Homotopic Paths and Spin-Statistics I am currently reading Schwartz' book on QFT, Section 12.2 on Spin and statistics. He shows, that in 3D there are only two inequivalent ways to exchange two indistinguishable particles. More formally, this means that the fundamental group of the configuration space (6D, identify (x1, x2) with (x2,x1)) is $Z_2$.
Now my question: Why can we deduce from the above that there are only two possible statistics/ ways the wavefunction can transform under this particle exchange? I think my question boils down to: If we tranform the state along two homotopic paths in the configuration space, why must the resulting state be the same for both paths?
 A: The wavefunction $\psi(x_1,x_2)$ assigns a complex number to each point in configuration space and is continuous, but possibly has many branches as far as phase is concerned. Consider a path $\gamma_0(\tau)$ in configuration space such that
$$\gamma_0(0)=(x_1,x_2),\quad \gamma_0(1)=(x_2,x_1).$$
Then the value of the complex number $\psi(\gamma_0(\tau))$ will change continuously from $\psi(\gamma_0(0))$ to $\psi(\gamma_0(1))=\eta_0\psi(\gamma_0(0))$, where $\eta_0$ is some phase. This last equality up to a phase follows from the indistinguishability of the particles.
We can repeat this with another path $\gamma_1$ with the same endpoints and get $\psi(\gamma_1(1))=\eta_1\psi(\gamma_1(0))$, where $\eta_1$ can be different from $\eta_0$ if the two paths go on different branches.
But if the paths are homotopic, i.e. there is a continuous function $\gamma_\sigma(\tau)$ of $\sigma,\tau$, which interpolates between $\gamma_0$ and $\gamma_1$ then we must have $\eta_0=\eta_1$ by continuity.
Moreover we can construct a loop which starts and ends on $(x_1,x_2)$ by traversing the path $\gamma_0$ and then the reverse of the path $\gamma_1$, and in general the endpoint of this loop would lead to $\eta_0\eta_1^*\psi(x_1,x_2)$, but if the two paths are homotopic this loop can be contracted to the trivial loop which never moves from $(x_1,x_2)$ so $\eta_0\eta_1^*=\eta_0\eta_0^*=1$, from which we see $\eta_0=\pm 1$.
