A question about the anyon Let's say we have two indistinguishable particles at positions (two spatial dimensional) $x^i_1$ and $x^i_2$ at some initial time and end up at positions $x^f_1$ and $x^f_2$ a time $T$ later. In the normal path integral representation for $\langle x^f_1,x^f_2|e^{-iHT}|x^i_1,x^i_2\rangle$ we have to sum over all paths. Now the paths can be divided into topologically distinct classes. Then it is stated that in A.Zee's QFT in a Nutshell that

Since the classes cannot be deformed into each other, the corresponding amplitudes cannot interfere quantum mechanically, and with the amplitudes in each class we are allowed to associate an additional phase factor $e^{i\alpha_n}$ beyond the usual factor coming from the action.

I find this statement hard to understand. Why the amplitudes corresponding different topological classes cannot interfere?
BTW, how to define quantum interfere here to make Zee's argument rigorous here? And why when two paths can not interfere quantum mechanically, we can assign different phases to them? 
 A: Consider first the case of 3 spatial dimensions, and just two particles for simplicity. If we are to "wind" the one around the other, it is clear that, due to our ability to deform the worldlines, the only considerations are the initial and final states of the particles. Therefore, we must necessarily identify winding angles $\phi$ which differ by $2\pi$, as they represent precisely the same physical situations. We cannot define the winding angle (and therefore the deformation) unambiguously, and so all we need are the initial and final states.
In 2 spatial dimensions, the worldlines representing angles $\phi + 2\pi k, k\in \mathbb{Z}$ cannot be deformed into each other. The winding angle $\phi$ is what characterises the "topological classes", but each of these actually represent a distinct physical situation. Wilczek (in Fractional Statistics and Anyon Superconductivity) points out that "it is impossible to take generic anyon statistics into account purely as a condition of the many-body wave function": you need more information than just the initial and final states to calculate statistics.
To carry out Feynman path integration, we must consider all paths that lead to the same situation, which is defined by quantities that do not necessarily correspond to observables (eg. rotation of the state vector).
