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I'm doing my master's degree and I'm starting to learn more about Anyons. I want to understand more deeply why they can exist and how. I've done some research on the internet and found this question here that was quite enlightening. From the references I took a read on this article (Leinaas and Myrheim 1977), and I also tried to read this one (Landsman 2013) but I think that the math is a little beyond my actual knowledge.

What I've understood (not necessarily correct):

When we are in 3D and 2D the topological representation of the configuration space is not simply connected, from (Leinaas and Myrheim 1977) it is shown that in 3D the space is doubly connected so there are two non-equivalent sets of paths in which the particles could be exchanged. As for 2D the configuration space is infinitely connected and there will be infinite sets of equivalent paths.

When Leinaas and Myrheim consider the quantization process they assign to each topological point (that corresponds to a set of equivalent states in the configuration space) a unidimensional Hilbert space. Any unit vector in this unidimensional space corresponds to a permutation-equivalent quantum state, and the effect of permutation is just a rotation around the complex unitary circle by $e^{i\theta}$.

In 3D a path encircling a singularity twice can be contracted to a point so in addition we require that the square of the permutation operator to be identity $P_x^2 = I$, so the rotations in the complex unitary circle can only be $e^{in\pi}$ with $n$ integer, leading only to bosonic and fermonic statistics. But in 2D it's not required and this gives space to the possibility of anyons existence.

My questions:

  1. Are those singularity points the points that represent the probability of finding two particles at the same location at the same time?

  2. Why do we need the concept of parallel displacement of vectors in the quantization process? What's the role of this idea in this process of quantization?

  3. There is no consideration of the nature of the identical particles, so why couldn't, for example, electrons behave like anyons in 2D or any other elementary particle? Why Anyons should be restricted to quasiparticles?

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    $\begingroup$ 1)- Singularity points are points where particles are in same state. In CM, that means being at same position in configuration space, for QM it may mean same position in phase space. 3)- Because electrons and other particles are excitations of fields in 3D which is our physical world. Excitations in 2D gives anyons. If our world was 2D, anyons could have been the fundamental particles instead of electrons and quarks. Please add to this as its not a complete answer. $\endgroup$
    – Siddharth Jain
    Commented May 21 at 19:04
  • $\begingroup$ Please narrow this down into a single question. $\endgroup$
    – Vincent Thacker
    Commented May 23 at 12:03

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