Fermions from photons: Bosonization of QED in 2+1 dimensions

TL;DR

This paper achieves a complete bosonization of 2+1 dimensional QED with one fermion flavor, explicitly constructing fermionic operators in terms of gauge fields and verifying Lorentz invariance.

Contribution

It introduces a novel Hamiltonian formalism for bosonizing 2+1D QED, explicitly constructs fermionic operators, and analyzes the algebra and invariance properties of the resulting bosonic theory.

Findings

Explicit fermionic operators in terms of gauge fields

Local, polynomial bosonic Hamiltonian consistent with Lorentz invariance

Analysis of Schwinger terms in the algebra of bilinears

Abstract

We perform the complete bosonization of 2+1 dimensional QED with one fermionic flavor in the Hamiltonian formalism. The Fermi operators are explicitly constructed in terms of the vector potential and the electric field. We carefully specify the regularization procedure involved in the definition of these operators, and calculate the fermionic bilinears and the energy - momentum tensor. The algebra of bilinears exhibits the Schwinger terms which also appear in perturbation theory. The bosonic Hamiltonian is a local, polynomial functional of $A_i$ and $E_i$, and we check explicitly the Lorentz invariance of the resulting bosonic theory. Our construction is conceptually very similar to Mandelstam's construction in 1+1 dimensions, and is dissimilar from the recent bosonization attempts in 2+1 dimensions, which hinge crucially on the presence of a Chern - Simons term.