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The best way to answer the question "How are anyons possible" is to use the "dynamical" path integral formalism, rather than the "static" wave function formalism. The permutation group action on the wave function is "static" in the sense that only initial and final states are specified. It will be ambiguous if there are more than one non-equivalent ways to ...

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The point $(11)$ is not correct, by doing $2$ successive "exchanges", you may have a global phase factor, such as $\psi'(x,y) = e^{i\alpha}\psi(x,y)$. The two wave functions describe the same physical state. The correct considerations are topological, inside of considering a discrete operation, consider a continuous operation, so that it is equivalent to ...

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How to obtain this braiding matrix from Non-Abelian Chern-Simon theory? To obtain braiding matrix $U^{ab}$ for particle $a$ and $b$, we first need to know the dimension of the matrix. However, the dimension of the matrix for Non-Abelian Chern-Simon theory is NOT determined by $a$ and $b$ alone. Say if we put four particles $a,b,c,d$ on a sphere, the ...

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The (unitary) "phase" factor for non-Abelian anyons satisfies the (non-Abelian) Knizhnik-Zamolodchikov equation: $$\big (\frac{\partial}{\partial z_{\alpha}} + \frac{1}{2\pi k} \sum_{\beta \neq \alpha} \frac{Q^a_{\alpha}Q^a_{\beta}}{z_{\alpha} - z_{\beta}}\big )U(z_1, ....,z_N) = 0$$ Where $z_{\alpha}$ is the complex plane coordinate of the particle ...

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Let me start from the beginning, also making explicit what I understand from user35388's answer. Consider a system of a couple of identical particles. Their states are pictured by normalized wavefunctions $\psi(x,y)$. However this representation is not one-to-one (this holds for every quantum system): states are wavefunctions up to phases. That is $\psi$ ...

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Check out the section about Chapter 17 Identical particles in Ballentines, he not only points out why looking at the Permutation operators of two particles in a multi particle setting is misleading but also discusses some errors in previous claims.

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