(1) As we know, we have theories of second quantization for both bosons and fermions. That is, let $W_N$ be the $N$ identical particle Hilbert space of bosons or fermions, then the "many particle" Hilbert space $V$ would be $V=W_0\oplus W_1\oplus W_2\oplus W_3\oplus...$ , and further we can define creation and annihilation operators which satisfy commutation(anticommutation) relations for bonsons(fermions).

So my first question is, do we also have a "second quantization theory" for anyons like bosons and fermions?

(2) Generally speaking, anyons can only happen in 2D. Is this conclusion based on the assumption that the particles are point-like?

In Kitaev's toric code model, the quasiparticles are point-like due to the local operators in the Hamiltonian. My question is, in 3D case, whether there exists a simple model whose Hamiltonian contains local operators and spatially extended operators, so that it has both poit-like quasiparticles(say, $\mathbf{e}$) and knot-like quasiparticles(say, $\mathbf{m}$), then the $\mathbf{e}$ and $\mathbf{m}$ particles have nontrivial mutual statistics in 3D?

• Regarding (1), anyon means that the wavefunction describing a two- or many-anyon state picks up a phase other than a multiple of $\pi$ upon exchange of two particles. To define a Fock space (i.e. many-particle Hilbert space), one needs to attach exclusion statistics to the constituent particles, which will act as a generalized Pauli principle. If you do that, then yes, you can devise a second-quantized theory of anyons, because the occupation-number basis would be well defined. – delete000 Apr 23 '13 at 3:37
• @ delete000 Thanks a lot. Do you mean the exclusion statistics is common to anyons(except bosons) like fermions? Do you have some related articles ? And do physicists now have a well developed theory of second quantization for anyons? – Kai Li Apr 23 '13 at 11:43
• Each type of anyons will have a different exclusion principle, which can be something between the Pauli principle (1 particle per state) and bosons (any number of particles per state). You may find this article useful: iopscience.iop.org/0305-4470/27/11/009 – delete000 Apr 23 '13 at 13:32
• Regarding your last point, there are indeed generalizations of Kitaev's toric code model with pointlike and stringlike quasiparticles. – Peter Shor Apr 27 '13 at 10:56
• This paper of Beni Yoshida proves theorems about a broad class of 2- and 3-dimensional generalizations of Kitaev's toric code. You may also want to look at the citations for earlier papers about these models. There are also a few papers about more general codes (corresponding to non-abelian anyons), but I can't easily locate these right now. – Peter Shor Apr 27 '13 at 20:09

No, “this conclusion” is based on the topological properties of rotation groups. Namely, for any $n > 2$ $\mathrm{Spin}(n)$ is the universal cover of $\mathrm{SO}(n)$, whereas for $n = 2$ it is not. That’s why in $n > 2$ any thing has to be controlled by a representation of the Spin group, whereas in $n = 2$ it has not.