Reformulation of gauge theories in terms of gauge invariant fields

Abstract

In this paper 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. We first consider the (1 + 1) dimensional case on the lattice, with both periodic and open boundary conditions. Then the procedure is generalized 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, pure abelian lattice gauge theories, the Lagrangian of quantum electrodynamics in (3+1) dimensions and 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.