Dirac's trick: A simple demonstration that SU(2) double-covers SO(3) in group theory

Introduction

: Mathematical Lie groups are abstract objects that are usually difficult to visualise and understand. However, these are embedded in physics and quantum mechanics. In SU(2), a 720 degree rotation is needed to return an object back to its original orientation. While this seems bizzare, the SU(2) group is commonly seen in quantum mechanics when modelling fermions (e.g. electrons, neutrons, etc.).

If we consider a cube that is connected on all faces with an oriented belt, we see that we need 2 rotations in order to arrive in its initial orientation. 2 rotations in our real world SO(3) space is required to constitute 1 rotation of an SU(2) object.

Fixed Belts

Similarly, we can consider belts fixed at both ends. The belts with an odd number of twists can never be untwisted to become a belt with an even number of twists.

Let us first look an animation involving an odd number of twists:

Conversely, consider a belt with zero or an even number of twists:

Implications

In a sense, the fact that fermion wavefunctions are negated when fermions are interchanged correspond to the SU(2) algebra demonstrated above. Instead of returning to its original state when rotated once, it becomes its additive inverse. Two interchanges are required in order to reach its original state.

References

1. https://en.wikipedia.org/wiki/Plate_trick
2. Pengelley, David; Ramras, Daniel (2017-02-21). "How Efficiently Can One Untangle a Double-Twist? Waving is Believing!". The Mathematical Intelligencer: 1–14. doi:10.1007/s00283-016-9690-x. ISSN 0343-6993.
3. Bolker, Ethan D. (November 1973). "The Spinor Spanner". The American Mathematical Monthly. 80 (9): 977. doi:10.2307/2318771.