As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the quasiparticals must be created or annihilated by pair.
On the other hand, consider the groups $SU(2)$ and $SO(3)$, they are the rotation groups for half-integer and integer spins, respectively. And we know that $SU(2)/\mathbb{Z}_2=SO(3)$, which means that each element in $SO(3)$ can be viewed as one pair $(U,-U)$, where $U\in SU(2)$ (otherwise put: the coset $\left\{U, -U\right\} \subset SU(2)$ in the quotient group $SU(2)/\mathbb{Z}_2$ is our element in $SO(3)$).
So I wonder that whether is there any underlying connection between the pair nature of quasiparticals in topological phase in physics side and the pair structure relating $SU(2)$ and $SO(3)$ in mathematics side?
Thank you very much.
Answer
The group of rotations of an $N$-dimensional space is $SO(N)$. Being a symmetry of nature, classical systems transform according to representations of $SO(N)$.
Quantum mechanics, on the other hand, allows systems which transform according to the universal covering groups of classical symmetries. This is the reason why we get in three dimensional quantum theory representations of $SU(2)$ which are not true representations of $SO(3)$, (the half integer spin representations). More generally, we have, in quantum theory, representations of $Spin(N) = SO(N) \ltimes \mathbb{Z}_2$.
However in the case of a two spatial dimensions, $SO(2) \cong U(1)$, and the universal covering of $U(1)$ is not $Spin(2)$ but rather $\mathbb{R}$.
In contrast to $SO(2)$ or $U(1)$ which allow discrete values of the two dimensional spin: $ u = e^{i n \theta}$ $n \in \mathbb{Z}$, $0\le \theta <2 \pi$, the universal covering $\mathbb{R}$ allows a continuum of spin values.
This is the basic reason of the fractionalization of spin in two dimensions.
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