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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 conformal symmetry ： ウィキペディア英語版 conformal symmetry In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. Conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation. Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. ==Generators and commutation relations== The conformal group has the following representation: : &P_\mu \equiv-i\partial_\mu \,, \\ &D \equiv-ix_\mu\partial^\mu \,, \\ &K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu) \,, \end where $M\_$ are the Lorentz generators, $P\_\backslash mu$ generates translations, $D$ generates scaling transformations (also known as dilatations or dilations) and $K\_\backslash mu$ generates the special conformal transformations. The commutation relations are as follows:〔 : &()=iP_\mu \,, \\ &()=2i\eta_D-2iM_ \,, \\ &(M_ ) = i ( \eta_ K_ - \eta_ K_\nu ) \,, \\ &() = i(\eta_P_\nu - \eta_P_\mu) \,, \\ &() = i (\eta_M_ + \eta_M_ - \eta_M_ - \eta_M_)\,, \end other commutators vanish. The definition of the tensor $\backslash eta\_$ is omitted. Additionally, $D$ is a scalar and $K\_\backslash mu$ is a covariant vector under the Lorentz transformations. The special conformal transformations are given by :$$ x^\mu \to \frac where $a^$ is a parameter describing the transformation. This special conformal transformation can also be written as $x^\backslash mu\; \backslash to\; x\text{'}^\backslash mu$, where :$$ \frac'^\mu}^2}= \frac - a^\mu, which shows that it consists of an inversion, followed by a translation, followed by a second inversion. In two dimensional spacetime, the transformations of the conformal group are the conformal transformations. There is an infinity of them. In more than two dimensions, Euclidean conformal transformations map circles to circles, and hyperspheres to hyperspheres with a straight line considered a degenerate circle and a hyperplane a degenerate hypercircle. In more than two Lorentzian dimensions, conformal transformations map null rays to null rays and light cones to light cones with a null hyperplane being a degenerate light cone. 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「conformal symmetry」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.