Question

The best way to answer the question "How are anyons possible" is to use the "dynamical" path integral formalism, rather than the "static" wave function formalism. The permutation group action on the wave function is "static" in the sense that only initial and final states are specified. It will be ambiguous if there are more than one non-equivalent ways to perform the exchange process, which is the key for the "possibility" of anyons.

Consider the amplitude from the initial state $|i\rangle$ to final state $|f\rangle$ in the path integral formalism $$\langle f|i\rangle = \int_\gamma \mathcal{D}x(t) e^{iS[x(t)]},$$ where $\gamma$ is a path from the initial configuration to the final configuration (they are set to the same). The confituration manifold will be discussed later. When two paths $\gamma_1$ and $\gamma_2$ are not equivalent to each other homotopically, we can assign a phase factor $e^{i\theta([\gamma])}$ to the path integral amplitude for each homotopy class $[\gamma]$: $$\langle f|i\rangle = \sum_{[\gamma]\in \pi_1(M)} e^{i\theta([\gamma])}\int_\gamma \mathcal{D}x(t) e^{iS[x(t)]},$$ where $\pi_1(M)$ denotes the fundamental group of the configuration space $M$. The phase factors $\{e^{i\theta([\gamma])}\}$ form an one dimensional representation of the group $\pi_1(M)$ because of the multiplication property of the propagator: $\langle f|i\rangle=\sum_m \langle f|m\rangle\langle m|i\rangle$. If we absorb the phase $\theta$ to the action $S$, it will be called a topological term as it depends only on the homotopy class.

The next task is to calculate the one dimensional representation of the fundamental group of the configuration space. For $N$ identical particles in $d$ space dimension, the configration space is $M=(\mathbb R^{Nd}\backslash D)/S_N$, where $D=\{(r_1,...,r_N)|\ \exists i\neq j,\ s.t. r_i=r_j\}$ is the space where two particles occupy the same point, and "$/S_N$" means the order of the particles is neglected.

(1) $d=1$. No exchange process can happen, and the notion of statistics is meaningless.

(2) $d=2$. $\pi_1(M)=B_N$ is the braiding group. The one dimension representation of $B_N$ is characterized by an angle $\theta$ which corresponds to the statistical angle of the Abelian anyon.

(3) $d\geq 3$. $\pi_1(M)=S_N$ is the permutation group. It means that, we only need to specify the order of particles in the initial and final states, to determine which homotopy class the path $\gamma$ belongs to. Therefore, only in this case, the wave function formalism can be used without ambiguity.

To describe the non-Abelian anyons, one only need to replace the phase factor $e^{i\theta}$ by an unitary matrix. The result is that non-Abelian anyons are determined by the higher dimension representations of the fundamental group of the configuration space.

This post imported from StackExchange Physics at 2014-04-11 15:20 (UCT), posted by SE-user Tengen