A Vector Matrices Realization of Hurwitz Algebras

• 作者: Daniel Sepunaru
• 来源期刊: 现代物理(英文)
• 学科: 数学
• 关键词: HILBERT SPACES HURWITZ ALGEBRAS Zorn MULTIPLICATION
• 年，卷（期）: 2018,（14）
• 页码范围: 2370-2377
• 页数: 8页
• 分类号: O1

摘要

We present here a realization of Hurwitz algebra in terms of 2 × 2 vector matrices which maintain the correspondence between the geometry of vector spaces that is used in the classical physics and the algebraic foundation underlying quantum theory. The multiplication rule we use is a 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. Extension to the calculus of the matrices (with Hurwitz algebra valued matrix elements) of the arbitrary dimensions is straightforward. We briefly discuss applications of the obtained results to extensions of standard Hilbert space formulation in quantum physics and to alternative wave mechanical formulation of the classical field theory.