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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?

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    $\begingroup$ related (possible dup?) physics.stackexchange.com/q/221881/84967 $\endgroup$ Mar 12, 2020 at 21:37
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    $\begingroup$ A projective rep of $SO(2)$ is described by any real number, not necessarily rational. This is because the universal cover of $SO(2)$ is $\mathbb R$. For $n>2$, the fundamental group of $SO(n)$ is $\mathbb Z_2$, which means there are two congruence classes of irreps: bosonic/fermionic (aka linear/projective, or tensor/spinor, etc.). For $n=2$, one has $\pi_1SO(2)=\mathbb Z$, thus an infinite number of congruence classes. $\endgroup$ Mar 12, 2020 at 21:41
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    $\begingroup$ Sorry, I misspoke! You do not want a rep of SO(3). You want a rep of ISO(3,1), i.e., of Poincare! (Let's just step back for a minute, :: sips coffee :: what we want is a unitary rep of the symmetry group of your theory, to wit, the Poincare group in four dimensions). How do you construct a rep of ISO(3,1)? You induce it from a rep of the little group of some reference momentum. This little group is SO(2). So you take a rep of SO(2), and use it to build a rep of ISO(3,1). Now the universal cover of Poincare is a two-cover, so $4\pi$ has to be trivial, and the rest follows. Is this clear? $\endgroup$ Mar 13, 2020 at 14:58
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    $\begingroup$ In $d=3+1$ the massless little group is indeed SO(2), and its irreps are indeed classified by a continuous real number. So the mathematics does not forbid massless particles with continuous spin, but usually such representations are thrown out the window since we have not observed such particles in nature. At least that is always how I've seen the argument go (pretty sure Weinberg argues this way in QFT vol 1). In fact, there are many modern research groups who are looking closer into the continuous reps of massless particles (e.g. arxiv.org/abs/1805.09706 and refs therein). $\endgroup$ Mar 15, 2020 at 5:40
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    $\begingroup$ @Student The "tangling picture" is how you prove that $\pi_1SO(2)=\mathbb Z$ and $\pi_1 SO(n)=\mathbb Z_2$ for $n>2$, or at least the intuitive idea behind the formal proof. The "two ideas" are basically the same: the "room for tangles to untangle" is nothing but a fact about the topology of the rotation group! $\endgroup$ Mar 15, 2020 at 15:00

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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.

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  • $\begingroup$ Good answer! Indeed, I should mention which theoretical framework I was considering. Otherwise, the ultimate answers in physics are always experimental. $\endgroup$
    – Student
    Mar 15, 2020 at 14:03
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    $\begingroup$ The question is clearly about any group theoretical reasons that particles are either bosons or fermions in d>2 (if the OP was asking about the QFT describing our universe the OP would not ask about d>2). The rest of the question hints at the same. So most of your answer about the specific QFT that might describe our universe is irrelevant. Are you just saying: There are no group theoretical reasons and that other representations (anyons) are allowed (but happen to be unobserved). That seems to just be outright wrong (see en.wikipedia.org/wiki/Anyon). $\endgroup$
    – Kvothe
    Jul 22, 2021 at 8:10
  • $\begingroup$ @Kvothe "This is the experimental physics answer:" Did you notice this ? $\endgroup$
    – anna v
    Jul 22, 2021 at 8:17
  • $\begingroup$ That is like saying this is the dermatological answer. The answer is completely irrelevant to this group theory question! (The only way an experimental answer could be useful was if it gave an example of the existence of a representation that was in question. On the other hand the absence of a field transforming in a certain representation in experiments says absolutely nothing about whether such an irrep could exist.) $\endgroup$
    – Kvothe
    Jul 22, 2021 at 8:21
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    $\begingroup$ Well this is clearly what you would call a "mathematics" question then. It does not ask what representations have been seen in our universe. It asks for d>2 (and so definitely more general than our universe) whether certain representations can exist. The answer is: No they cannot for group theory reasons. And yet you answer: Yes it is possible but it hasn't been detected. $\endgroup$
    – Kvothe
    Jul 22, 2021 at 12:36

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