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In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of matrices. They generalize to dimensions the Pauli matrices which relate to 3-dimensional Euclidean space. They are named for Richard Brauer and Hermann Weyl,〔.〕 and were one of the earliest systematic constructions of spinors from a representation theoretic standpoint. The matrices are formed by taking tensor products of the Pauli matrices, and the space of spinors in dimensions may then be realized as the column vectors of size on which the Weyl–Brauer matrices act. ==Construction== Suppose that ''V'' = Rn is a Euclidean space of dimension ''n''. There is a sharp contrast in the construction of the Weyl–Brauer matrices depending on whether the dimension ''n'' is even or odd. Let = 2 (or 2+1) and suppose that the Euclidean quadratic form on is given by : where (''p''i, ''q''i) are the standard coordinates on R''n''. Define matrices 1, 1'=''σ''3, ''P''=''σ''1, and ''Q''=−''σ''2 by :. In even or in odd dimensionality, this quantization procedure amounts to replacing the ordinary ''p'', ''q'' coordinates with non-commutative coordinates constructed from ''P'', ''Q'' in a suitable fashion. 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「Weyl–Brauer matrices」の詳細全文を読む スポンサード リンク
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