Why is it true that anyons can have angular momentum taking any real value? Why aren't they restricted to the $j(j+1)$ integer values most are familar with?
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Starting in a 3 dimensional space, any path where one particle traces a closed loop around another can be trivially contracted to a point where no motion occurred. This then means that the wavefunction before and after the motion must be the same and so the wavefunction can only be multiplied by a phase of $e^{i2\pi n}$ where $n$ is an integer. In 2 dimensions, however, the closed path around another particle cannot be contracted to a point. Thus, the wavefunction does not need to return to its original form and may be multiplied by a phase of the form $e^{i\theta}$ where $\theta$ is a real number.
More detail is available in the book Anyons by Jon Magne Leinaas.
