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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 Bogoliubov inner product ： ウィキペディア英語版 Bogoliubov inner product The Bogoliubov inner product (''Duhamel two-point function'', ''Bogolyubov inner product'', ''Bogoliubov scalar product'', ''Kubo-Mori-Bogoliubov inner product'') is a special inner product in the space of operators. The Bogoliubov inner product appears in quantum statistical mechanics〔D. Petz and G. Toth. (The Bogoliubov inner product in quantum statistics ), ''Letters in Mathematical Physics'' 27, 205-216 (1993).〕〔D. P. Sankovich. (On the Bose condensation in some model of a nonideal Bose gas ), ''J. Math. Phys.'' 45, 4288 (2004).〕 and is named after theoretical physicist Nikolay Bogoliubov. ==Definition== Let $A$ be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as :$\backslash langle\; X,Y\backslash rangle\_A=\backslash int\backslash limits\_0^1\; (^\; X^\backslash dagger^Y)dx$ The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., $\backslash langle\; X,X\backslash rangle\_A\backslash ge\; 0$), and satisfies the symmetry property $\backslash langle\; X,Y\backslash rangle\_A=\backslash langle\; Y,X\backslash rangle\_A$. In applications to quantum statistical mechanics, the operator $A$ has the form $A=\backslash beta\; H$, where $H$ is the Hamiltonian of the quantum system and $\backslash beta$ is the inverse temperature. With these notations, the Bogoliubov inner product takes the form :$\backslash langle\; X,Y\backslash rangle\_=\; \backslash int\backslash limits\_0^1\; \backslash langle^\; X^\backslash dagger^Y\backslash rangle\; dx$ where $\backslash langle\; \backslash dots\; \backslash rangle$ denotes the thermal average with respect to the Hamiltonian $H$ and inverse temperature $\backslash beta$. In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum: :$\backslash langle\; X,Y\backslash rangle\_=\backslash frac\backslash ,^\; \backslash bigg\backslash vert\_$ 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「Bogoliubov inner product」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.