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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 Representation theory of SU(2) ： ウィキペディア英語版 Representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1. SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter. This also specifies importance of SU(2) for description of non-relativistic spin in theoretical physics; see below for other physical and historical context. As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a integer or half-integer , with dimension . ==Lie algebra representations== The representations of the group are found by considering representations of $\backslash mathfrak(2)$, the Lie algebra of SU(2). In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups consist of 'integrated' transformations. In what follows, we shall consider the complex Lie algebra (i.e. the complexification of the Lie algebra), which doesn't affect the representation theory. The Lie algebra is spanned by three elements , and with the Lie brackets :$()=e$ :$()=-f$ :$()=h$ (These elements may be expressed in terms of matrices , and which are related to the Pauli matrices by multiplication by a factor of . , , and .) Since $\backslash mathfrak(2)$ is semisimple, the representation is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights. Its eigenvectors can be taken as a basis for the vector space the group acts upon. The dimension of the representation can be determined by counting the number of these eigenvectors. Suppose is an eigenvector of weight . Then, :$h()=\backslash alpha\; x$ :$h$ :$h$ In other words, raises the weight by one and reduces the weight by one. and are referred to as ladder operators, taking us between eigenvectors or to 0. A consequence is that :$h^2+ef+fe$ is a Casimir invariant and commutes with the generators of the algebra. By Schur's lemma, its action is proportional to the identity map, for irreducible representations. It is convenient to write the constant of proportionality as . (The expression $h^2+ef+fe$ is equal to $I^2$ defined as $I\_1^2\; +\; I\_2^2\; +\; I\_3^2$, which is related to the magnitude of angular momentum operator in quantum physics.) 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「Representation theory of SU(2)」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.