Hurwitz代数的向量矩阵实现
Vector Matrices Realization of Hurwitz Algebras
摘要：
我们给出了Hurwitz代数在2 × 2向量矩阵中的实现，它保持了经典物理中使用的向量空间的几何与量子理论的代数基础之间的对应关系。所用的乘法规则是对最初由M. Zorn引入的乘法规则的修改。我们证明，我们的乘法不是本质上非结合的实和复数的实现是交换和关联，真正的四元数保持结合性和真正的八元数矩阵形成一个替代代数。扩展到任意维数的矩阵（具有Hurwitz代数值的矩阵元素）的微积分是简单的。我们简要地讨论了所得结果在量子物理标准希尔伯特空间公式的推广和经典场论的交替波动力学公式中的应用。
英文摘要：
We present the realization of Hurwitz algebras in terms of 2x2 vector matrices, which maintain the correspondence between the geometry of the vector spaces used in the classical physics and the underlined algebraic foundation of the quantum theory. The multiplication rule used is modification of the one originally introduced by M.Zorn. We demonstrate that our multiplication is not intrinsically non-associative; the realization of the real and complex numbers is commutative and associative, the real quaternions maintain associativity and the real octonion matrices form an alternative algebra. The extension to the calculus of the matrices (with Hurwitz algebra valued matrix elements) of the arbitrary dimensions is straightforward. We discuss briefly the applications of the obtained results to the extensions of the standard Hilbert space formulation of the quantum physics and to the alternative wave mechanical formulation of the classical field theory.
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