Why are particles either bosons or fermions in spatial dimensions $d>2$? (in Wigner classification) This questions might have been asked several times, but I haven't seen a mathematical point of view, so here it is.
Based on Wigner classfication: A particle is a representation, because any theory that describes a particle in a space must teach us how to describe the change of state as we change coordinates, e.g. gradually rotating the resting frame. Therefore, a massive particle is at least a (projective) representation of $SO(3)$, and a massless particle is at least a (projective) representation of $SO(2)$. In this question, I focus on the later.
A projective representation of $SO(2)$ can be described in terms of a rational number $\frac{r}{s} \in \mathbb{Q}$, so it is natural to consider massless particles of $1/3, 1/4$ .. etc. My question is, why not?
A typical answer I got from my physics friends and profs is that

Yes, you can consider it, but they only exist in $2+1$ space-time. This is because in $3+1$ or above, exchanging two particles draws you a tangle in a $4$-space, which is trivial!

I understand you can un-tangle any tangles in $4$-spaces. What I fail to see is the relation between this reason and my question. I was never considering two particles! Why would everyone tell me the picture with 2 particles winding around with each other (even wikipedia:anyon does that)?
After all, what $1/3$ really means mathematically is: if you focus on that single particle, and slowly change coordinates with that particle fixed at the origin, you will find the the state got changed by a scalar multiplication by $\exp(2\pi i/3)$ after a full turn. This, to me, seems to work in any dimension. What's the fundamental difference for $2+1$, without invoking that un-tangling business? Or do I miss something?
 A: The question and the comments discuss the mathematical models devised to describe particles. This is the experimental physics answer:
Physics theoretical models are  modeling measurements and predicting new situations.
The  answer to why are particles either bosons or fermions is that that is what we have observed in our measurements of particle interactions.
In order for angular momentum conservation to hold as a true law at the level of particle physics ( laws are physics axioms) when interacting and decaying, an angular momentum as specific as mass and charge had to be assigned to each particle. This has resulted in the symmetries seen in the quark model and the ability of the models to be predictive. They are validated continuously with any new experiments, up to now.
Thus the mathematical models you are discussing are necessary in order to fit data and observations and be predictive of new states.
If future experiments  discover new particles where the assignment of spin in order to obey angular momentum conservation in its interactions, need a different spin , a new quantum mechanical wave equation has to be found other than Dirac,Klein Gordon and quantized Maxwell to model its wavefunctions and keep angular momentum conservation as a  law.
