Anyonic States in Chern-Simons Theory

TL;DR

This paper analyzes the canonical quantization of Chern-Simons theory coupled to Dirac fermions, revealing two types of charged states with distinct rotational properties, and clarifies their relation to gauge constraints and statistics.

Contribution

It demonstrates the existence of two classes of charged states with different rotational anomalies in Chern-Simons theory coupled to fermions, and clarifies their origin and statistical behavior.

Findings

Two types of charged states with and without rotational anomalies.

Rotational anomalies are independent of Gauss's law implementation.

States with or without anomalies obey Fermi statistics.

Abstract

We discuss the canonical quantization of Chern-Simons theory in $2+1$ dimensions, minimally coupled to a Dirac spinor field. Gauss's law and the gauge condition, $A_0 = 0$, are implemented by embedding the formulation in an appropriate physical subspace. We find two kinds of charged particle states in this model. One kind has a rotational anomaly in the form of arbitrary phases that develop in $2\pi$ rotations; the other kind rotates ``normally''---i.e., charged states only change sign in $2\pi$ rotations. The rotational anomaly has nothing to do with the implementation of Gauss's law. It is possible to inadvertently produce these anomalous states in the process of implementing Gauss's law, but it is also possible to implement Gauss's law without producing rotational anomalies. Moreover, states with or without rotational anomalies obey ordinary Fermi statistics.