  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.
