Gauss's Law, Gauge-Invariant States, and Spin and Statistics In Abelian Chern-Simons Theories

TL;DR

This paper explores topologically massive QED in (2+1) dimensions with a Chern-Simons term, constructing gauge-invariant states, analyzing their interactions, and clarifying the role of spin and statistics in the theory.

Contribution

It introduces a method to embed topologically massive QED into a Fock space with gauge-invariant states, revealing nonlocal interactions and clarifying spin and statistics.

Findings

Gauge-invariant charged particles obey original anticommutation rules.

Interactions include nonlocal potentials like Hopf-like and Coulomb.

Unitarity is manifest when using gauge-invariant states.

Abstract

We discuss topologically massive QED --- the Abelian gauge theory in which (2+1)-dimensional QED with a Chern-Simons term is minimally coupled to a spinor field. We quantize the theory in covariant gauges, and construct a class of unitary transformations that enable us to embed the theory in a Fock space of states that implement Gauss's law. We show that when electron (and positron) creation and annihilation operators represent gauge-invariant charged particles that are surrounded by the electric and magnetic fields required by Gauss's law, the unitarity of the theory is manifest, and charged particles interact with photons and with each other through nonlocal potentials. These potentials include a Hopf-like interaction, and a planar analog of the Coulomb interaction. The gauge-invariant charged particle excitations that implement Gauss's law obey the identical anticommutation rules as do the original gauge-dependent ones. Rotational phases, commonly identified as planar `spin', are arbitrary, however.