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 2⌊n/2⌋ × 2⌊n/2⌋ matrices. They generalize the Pauli matrices to n dimensions, and are a specific construction of higher-dimensional gamma matrices. 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 n dimensions may then be realized as the column vectors of size 2⌊n/2⌋ 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 n = 2k (or 2k+1) and suppose that the Euclidean quadratic form on V is given by
q 1 2 + ⋯ + q k 2 + p 1 2 + ⋯ + p k 2 ( + p n 2 ) , {\displaystyle q_{1}^{2}+\dots +q_{k}^{2}+p_{1}^{2}+\dots +p_{k}^{2}~~(+p_{n}^{2})~,}
where (pi, qi) are the standard coordinates on Rn. Define matrices 1, 1', P, and Q by
1 = σ 0 = ( 1 0 0 1 ) , 1 ′ = σ 3 = ( 1 0 0 − 1 ) , P = σ 1 = ( 0 1 1 0 ) , Q = − σ 2 = ( 0 i − i 0 ) {\displaystyle {\begin{matrix}{\mathbf {1} }=\sigma _{0}=\left({\begin{matrix}1&0\\0&1\end{matrix}}\right),&{\mathbf {1} }'=\sigma _{3}=\left({\begin{matrix}1&0\\0&-1\end{matrix}}\right),\\P=\sigma _{1}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right),&Q=-\sigma _{2}=\left({\begin{matrix}0&i\\-i&0\end{matrix}}\right)\end{matrix}}} . 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.
Even case In the case when n = 2k is even, let
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