# The mathematics of anyons

I've recently learnt about particles called anyons which exist within a two dimensional framework. Which I find quite strange since we, well live in a three dimensional world. 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 come 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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This is not an answer, but rather a rather long comment which I hope is helpful.

I think you have three distinct inquiries. 1) We live in a 3+1 (space+time) dimensional world, how come there are particles called anyons which appear only in two dimensional framework? 2) Why do these particles can have any angular momentum, and not just the discrete values as in regular quantum mechanics? 3) In usual QM we have only two statistics, namely Bose-Einstein or Fermi-Dirac, the so called Spin-Statistics theorem, but anyons violate this theorem and possess different statistics. Why so?

1)Although we live in 3+1 there are experiments where one of the spatial dimensions is heavily constrained, be it by layers of materials or by strong external fields. Physics works differently if you start with only 2 dimensions, but to the extent of my knowledge there is not a complete theoretical argument explaining how confining one dimension leads to the appearance of anyons. On the other hand physicists have the advantage of being able to perform experiments. The fractional quantum Hall effect seems to be an instance of appearance of anyons, so we can assume that by constraining we do indeed have two dimensional physics.

Altland and Simons is a good reference for fractional quantum Hall effect and its relation to topology.

2)The angular momentum is easier to explain. The fact that usual quantum mechanics only admits $j(j+1)$ values is tied to the representation theory of $SO(3)$ (or $SU(2)$ in more generality), where $j$ is the weight of different representations. In two dimensions in order to find the allowed values of angular momentum one has to look instead at the representation theory of $SO(2)$ (respectively $U(1)$), and the abelian nature of this group accounts for the difference.