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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 Spinors in three dimensions ： ウィキペディア英語版 Spinors in three dimensions In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3). ==Formulation== This algebra admits a convenient description, due to William Rowan Hamilton, by means of quaternions. In detail, given a vector x = (''x''1, ''x''2, ''x''3) of real (or complex) numbers, one can associate the matrix of complex numbers: : Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space: * det ''X'' = - (length x)2. * ''X''2 = (length x)2''I'', where ''I'' is the identity matrix. * $\backslash frac(XY+YX)=(\backslash cdot)I$ * $\backslash frac(XY-YX)=iZ$ where ''Z'' is the matrix associated to the cross product z = x × y. * If u is a unit vector, then ''−UXU'' is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u. * It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if ''R'' is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u1 followed by the plane perpendicular to u2, then the matrix ''U''2''U''1''XU''1''U''2 represents the rotation of the vector x through ''R''. Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector :$\backslash xi=\backslash left(),$ with complex entries ''ξ''1 and ''ξ''2. The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if ''X'' → ''RXR''−1 is a representation of a rotation, then replacing ''R'' by -''R'' will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always ''double-valued.'' 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「Spinors in three dimensions」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.