By a curious physics student
Quantum theory: Topology of histories and spin-1/2
If we want to rotate an object in a particular way, it is always possible to effect the rotation by choosing an axis and rotating by an angle about that axis. Using Euler angles, we can parametrise any rotation by giving two angles to fix the unit vector that is the axis of rotation and a third angle that specifies by how much we want to rotate.
This means that the space of all orientation of a 3D object is a sphere of radius , as shown. A point on the sphere is an orientation of the object.
Now let’s suppose that I rotate my object through small angles – a little around the y-axis, a little around the z-axis etc. Then its trajectory in orientation space will be a small loop around the origin. The orientation doesn’t change very much.
Now suppose that I rotate the object by . Then the history of the object in orientation space will look like this:
This is a closed path, but it is not homotopic to zero. By this I mean the following: in the case of the green loop corresponding to small rotations (i.e. rotations close to the identity) I can continuously deform the loop and contract it to the identity. On the other hand, however I modify the trajectory for rotation by it always forms a complete loop. There is a topological difference between a history of the object where the object stays close to its original orientation and a history where the object makes a complete rotation around an axis.
I can deform the orange trajectory continuously to an orientation history that is the identity throughout. So a rotation by is topologically equivalent to no rotation at all.
This makes it in some way plausible that a rotation by is different from the identity, but then a rotation by must be the identity. Indeed, defining the operator that effects rotations about some fixed axis, we have for fermions and .