quantum mechanics - direct sum of anyons?

In the topological phase of a fractional quantum Hall fluid, the excitations of the ground state (quasiparticles) are anyons, at least conjecturally. There is then supposed to be a braided fusion category whose irreducible objects are in 1-1 correspondence with the various types of elementary quasiparticles.

The tensor product of objects has an obvious physical meaning: it's the operation of colliding (fusing) quasiparticles...

... but what about direct sum?

• The tensor product of two irreducible objects might be a direct sum of irreducible ones: what does this means physically in terms of the outcome of a collision of quasiparticles?

• Let $X$ be an irreducible object of the fusion category. Is there any physical difference between (the physical states corresponding to) $X$ and to $X\oplus X$?

Answer

The simple objects in the braided fusion category correspond to the possible particle types. In the simplest important example there are two particle types 1 and $\phi$. (Well, 1 is the vacuum so it's a slightly odd sort of particle type.)

The non-simple objects don't have any intrinsic physical meaning, $\phi \oplus \phi$ just means any system "that can be a single particle but in two different ways" but makes no claims about what those two different ways are.

Tensor product of simple objects does have an intrinsic meaning, it means looking at a system with several particles in it.

Since the underlying category only has finitely many objects, any time you have a multi-particle system you can break up the Hilbert space as a direct sum of states where you've fused them all together into a single particle (either 1 or $\phi$). For example, since $\phi \otimes \phi \otimes \phi \cong \phi \oplus \phi \oplus 1$ this means that the Hilbert space for the 3 particle system is 3-dimensions, and splits up into a two-dimensional space of things that behave like a single particle (this is the $\phi \oplus \phi$ part) and a one-dimensional space of things that behave like the vacuum (this is the 1 part). In this case $\phi \oplus \phi$ has a physical meaning imbued by virtue of its appearing as a summand of $\phi^{\otimes 3}$, but other appearances of $\phi \oplus \phi$ inside other tensor products have different physical meanings.

In general, the Hilbert space assigned to the system of k particles $X_{a_1} \otimes X_{a_2} \otimes \ldots \otimes X_{a_k}$ is the direct sum over all particle types $X_i$ $$\bigoplus_{X_i} \mathrm{Hom}(X_{a_1} \otimes X_{a_2} \otimes \ldots \otimes X_{a_k}, X_i).$$