The problem of the diagonal and the evolution operator in the theory of anyons

The system of N free anyons moving on a plane and having the fractional statistics ν is considered. It is shown that the space of all ν-equivariant complex valued functions over the N-particle configuration space cannot be identified with the space of quantum states of the system, since the former is not invariant under the evolution operator. This non-invariance manifests itself as a vanishing of the singularities of the phase of the wave function on the fat diagonal.