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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 nonlinear realization ： ウィキペディア英語版 nonlinear realization In mathematics, nonlinear realization of a Lie group $G$ possessing a Cartan subgroup $H$ is a particular induced representation of $G$. In fact it is a representation of a Lie algebra $\backslash mathfrak\; g$ of $G$ in a neighborhood of its origin. A nonlinear realization technique is part and parcel of many field theories with spontaneous symmetry breaking, e.g., nonlinear sigma model, chiral symmetry breaking, Goldstone boson theory, classical Higgs field theory, gauge gravitation theory and supergravity. Let $G$ be a Lie group and $H$ its Cartan subgroup which admits a linear representation in a vector space $V$. A Lie algebra $\backslash mathfrak\; g$ of $G$ is split into the sum $\backslash mathfrak\; g=\backslash mathfrak\; h\; +\backslash mathfrak\; f$ of the Cartan subalgebra $\backslash mathfrak\; h$ of $H$ and its supplement $\backslash mathfrak\; f$ so that : $(f,\backslash mathfrak\; f)\backslash subset\; \backslash mathfrak\; h,\; \backslash qquad\; (f,\backslash mathfrak\; h$ )\subset \mathfrak f. There exists an open neighbourhood $U$ of the unit of $G$ such that any element $g\backslash in\; U$ is uniquely brought into the form : $g=\backslash exp(F)\backslash exp(I),\; \backslash qquad\; F\backslash in\backslash mathfrak\; f,\; \backslash qquad\; I\backslash in\backslash mathfrak\; h.$ Let $U\_G$ be an open neighborhood of the unit of $G$ such that $U\_G^2\backslash subset\; U$, and let $U\_0$ be an open neighborhood of the $H$-invariant center $\backslash sigma\_0$ of the quotient $G/H$ which consists of elements : $\backslash sigma=g\backslash sigma\_0=\backslash exp(F)\backslash sigma\_0,\; \backslash qquad\; g\backslash in\; U\_G.$ Then there is a local section $s(g\backslash sigma\_0)=\backslash exp(F)$ of $G\backslash to\; G/H$ over $U\_0$. With this local section, one can define the induced representation, called the nonlinear realization, of elements $g\backslash in\; U\_G\backslash subset\; G$ on $U\_0\backslash times\; V$ given by the expressions : $g\backslash exp(F)=\backslash exp(F\text{'})\backslash exp(I\text{'}),\; \backslash qquad\; g:(\backslash exp(F)\backslash sigma\_0,v)\backslash to\; (\backslash exp(F\text{'})\backslash sigma\_0,\backslash exp(I\text{'})v).$ The corresponding nonlinear realization of a Lie algebra $\backslash mathfrak\; g$ of $G$ takes the following form. Let $\backslash$, $\backslash$ be the bases for $\backslash mathfrak\; f$ and $\backslash mathfrak\; h$, respectively, together with the commutation relations : $()=\; c^d\_I\_d,\; \backslash qquad\; ()=\; c^d\_I\_d,$ \qquad ()= c^\beta_F_\beta. Then a desired nonlinear realization of $\backslash mathfrak\; g$ in $\backslash mathfrak\; f\backslash times\; V$ reads : $F\_\backslash alpha:\; (\backslash sigma^\backslash gamma\; F\_\backslash gamma,v)\backslash to\; (F\_\backslash alpha(\backslash sigma^\backslash gamma)F\_\backslash gamma,$ F_\alpha(v)), \qquad I_a: (\sigma^\gamma F_\gamma,v)\to (I_a(\sigma^\gamma)F_\gamma,I_av), , : $F\_\backslash alpha(\backslash sigma^\backslash gamma)=$ \delta^\gamma_\alpha + \frac(c^\beta_c^\gamma_ - 3 c^b_c^\gamma_)\sigma^\mu\sigma^\nu, \qquad I_a(\sigma^\gamma)=c^\gamma_\sigma^\nu, up to the second order in $\backslash sigma^\backslash alpha$. In physical models, the coefficients $\backslash sigma^\backslash alpha$ are treated as Goldstone fields. Similarly, nonlinear realization of Lie superalgebras is considered. == See also == *Induced representation 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「nonlinear realization」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.