  Starting from the Weyl gauge formulation of quantum electrodynamics (QED),
the formalism of quantum-mechanical gauge fixing is extended using techniques
from nonrelativistic QED. This involves expressing the redundant gauge degrees
of freedom through an arbitrary functional of the gauge-invariant transverse
degrees of freedom. Particular choices of functional can be made to yield the
Coulomb gauge and Poincar\'{e} gauge representations. The Hamiltonian we derive
therefore serves as a good starting point for the description of atoms and
molecules by means of a relativistic Dirac field. We discuss important
implications for the ontology of noncovariant canonical QED due to the gauge
freedom that remains present in our formulation.
