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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 Symplectic spinor bundle ： ウィキペディア英語版 Symplectic spinor bundle In differential geometry, given a metaplectic structure $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ on a $2n$-dimensional symplectic manifold $(M,\; \backslash omega),\backslash ,$ one defines the symplectic spinor bundle to be the Hilbert space bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group —the two-fold covering of the symplectic group— gives rise to an infinite rank vector bundle, this is the symplectic spinor construction due to Bertram Kostant. A section of the symplectic spinor bundle $\backslash ,$ is called a symplectic spinor field. ==Formal definition== Let $(,F\_)$ be a metaplectic structure on a symplectic manifold $(M,\; \backslash omega),\backslash ,$ that is, an equivariant lift of the symplectic frame bundle $\backslash pi\_\backslash colon\backslash to\; M\backslash ,$ with respect to the double covering $\backslash rho\backslash colon\; )\backslash to\; ).\backslash ,$ The symplectic spinor bundle $\backslash ,$ is defined 〔 page 37 〕 to be the Hilbert space bundle : $=\backslash times\_L^2(^n)\backslash ,$ associated to the metaplectic structure $$ via the metaplectic representation $\backslash colon\; )\backslash to\; (L^2(^n)),\backslash ,$ also called the Segal-Shale-Weil 〔 〕 representation of $).\backslash ,$ Here, the notation $()\backslash ,$ denotes the group of unitary operators acting on a Hilbert space $.\backslash ,$ The Segal-Shale-Weil representation 〔 〕 is an infinite dimensional unitary representation of the metaplectic group $)$ on the space of all complex valued square Lebesgue integrable functions $L^2(^n).\backslash ,$ Because of the infinite dimension, the Segal-Shale-Weil representation is not so easy to handle. 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「Symplectic spinor bundle」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.