In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-s Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.

在物理学中，自旋通常只能通过其现象学特征的镜头来看待：作为角动量的一种内在形式。然而，越来越多的证据表明，自旋从根本上起源于几何学的性质，而不是动力学的性质，并且最近的工作进一步表明，非相对论性欧几里得三空间的结构足以定义它。在本文中，我们通过构造包含任意自旋系统结构的位置算子的非交换代数，直接解释了自旋的这种基本非相对论几何性质。这些“自旋位置代数”是通过基本方法和欧几里得三空间的性质来定义的，并构成了具有非零自旋的量子力学系统的全新模型，其中位置和自旋自由度都没有，也不在自己内部设置自由度，通勤。这表明，具有自旋的系统的可观测量可以完全在几何上描述为定向平面元素的张量，并且系统中非零自旋的存在自然会在其中生成非交换几何。我们还将讨论自旋位置代数为 Clifford 和 Duffin-Kemmer-Petiau 代数的任意自旋的泛化奠定基础的潜力。