Understanding quaternions and the Dirac belt trick
Abstract
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 fourdimensional parameter space for rotations that is simply connected. An investigation into the geometry of this fourdimensional space leads directly to the system of quaternions, and to an interpretation of threedimensional vectors as the generators of rotations in this larger fourdimensional 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.
 Publication:

European Journal of Physics
 Pub Date:
 May 2010
 DOI:
 10.1088/01430807/31/3/004
 arXiv:
 arXiv:1001.1778
 Bibcode:
 2010EJPh...31..467S
 Keywords:

 Physics  Popular Physics;
 Physics  Physics Education
 EPrint:
 19 pages, 4 figures