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I have been studying anyons and I have found the algebraic approach rather abstract and I am struggling to understand it as it seems quite different to the usual procedure of quantum mechanics.

  1. I do not understand why the Hilbert spaces describing anyonic states are labelled with a particular fusion process. Why aren't fusion processes represented by operators on the Hilbert space $\mathcal{H}$ of our system? In other words, if I was to fuse anyons $a$ and $b$ to produce $c$, my first thought would be I would have some initial state $|a,b\rangle \in \mathcal{H}$ which is then mapped to $|c \rangle \in \mathcal{H}$. Why isn't this done? What if I didn't want to fuse my anyons, then what state would they be represented by?
  2. If I accept that anyonic states are represented by fusion processes, then for the case of three anyons $a,b$ and $c$, why are the states $|(ab)c \rightarrow d \rangle$ and $|a(bc) \rightarrow d \rangle$ not considered orthogonal and distinguishable? It seems that we are happy to say that the intermediate fusion pathways are orthogonal (despite having the same outcome) but fusing the initial anyons in a different order isn't. Why is this?

Reference: Section 9.12 of John Preskill's quantum computing lecture notes

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The way I think of fusion is as an anlogue of tensor producting group representations. Indeed in the case of affine Lie groups it is exactly that. The physical systems that I understand best are the exotic quantum Hall phases where the anyons are generalized quasiholes. If you have a droplet of QHE fluid containing quasiholes, the the surface of the droplet carries an infinite representation of the affine algebra as the Hilbert space of its edge states. That infinite dimensional representation is the result of combining the represtations of the algebra on the inner edge of the quasiholes via the geometric co-product. In this algebraic picture the chiral vertex operators that create the quasiholes in he fluid are labeled by integrable representations of the affine algebra and their matrix elements are infinite dimensional analoges of the Clebsh-Gordan coefficients. The $F$ matrices that give you the morphisms between $(ab)c$ and $a(bc)$ are then direct analogues of the 6-j symbols familiar from the SU(2) anguar momentum "addition" rules.

Although I am a physicst, this is one of the areas in which I find the mathematicians account of how the Hilbert space stuff works is much clearer. I was really puzzled by it all until I read the first three chapters of the thin little book by book Kohno called "Conformal field theory and topology". When you read his precise definition of a conformal block, you immediately recognize that this is a mathematical account of how a physical QHE quasihole carries an entire Hilbert space along with it.

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