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.

This post imported from StackExchange Physics at 2014-03-30 15:49 (UCT), posted by SE-user David Bar Moshe