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There have been previous questions on this, for example see this and this question, but my question is different.

I get that in 2+1D, mathematically speaking, exchanging two identical particles twice may not be equivalent to not exchanging the particles. All good. But since we do not live in a 2+1D world, but actually in a 3+1D world, why should I care about this fact? How can anyons exist in our world?

But recently, anyons have been been experimentally confirmed. Although I do not understand these discoveries, I am ready to accept that in many body physics, emergent particles with non-trivial properties can arise. So, anyons may serve as a model to efficiently study these systems, with the realization that one could also have, in principle, studied these systems in terms of Fermions and Bosons, without mentioning anything about anyons.

But if I am studying a system with 3 or 4 fermions, will I really run into an issue if I do not know what anyons are, given that my system lives in a 3+1D world? That is, if my system is simple enough that I can correctly treat the particles as Fermionic and Bosonic, will there be any issue, given that we do not live in 2+1D?

Qmechanic
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  • Even with 3 or 4 fermions, you're going to have to resort to approximations to match experiment. So, you cannot use a mathematical concept of "truth" here. – John Doty Feb 19 '24 at 14:27

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The short answer is that no, you will in principle never run into issues describing your system with only fermions and bosons, as physical reality indeed only consists of these two particles.

Whether this is a feasible description is a different question. Usually, quasiparticles such as anyons are introduced to simplify the description of systems of strongly interacting particles by treating them in terms of weakly interacting quasiparticles instead. Strong interactions often make calculations practically impossible to perform, while we have plenty of tools to treat weak interactions.

That anyons have been discovered means that we have found a system where the anyonic description is an accurate one. While it is indeed only an approximate description, it is a very accurate one for that system, and successfully predicts a number of interesting physical phenomena exhibited by the system.

Codename 47
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    Our description of reality in terms of fermions and bosons is also only approximate. "All models are wrong, but some models are useful". – John Doty Feb 19 '24 at 13:58
  • "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful"-George Box – hyportnex Feb 19 '24 at 14:42
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    I’m not sure about saying that the anyonic description is “approximate”. The statement that the low-energy excitations of say, a Fractional Quantum Hall system are anyons is exact in the thermodynamic limit, at least. – Dominic Else Feb 19 '24 at 14:55