Reformulation of gauge theories in terms of gauge invariant fields

TL;DR

This paper introduces a reformulation of Abelian gauge theories using gauge invariant fields, simplifying the description of such theories across various models and dimensions, with potential implications for interacting gauge theories.

Contribution

It presents a novel gauge invariant reformulation applicable to multiple models, including lattice and continuum theories, extending from 1+1 dimensions to higher dimensions and continuum limits.

Findings

Reformulation applied to single particle in magnetic field

Reformulation used in pure Abelian lattice gauge theories

Application to QED in 3+1 dimensions and Hofstadter models

Abstract

We present a reformulation of gauge theories in terms of gauge invariant fields. Focusing on Abelian theories, we show that the gauge and matter covariant fields can be recombined to introduce new gauge invariant degrees of freedom. Starting from the $(1+1)$ dimensional case on the lattice, with both periodic and open boundary conditions, we then generalize to higher dimensions and to the continuum limit. To show explicit and physically relevant examples of the reformulation, we apply it to the Hamiltonian of a single particle in a (static) magnetic field, to pure abelian lattice gauge theories, to the Lagrangian of quantum electrodynamics in $(3+1)$ dimensions and to the Hamiltonian of the $2d$ and $3d$ Hofstadter model. In the latter, we show that the particular construction used to eliminate the the gauge covariant fields enters the definition of the magnetic Brillouin zone. Finally, we briefly comment on relevance of the presented reformulation to the study of interacting gauge theories.