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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 Bogomol'nyi–Prasad–Sommerfield bound ： ウィキペディア英語版 Bogomol'nyi–Prasad–Sommerfield bound The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny,〔E. B. Bogomolny, Sov.J.Nucl.Phys. 24 (1976) 449; Yad.Fiz. 24 (1976) 861〕 Manoj Prasad, and Charles Sommerfield〔M.K. Prasad & C. M. Sommerfield, Phys.Rev.Lett. 35 (1975) 760.〕) is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied (called "saturated"), one can come up with a simpler set of partial differential equations to solve, the Bogomol'nyi equations. Solutions saturating the bound are called BPS states and play an important role in field theory and string theory. ==Example== In a theory of U(1) Yang-Mills-Higgs, the energy at a given time ''t'' is given by :$E=\backslash int\; d^3x\backslash ,\; \backslash left$ where ''D'' is the covariant derivative and ''V'' is the potential. If we assume that ''V'' is nonnegative and is zero only for the Higgs vacuum and that the Higgs field is in the adjoint representation, then :$$ \begin E & \geq \int d^3x\, \left \right] \\ & \geq \int d^3x\, \operatorname\left( \frac\left(\overrightarrow\mp\frac\vec\right)^2 \pm\frac\overrightarrow\cdot \vec\right ) \\ & \geq \pm \frac\int d^3x\, \operatorname\left(\vec\right ) \\ & = \pm\frac\int_\left(\vec\cdot d\vec\right ). \end Therefore, :$E\backslash geq\; \backslash left\backslash |\backslash int\_\; \backslash operatorname\backslash left(\backslash vec\backslash cdot\; d\backslash vec\backslash right)\backslash right\; \backslash |.$ Saturation happens when $\backslash pi\; =\; 0$ and :$\backslash overrightarrow\backslash mp\backslash frac\backslash vec\; =\; 0$ The Bogomol'nyi equation. The other condition for saturation is the Higgs mass and self-interaction are zero, which is the case in N=2 supersymmetric theories. This quantity is the absolute value of the magnetic flux. A slight generalization applying to dyons also exists. For that, the Higgs field needs to be a complex adjoint, not a real adjoint. 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「Bogomol'nyi–Prasad–Sommerfield bound」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.