Braids and the Wigner-Racah calculus

The Wigner-Racah approach to the system of identical particles is compared with the theory of N anyons for the case when each hard-core particle is moving on an orientable two-dimensional manifold. It results that these two approaches are non-equivalent, and that the main source of this fact is a distinction in the meaning of the notion of indistinguishability, represented by the symmetric and the braid group for the Wigner-Racah and anyon theory, respectively.