Elementary texts on quantum mechanics justify the existence of fermions and bosons using the simple argument that if we have a state of two indistinguishable particles $|a,b \rangle$, where $a$ and $b$ each label all quantum numbers of the two particles, then

$$ |a,b \rangle = \pm |b,a \rangle$$

This means that our states are (anti)symmetric under exchange of quantum numbers.

Well, in the literature on anyons, it is stated the act of swapping particles should really be though of a physical adiabatic exchange of particles through spacetime. This leads us to the Braid group and anyons when we discuss particles in $(2+1)$-dimensions .

My Confusion

Suppose our anyons had internal degrees of freedom such as spin. I could distinguish between my otherwise identical particles by virtue of their internal states, i.e. if I had to electrons of opposite spin, I could distinguish between them by measuring their spin states. In this case, my braiding through spacetime should no longer pick up a phase because the before and after states are distinguishable, and hence not in the same ray. As anyonic statistics rests on the fact that we're dealing with indistinguishable particles it will therefore break down.

My Question

As stated above, discussions of fermions and bosons (a subset of anyons) makes it very clear that we have to swap all the properties of our particles for the before and after states to be truly indistinguishable. Should the braiding of anyons not just involve braiding of anyons in spacetime, but also the braiding of anyons within their internal spaces? Why does it appear to me that anyonic statistics only arise from spatial degrees of freedom in the literature?


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