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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 Liouville field theory ： ウィキペディア英語版 Liouville field theory In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional quantum field theory whose classical equation of motion resembles the Joseph Liouville's non-linear second order differential equation that appears in the classical geometrical problem of uniformizing Riemann surfaces.〔Lützen, J. ''Joseph Liouville, 1809–1882: Master of pure and applied mathematics'', Springer 1990 ISBN 0387971807.〕 The field theory is defined by the local action ::$$ S = \frac \int d^2x \sqrt (g^ \partial _\mu \phi \partial _ \phi + (b+b^) R \phi + 4\pi e^), where $\backslash partial\; \_\backslash mu\; =\; \backslash partial\; /\backslash partial\; x^\backslash mu\; ,\backslash \; g\_$ is the metric of the two-dimensional space on which the field theory is formulated, $R$ is the Ricci scalar of such space, and $b$ is a real coupling constant. The field $\backslash phi$ is consequently dubbed the Liouville field. The equation of motion associated to this action is ::$$ \Delta \phi(x) = \frac (b+b^) R(x) + 4\pi b e^ where $\backslash Delta\; =\; g^\; \backslash partial\; \_\; (g^\; g^\; \backslash partial\_\; )$ is the D'Alembert operator in such space (see also Laplace–Beltrami operator). In the case the metric of the space being the Euclidean metric, and using standard notation, it becomes the classical Liouville equation ::$$ \left(\frac + \frac \right) \phi (x,y) = 4\pi b e^ Liouville field theory is a conformal field theory that incarnates Weyl symmetry in a very special way. Its central charge $c$ is given in terms of the parameter appearing in the action through the expression $c=1+6(b+1/b)^$. Liouville theory appears in the context of string theory when trying to formulate a non-critical version of the theory in the path integral formulation. Also in the string theory context, if coupled to a free bosonic field Liouville field theory can be thought of as the theory describing string excitations in a two-dimensional space(time). Liouville field theory is one of the best understood examples of what is called a non-rational conformal field theory, for which some observables have been computed explicitly. Such is the case of two-point and three-point correlation functions of primary operators on the topology of the sphere. Explicit expressions for observables of the theory defined on other topologies, like the partition function on the torus and the one-point function on the disk, were also calculated in the recent years. Liouville theory is also closely related to other problems in physics and mathematics, like two-dimensional quantum gravity, two-dimensional string theory, three-dimensional general relativity in negatively curved spaces, four-dimensional superconformal gauge theories, the uniformization problem of Riemann surfaces, and other problems in conformal mapping. It is also connected to other two-dimensional non-rational conformal field theories with affine symmetry, like the Wess–Zumino–Novikov–Witten theory for the group $SL(2,R)$, and, besides, it can be regarded as a special case (namely the case $N=1$) of the family of $A\_$ Toda field theories. Liouville theory also admits supersymmetric extension. == See also == * Liouville gravity * Liouville's theorem * Conformal field theory * String theory 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「Liouville field theory」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.