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What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that rotating in real space. I think it is the state vector that rotates in spin space. Am I right? Is it the property of the "weird" spin space (Hilbert space) that rotation by $4\pi$, brings it back to where it started?

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Most of this question is already addressed from more than one point of view in answers (and comments) to the OP's related question:

Idea of Covering Group

In particular, two approaches to seeing why $\mathrm{SU}(2)$ arises are discussed in detail. The first uses the idea of projective representations, and the second involves algebra of quantum observables.

However, there is one part of the question that is not directly addressed therein.

What really rotates and which space it rotates?...I think it is the state vector that rotates in spin space. Am I right?

Yes, you are right. When you rotate a spin-1/2 system in the real world, this corresponds to acting on the quantum state of the system with a unitary operator that represents that rotation on the Hilbert space.

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