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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 Feynman parametrization ： ウィキペディア英語版 Feynman parametrization Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well. Richard Feynman observed that: :$\backslash frac=\backslash int^1\_0\; \backslash frac$ which simplifies evaluating integrals like: :$\backslash int\; \backslash frac=\backslash int\; dp\; \backslash int^1\_0\; \backslash frac=\backslash int^1\_0\; du\; \backslash int\; \backslash frac.$ More generally, using the Dirac delta function: :$\backslash begin$ \frac&=(n-1)!\int^1_0 du_1 \cdots \int^1_0 du_n \frac \\ &=(n-1)! \int^1_0 du_1 \int^_0 du_2 \cdots \int^ \frac">(A_n-A_)\right )^n}. \end Even more generally, provided that $\backslash text\; (\; \backslash alpha\_\; )\; >\; 0$ for all $1\; \backslash leq\; j\; \backslash leq\; n$: :$\backslash frac\}\backslash cdots\; A\_^=\backslash frac)\})\}\backslash int\_^du\_\backslash cdots\backslash int\_^du\_\backslash fracu\_-1)u\_^\backslash cdots\; u\_^\}+\backslash cdots+u\_A\_\backslash right"\; TITLE="u\_A\_+\backslash cdots+u\_A\_\backslash right">)^\backslash alpha\_\}\}$ . 〔 〕 See also Schwinger parametrization. ==Derivation== :$\backslash frac\; =\; \backslash frac\backslash left(\backslash frac-\backslash frac\backslash right)=\backslash frac\backslash int\_B^A\; \backslash frac.$ Now just linearly transform the integral using the substitution, :$u=(z-B)/(A-B)$ which leads to $du\; =\; dz/(A-B)$ so $z\; =\; uA\; +\; (1-u)B$ and we get the desired result: :$\backslash frac\; =\; \backslash int\_0^1\; \backslash frac.$ In more general cases, derivations can be done very efficiently using the Schwinger parametrization. For example, in order to derive the Feynman parametrized form of :$\backslash frac$, we first reexpress all the factors in the denominator in their Schwinger parametrized form: :$\backslash frac=\; \backslash int^\backslash infty\_0\; ds\_i\; \backslash ,\; e^,\; \backslash forall\; i\; =1,...,n$ and rewrite, :$$ \frac=\int_0^\infty ds_1...ds_n \exp\left(). Then we perform the following change of integration variables, :$$ \alpha = s_1+...+s_n, :$$ \alpha_ = \frac; i=1,\ldots,n-1, to obtain, :$$ \frac = \int_^d\alpha_1...d\alpha_ \int_^d\alpha \alpha^\exp\left(-\alpha\left\+ \left(1-\alpha_-...\alpha_\right)A_\right\} \right). The next step is to perform the $\backslash alpha$ integration. :$$ \int_^d\alpha \alpha^\exp(-\alpha x)= \frac}\left(\int_^d\alpha\exp(-\alpha x)\right)=\fracA_+ \left(1-\alpha_-...\alpha_\right)A_. Substituting this result, we get to the penultimate form, :$$ \frac=\left(n-1\right)!\int_^d\alpha_1...d\alpha_\frac+" TITLE="\alpha_1A_1+...+\alpha_A_+">\left(1-\alpha_-...\alpha_\right)A_ )^n} , and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely, :$$ \frac=\left(n-1\right)!\int_^d\alpha_1...d\alpha_\frac" TITLE="\alpha_1A_1+...+\alpha_A_">)^n} . Similarly, in order to derive the Feynman parametrization form of the most general case, :$\backslash frac\}$ one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely, :$$ \frac\int^\infty_0 ds_1 \,s_1^ e^ = \frac\frac}\left(\int_^ds_1 e^\right) and then proceed exactly along the lines of previous case. 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「Feynman parametrization」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.