Title: A Relationship Between Spin and Geometry Show Chinese title

Title: 自旋与几何之间的关系

Abstract: In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean three-space was drawn. However, the details of this relationship and the extent to which it can be developed by elementary means were not expounded. In this paper, we will reveal the geometric content of the spin algebras by realising them within a novel, generalised form of Clifford-like algebra. In so doing, we will demonstrate a natural connection between spin and non-commutative geometry, and discuss the impact of this on the measurement of hypervolumes and on quantum mechanics. Show Chinese abstract

Abstract: 在最近的一篇论文中，所有非相对论自旋的代数描述都是通过基本方法直接从李代数$\specialorthogonalliealgebra{3}$得出的，并且建立了自旋与欧几里得三维空间几何之间的联系。 然而，这种关系的细节以及通过基本方法可以发展到何种程度并未详细说明。 在本文中，我们将通过在一个新颖的、广义的类似克利福德代数中实现这些自旋代数，揭示自旋代数的几何内容。 在此过程中，我们将展示自旋与非交换几何之间的自然联系，并讨论这对超体积测量和量子力学的影响。