Defining exchange statistics of anyons in terms of Berry phase

[throneoo]

In 2D, if we define exchange statistics in terms of the phase change of the wavefunction of two identical particles when there are exchanged via adiabatic transport (https://arxiv.org/abs/1610.09260), we would discover that this phase can be arbitrary due to the topology of relative configuration space in 2D. (in 3D the phase is either 0 or pi)

However, what I'm not entirely clear about is the mechanism of the generation of such a phase. Since I'm not at all familiar with the path integral formulation of quantum mechanics, I am trying to understand it purely in the Hamiltonian formalism. According to this document (http://users.physik.fu-berlin.de/~pelster/Anyon1/hansson.pdf), we can interpret the phase as the Berry phase that arises when we adiabatically exchange these particles by varying the localized potential traps for real.

That is all fine. However what bothers me is that when we move the particles around, they follow classical trajectories. The reason why true classical indistinguishability isn't well defined is because we can distinguish the particles by their non-intersecting trajectories, which is precisely the case here. In short, I feel like we are not dealing with quantum-mechanically indistinguishable particles anymore, and that it's not an entirely correct formulation of quantum statistics.

 

[azntoon]

So my question is, how can we explain the emergence of an arbitrary phase when exchanging two identical particles in 2D, in terms of the Hamiltonian formalism? Is my understanding of the situation correct?