Understanding quaternions and the Dirac belt trick
Understanding quaternions and the Dirac belt trick
复制标题
理解四元数和狄拉克带技巧
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Mark Staley
中科院分区：
文献类型：
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作者：
Mark Staley
狄拉克带技巧经常在物理课堂上使用，以表明2π旋转不等于没有旋转，而4π旋转是拓扑等价的，反映了四元数及其同构表兄弟旋量的一个关键性质。腰带戏法可能会让学生怀疑是否已经实现了对四元数和旋量的真实的理解，或者这个戏法只是一个有趣的类比。本文的目标是揭开腰带戏法的神秘面纱，并表明它暗示了一个简单连通的旋转四维参数空间。对这个四维空间的几何学的研究直接导致了四元数系统，并解释了三维向量作为这个更大的四维世界中旋转的生成元。本文还说明了为什么四元数是复数到四维的自然扩展。论文的水平适合物理学本科生。
The Dirac belt trick is often employed in physics classrooms to show that a 2π rotation is not topologically equivalent to the absence of rotation whereas a 4π rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it suggests an underlying four-dimensional parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.