backslash right)\; \backslash ,\; \backslash mathrm\backslash theta=\backslash int\_\backslash left(\; \backslash cdots\; \backslash int\_\backslash left(\; \backslash int\_f\backslash left(\backslash theta\backslash right)\; \backslash ,\; \backslash mathrm\backslash theta\_\backslash right)\; \backslash ,\; \backslash mathrm\backslash theta\_2\; \backslash cdots\; \backslash right)\backslash mathrm\backslash theta\_n$ expresses the Fubini law. On the right-hand side, the interior integral of a monomial $f=g\backslash left(\; \backslash theta^\backslash right)\; \backslash theta\_$ is set to be $g\backslash left(\; \backslash theta^\backslash right)\; ,\backslash$ where $\backslash theta^=\backslash left(\backslash theta\_,...,\backslash theta\_\backslash right)$; the integral of $f=g\backslash left(\; \backslash theta^\backslash right)$ vanishes. The integral with respect to $\backslash theta\_$ is calculated in the similar way and so on. 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「Berezin integral」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.