# Topological proof of spin-statistics theorem confusion

I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as:

This is a little grab-bag of proofs of the spin-statistics theorem. Quantum mechanics says that if you turn a particle around 360°, its wavefunction changes by a phase of either +1 (that is, not at all) or -1. It also says that if you interchange two particles of the same type, their joint wavefunction changes by a phase of +1 or -1. The spin-statistics theorem says that these are not independent choices: you get the same phase in both cases! The phase you get by rotating a particle is related to its spin, while the phase you get by switching two goes by the funny name of "statistics". The spin-statistics theorem says how these are related.

Then, he states the proof as:

The "proof" is topological, in that if you rotate a thing (particle) with strings all over it, the strings are all twisted after 360 degrees of rotation, but after 720 degrees the strings can be untangled without moving the object. Similarly, if two things (particles) connected by lots of strings are interchanged, the strings are left twisted up exactly as if one particle had been rotated by 360 degrees. So the conclusion is that interchanging two particles is topologically indistinguishable from a rotation of one particle by 360 degrees — a particle which changes sign after a rotation will be antisymmetric with respect to pairwise interchange.

However, I think I am vastly misunderstanding this argument. First, I do not understand why, if particles were connected by strings, interchanging them would result in the strings being twisted. Secondly, I don't understand why after 720 degrees the strings can be untangled without moving the object. Third, which might be the general underlying confusion, I don't see what axis these rotations are about in general.

Is there a way to understand this argument more explicitly?

That these strings always become untangled after 720° rotations is the "topological" part: The rotation group $$\mathrm{SO}(3)$$ is not simply connected, but its double cover is. When we look at the orientation of objects over time, we're looking at paths in the rotation group because an orientation is specified by an ordered orthonormal basis (e.g. the three vectors normal to the cube faces).
The path traced out by a 360° rotation arrives back at the same orientation, i.e. is a loop in $$\mathrm{SO}(3)$$ but it is not null-homotopic. Meanwhile, the path traced out by a 720° rotation is, meaning there is a continuous deformation from this loop to the do-nothing loop. Moving along this homotopy gives a way to restore the original untangled state of the strings without rotating the cube again (see also WillO's explanation of this for the case of the plate trick here), so the strings are not really tangled after 720° rotations.