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I am an undergraduate wanting to understand anyons (in order to understand topological quantum computing). The foundational observation upon which the existence of anyons rests seems to be that the effect of permutations on identical systems is of a different nature when considering space of dimension $d \leq 2$ versus when considering space of dimension $d > 2$.

In order to even say "is of a different nature" we must first have a notion of what happens when we permute an identitcal system embedded in 3D space. In chapter 7 of Sakurai's Modern Quantum Mechanics 3ed., it is presented as a trivial result that acting on a system of identical particles with the permutation operation twice is the same as applying the identity operator. The result is of course trivial given that we accept the definition Sakurai provides for the permutation operator. However, one of the original papers on anyons finds fault with this definition (https://www.ifi.unicamp.br/~cabrera/teaching/referencia.pdf).

Question: So, what is the rigorous, theoretical explanation for what the permutation operator is and how its action depends on the dimension of space we are considering?

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  • $\begingroup$ Any treatment of anyons would talk about this different nature, because we have the spin-statistics theorem that guarantees that 3+1D, we just have fermions and bosons, and this is the basic stuff we care most about. Spin-stats thm is a complicated thing that we are sure of its truth, but have no idea how to intuitively understand it, as Feynman famously lamented about it. That is, we are sure we are correct about fermions and bosons and the permutation operator, but leave open the possibility of anyons for 2+1D and lower. $\endgroup$
    – naturallyInconsistent
    Commented May 9, 2023 at 2:44
  • $\begingroup$ There are several posts here regarding (the existence of) anyons, which, IRRC; also talk about the nature of this particle permutation. Other than that, mathematically the situation is clear - or is this your question? $\endgroup$
    – Tobias Fünke
    Commented May 9, 2023 at 2:45
  • $\begingroup$ @TobiasFünke Would it be possible to point me towards the posts whose answers mathematically describe the permutation operation. I find Sakurai's explanation incorrect in light of the linked paper and to find a correct explanation is my primary goal. I assume the existence of anyons should follow from a precise and sufficiently general definition of the permutation operation, so the existence of anyons is not my direct interest with this question. $\endgroup$
    – Silly Goose
    Commented May 9, 2023 at 2:48
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    $\begingroup$ Ah I see. I am interested to know the more general kind of permutation operator that is consistent with the Sakurai case and the anyons case @naturallyInconsistent. $\endgroup$
    – Silly Goose
    Commented May 9, 2023 at 3:19
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    $\begingroup$ Yes, I know that, which is why my first comment left that open. I was only commenting for that one important technicality that we must first get correct. I did not want you to be, in the rare chance, left in limbo over the correctness of basic quantum theory. $\endgroup$
    – naturallyInconsistent
    Commented May 9, 2023 at 4:06

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