四元数、旋量与霍普夫纤维化：经典力学中的隐变量
Quaternions, Spinors and the Hopf Fibration: Hidden Variables in Classical Mechanics
摘要：
三维空间中的旋转同样可以用SU（2）和SO（3）群来描述。这些同构群使用单位四元数的不同代数结构生成相同的3D运动学。霍普夫纤维化是四维空间中四元数的超球面$\mathbb{S}^3$和三维空间中单位球面$\mathbb{S}^2$之间的投影。$\mathbb{S}^3$中的大圆通过6个霍普夫映射映射到$\mathbb{S}^2$中的点，并通过赤平投影示出。高、低维空间通过$\mathbb{S}^1$纤维丛连接，该纤维丛由全局相、几何相和动力相组成。整体相位被量化为$2π$的整数倍，并呈现为经典力学的自然隐变量。
英文摘要：
Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection between the hypersphere $\mathbb{S}^3$ of the quaternion in 4D space, and the unit sphere $\mathbb{S}^2$ in 3D space. Great circles in $\mathbb{S}^3$ are mapped to points in $\mathbb{S}^2$ via the 6 Hopf maps, and are illustrated via the stereographic projection. The higher and lower dimensional spaces are connected via the $\mathbb{S}^1$ fibre bundle which consists of the global, geometric and dynamic phases. The global phase is quantized in integer multiples of $2π$ and presents itself as a natural hidden variable of Classical Mechanics.
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