Anyonic States in Chern-Simons Theory

TL;DR

This paper investigates the canonical quantization of Chern-Simons theory coupled to Dirac spinors in 2+1 dimensions, demonstrating that charged states obey fermionic statistics and are not fractional, across different gauge choices.

Contribution

It provides a detailed analysis showing that charged states in Chern-Simons theory obey fermionic statistics, challenging the common expectation of fractional statistics in such theories.

Findings

Charged states obey fermionic, not fractional, statistics.

Gauss's law implementation does not require fractional statistics fields.

Hamiltonians in different gauges are equivalent for charged states.

Abstract

We discuss the canonical quantization of Chern-Simons theory in $2+1$ dimensions, minimally coupled to a Dirac spinor field, first in the temporal gauge and then in the Coulomb gauge. In our temporal gauge formulation, Gauss's law and the gauge condition, $A_0 = 0$, are implemented by embedding the formulation in an appropriate physical subspace. We construct a Fock space of charged particle states that satisfy Gauss's law, and show that they obey fermion, not fractional statistics. The gauge-invariant spinor field that creates these charged states from the vacuum obeys the anticommutation rules that generally apply to spinor fields. The Hamiltonian, when described in the representation in which the charged fermions are the propagating particle excitations that obey Gauss's law, contains an interaction between charge and transverse current densities. We observe that the implementation of Gauss's law and the gauge condition does not require us to use fields with graded commutator algebras or particle excitations with fractional statistics. In our Coulomb gauge formulation, we implement Gauss's law and the gauge condition, $\partial_lA_l=0$, by the Dirac-Bergmann procedure. In this formulation, the constrained gauge fields become functionals of the spinor fields, and are not independent degrees of freedom. The formulation in the Coulomb gauge confirms the results we obtained in the temporal gauge: The ``Dirac-Bergmann'' anticommutation rule for the charged spinor fields $\psi$ and $\psi^\dagger$ that have both been constrained to obey Gauss's law, is precisely identical to the canonical spinor anticommutation rule that generates standard fermion statistics. And we also show that the Hamiltonians for charged particle states in our temporal and Coulomb gauge formulations are identical, once Gauss's law