Bosonization in 2+1 dimensions without Chern - Simons attached

TL;DR

This paper presents a complete Hamiltonian bosonization of 2+1 dimensional QED with one fermion flavor, explicitly constructing fermionic operators in terms of gauge fields without relying on Chern-Simons terms.

Contribution

It introduces a novel bosonization approach in 2+1 dimensions that avoids the use of Chern-Simons terms and explicitly constructs fermionic operators in the Hamiltonian formalism.

Findings

Fermionic operators are explicitly constructed from gauge fields.

The bosonic Hamiltonian is a local polynomial in gauge fields.

Lorentz invariance of the bosonic theory is explicitly verified.

Abstract

We perform the complete bosonization of 2+1 dimensional QED with one fermionic flavor in the Hamiltonian formalism. The fermion 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 density is a local polynomial function 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 existence of a Chern - Simons term.