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I've been about particles called anyons which exist within a two dimensional framework. I've also found out that these particles can have an angular momentum equal to any real number. Normally, in quantum physics in a three dimensional space angular momentum can only take values equal to $j(j+1)$ where $j\in\mathbb{N}$. I've asked around and the I believe this answer probably gives the best reason why this is true:

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.

Now, I coming from a pure math background and I've only recently begin to do research in physics. What are the topological 'reasons' for why this is true? Why is this not possible in a 2 dimensional space.

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    $\begingroup$ I'm not seeing a mathematical question here. It looks like your question has more to do with a specific model, and you haven't quite made clear what model you're thinking about. $\endgroup$ May 29, 2014 at 22:16
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    $\begingroup$ you can find the answer on Wikipedia: en.wikipedia.org/wiki/Anyon#Topological_basis $\endgroup$ May 30, 2014 at 8:09

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