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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 Quantum dilogarithm ： ウィキペディア英語版 Quantum dilogarithm In mathematics, the quantum dilogarithm also known as q-exponential is a special function defined by the formula :$$ \phi(x)\equiv(x;q)_\infty=\prod_^\infty (1-xq^n),\quad |q|<1 Thus in the notation of the page on q-exponential mentioned above, $\backslash phi(x)=E\_q(x)^$ . Let $u,v$ be “q-commuting variables”, that is elements of a suitable noncommutative algebra satisfying Weyl’s relation $uv=qvu$. Then, the quantum dilogarithm satisfies Schützenberger’s identity :$$ \phi(u) \phi(v)=\phi(u + v) Faddeev-Volkov's identity :$$ \phi(v) \phi(u)=\phi(u +v -vu) and Faddeev-Kashaev's identity :$$ \phi(v) \phi(u)=\phi(u)\phi(-vu)\phi(v) The latter is known to be a quantum generalization of Roger's five term dilogarithm identity. Faddeev's quantum dilogarithm $\backslash Phi\_b(w)$ is defined by the following formula: : $\backslash Phi\_b(z)=\backslash exp$ \left( \frac\int_C \frac} \frac \right) where the contour of integration $C$ goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. Ludvig Faddeev discovered the quantum pentagon identity: : $\backslash Phi\_b(\backslash hat\; p)\backslash Phi\_b(\backslash hat\; q)$ = \Phi_b(\hat q) \Phi_b(\hat p+ \hat q) \Phi_b(\hat p) where $\backslash hat\; p$ and $\backslash hat\; q$ are (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation :$(p,\backslash hat\; q)=\backslash frac1\}(-e^)\})\}$ valid for Im $b^2>0$. == References == * * * * * * 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「Quantum dilogarithm」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.