Spin 1/2 as belt trick in a smooth field In the (English) Wikipedia article on Spinor, there is an animation, demonstrating the Dirac belt trick as a model for Spin 1/2.
My interpretation of that animation goes like this: If you rotate an object within space by 360°, obviously in the end you don't change anything. If, on the other hand, you have a field, the situation is a different one. Imagine, you grab a curtain at one point and twist your hand. After a rotation by 360°, the curtain is by no means in its original state, instead you twisted the curtain round your finger. I.e. a rotation of a signle point of a field by 360° is by no means the identity. There is a (non homeomorphic-to-$\mathbb{R}^3$) way, though, to smoothly connect a single point to it's vicinity that admits rotation of that point without coiling up the vicinity. This construction exactly yields the 720° symmetry that spin 1/2 particles have.
My questions are:

*

*Is this a viable explanation for spin or merely suitable as a loose visualization of a 720° symmetry?

*If it is a viable microscopic description of a spin field, how do $SU(2)$ and spinor fields emerge from this microscopic description?

 A: From a mathematical standpoint, the belt trick should really be understood as a path inside the space of rotations $SO(3)$. You can continuously parametrize a path of rotations (around some axis) from 0º (the identity transformation) all the way up to 360º (the identity transformation again). The belt trick is a visualization of the fact that this path of rotations cannot be deformed, within $SO(3)$, to a constant path; in terms of the belt trick, it means you cannot wiggle the (once-twisted) belt so that it is untwisted (the twisting throughout the belt represents each stage in your path through rotation space -- the untwisted belt would then represent a constant path). But if you run through that path twice, it turns out that it can be deformed to a constant path -- this is the belt trick.
Now $SU(2)$ is what topologists call the double cover of $SO(3)$, which means it "unwinds" this closed loop (the path from 0º rotation up to to 360º rotation) into a non-closed path. In $SU(2)$ we can then understand that what looked like a closed loop inside $SO(3)$ in some sense "doesn't really land us back where we started". More specifically, there is a morphism $SU(2) \to SO(3)$ such that if we take our closed loop described above, break it up into many little pieces, and then "lift" each piece up to $SU(2)$ (which can be done because that morphism is locally injective), we will end up with a non-closed path inside $SU(2)$ (which in this case will end at $-I$, provided we lifted the $0º$ rotation to $I$).
(NB: when I say "the path from 0º rotation up to 360º rotation", this is an imprecise short version of parametrizing the path within $SO(3)$ that at time $t$ rotates space (around some fixed axis) by $2 \pi t$ radians -- strictly speaking, a 360º rotation is just the identity transformation).
Physically, the spin of a spin 1/2 particle corresponds to the defining (also called spinor) representation of $SU(2)$ on $\mathbb{C}^2$ (given by multiplication of matrices). Since in this representation $I$ does not act the same as $-I$, it will follow that the path from 0º through 360º, lifted to $SU(2)$, will end in a group element that acts differently from the initial one.
