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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 Coherent states in mathematical physics ： ウィキペディア英語版 Coherent states in mathematical physics Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also〔J-P. Gazeau,''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.〕). However, they have generated a huge variety of generalizations, which have led to a tremendous literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys .〔S.T. Ali, J-P. Antoine, J-P. Gazeau, and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, ''Reviews in Mathematical Physics'' 7 (1995) 1013-1104.〕〔S.T. Ali, J-P. Antoine, and J-P. Gazeau, ''Coherent States, Wavelets and Their Generalizations'', Springer-Verlag, New York, Berlin, Heidelberg, 2000.〕〔S.T. Ali, Coherent States, ''Encyclopedia of Mathematical Physics'', pp. 537-545; Elsevier, Amsterdam, 2006.〕 == A general definition == Let $\backslash mathfrak\; H\backslash ,$ be a complex, separable Hilbert space, $X$ a locally compact space and $d\backslash nu$ a measure on $X$. For each $x$ in $X$, denote $|x\backslash rangle$ a vector in $\backslash mathfrak\; H$. Assume that this set of vectors possesses the following properties: # The mapping $x\; \backslash mapsto\; |\; x\; \backslash rangle$ is weakly continuous, i.e., for each vector $|\backslash phi\backslash rangle$ in $\backslash mathfrak\; H$, the function $\backslash Psi\; (x)\; =\; \backslash langle\; x|\; \backslash phi\backslash rangle$ is continuous (in the topology of $X$). # The resolution of the identity :$\backslash int\_X\; |\; x\backslash rangle\backslash langle\; x|\backslash ;\; d\backslash nu\; (x)\; =\; I\_$ holds in the weak sense on the Hilbert space $\backslash mathfrak\; H$, i.e., for any two vectors $|\; \backslash phi\backslash rangle\; ,\; |\; \backslash psi\; \backslash rangle$ in $\backslash mathfrak\; H$, the following equality holds: :$\backslash int\_X\; \backslash langle\backslash phi|\; x\backslash rangle\backslash langle\; x|\backslash psi\backslash rangle\backslash ;\; d\backslash nu\; (x)\; =\; \backslash langle\backslash phi|\backslash psi\backslash rangle\backslash ;.$ A set of vectors $|\; x\backslash rangle$ satisfying the two properties above is called a family of ''generalized coherent states''. In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take $X\backslash equiv\backslash mathbb$, the complex plane, $x\; \backslash equiv\; \backslash alpha$ and $d\backslash nu\; (x)\; \backslash equiv\; \backslash frac\; d^2\backslash alpha.$ Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors $|\; x\; \backslash rangle$ simply forming a total set in $\backslash ,$ and the functions $\backslash Psi\; (x)\; =\; \backslash langle\; x\; |\; \backslash psi\backslash rangle$, as $|\; \backslash psi\; \backslash rangle$ runs through $$, forming a ''reproducing kernel Hilbert space''. The objective in both cases is to ensure that an arbitrary vector $|\; \backslash psi\; \backslash rangle$ be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that :$$ | \psi \rangle = \int_X \Psi (x)| x\rangle\; d\nu (x)\; , where $\backslash Psi\; (x)\; =\; \backslash langle\; x\; |\; \backslash psi\backslash rangle$. These vectors $\backslash Psi$ are square integrable, continuous functions on $X$ and satisfy the ''reproducing property'' :$$ \int_X K (x,y )\Psi (y)\; d\nu (y) = \Psi (x)\, , where $K\; (x,\; y)\; =\; \backslash langle\; x\; |\; y\; \backslash rangle$ is the reproducing kernel, which satisfies the following properties :$\backslash quad\; K\; (x,\; y)\; =\; \backslash overline\backslash ;\; ,\; \backslash qquad\; K\; (x,\; x)\; >\; 0\backslash ;\; ,$ :$\backslash int\_X\; K(x,z)\backslash ;\; K(z,\; y)\; \backslash ;\; d\backslash nu\; (z)\; =\; K(x,y)\backslash ;\; .$ 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「Coherent states in mathematical physics」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.