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To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\mathfrak{so}(3)$. Now, $SU(2)$ also happens to be a double cover of $SO(3)$, but this seems coincidental/unimportant. It is precisely the universal cover properties which, together with some Lie theory and Bargmann's theorem, lead to a classification of the irreducible projective representations of $SO(3)$.

So, what is the physical importance of $SU(2)$ being the double cover of $SO(3)$? I prefer an answer in terms of the relevant mathematics as alluded to in the comments below this post.

Silly Goose
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1 Answers1

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It's actually Spin(n) which is the universal and double cover of SO(n) for n>=3.

However, it turns out that Spin(3) is isomorphic to SU(2). This is important because SU(2) is defined via complex 2 × 2 matrices. Thus we can use complex matrices to represent elements of Spin(3). This is how the Pauli matrices come into the picture.

Mozibur Ullah
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