# Is it possible to make statements about bosonic/fermionic systems by taking the limit $\theta\to \pi$ or $\theta\to 0$, of an anyonic system?

One might naïvely write the (anti-)commutation relations for bosonic/fermionic ladder operators as limits

$$\delta_{k,\ell} = \bigl[ \hat{b}_{k}, \hat{b}_{\ell}^\dagger \bigr] = \hat{b}_{k} \hat{b}_{\ell}^\dagger - \hat{b}_{\ell}^\dagger \hat{b}_{k} = \lim_{\theta\to\pi} \Bigl( \hat{b}_{k} \hat{b}_{\ell}^\dagger + e^{i\theta}\cdot\hat{b}_{\ell}^\dagger \hat{b}_{k} \Bigr)$$ $$\delta_{k,\ell} = \bigl\{ \hat{c}_{k}, \hat{c}_{\ell}^\dagger \bigr\} = \hat{c}_{k} \hat{c}_{\ell}^\dagger + \hat{c}_{\ell}^\dagger \hat{c}_{k} = \lim_{\theta\to 0} \Bigl( \hat{c}_{k} \hat{c}_{\ell}^\dagger + e^{i\theta}\cdot\hat{c}_{\ell}^\dagger \hat{c}_{k} \Bigr).$$ I.e. as limits of Abelian anyonic commutation relations. Assuming now that some system could be solved for anyons with $0 < \theta < \pi$, would taking the limits of e.g. the energy eigenstates for $\theta\to \pi$ yield in general the correct eigenstates of the bosonic system (which might be harder to solve directly)?

I'm inclined to think it would work, but after all, the whole Fock space looks different depending on $\theta$, with all kinds of possible topological nontrivialities.

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Quite honestly, I've never actually dealt with anyons, so I can't much judge what you say here. It would be nice if you could make your points a bit clearer. — Anyway, the main question wasn't really about Fock space / creator/annihilator commutations, but about the $\theta\to k\pi$-limit of observables in anyonic systems. How about that? –  leftaroundabout Oct 31 '12 at 23:33
Fock space picture only lead to totally symmetric or total antisymmetric many-body wave-function. So it fails on anyons. Also, $\theta$ is always a rational number for anyons and the properties of an anyon system is not a continuous function of $\theta$. So we cannot take the limit. –  Xiao-Gang Wen Nov 1 '12 at 6:05