Nonabelian braid statistics versus projective permutation statistics

TL;DR

This paper critically examines the proposal of nonabelian projective permutation statistics as a new particle statistic, demonstrating its incompatibility with locality and clarifying the nature of braid group representations in quantum Hall systems.

Contribution

It refutes the possibility of projective permutation statistics in local quantum field theories and clarifies the structure of braid group representations in quantum Hall contexts.

Findings

Projective permutation statistics violate locality in quantum field theory.

Nayak and Wilczek's braid group representation is not a projective permutation representation.

Finite image of the braid group has a specific structure in a 2^{n/2-1}-dimensional representation.

Abstract

Recent papers by Finkelstein, Galiautdinov, and coworkers {[J. Math. Phys. 42, 1489, 3299 (2001)]} discuss a suggestion by Wilczek that nonabelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek's suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the nonabelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2^{n/2-1}-dimensional representation is obtained.