  Atomic high-precision measurements have become a competitive and essential
technique for tests of fundamental physics, the Standard Model, and our theory
of gravity. It is therefore self-evident that such measurements call for a
consistent relativistic description of atoms that eventually originates from
quantum field theories like quantum electrodynamics. Most quantum-metrological
approaches even postulate effective field-theoretical treatments to describe a
precision enhancement through techniques like squeezing. However, a consistent
derivation of interacting atomic quantum gases from an elementary quantum field
theory that includes both the internal structure as well as the center of mass
of atoms, has not yet been addressed. We present such a subspace effective
field theory for interacting, spin carrying, and possibly charged ensembles of
atoms composed of nucleus and electron that form composite bosons called
cobosons, where the interaction with light is included in a multipolar
description. Relativistic corrections to the energy of a single coboson,
light-matter interaction, and the scattering potential between cobosons arise
in a consistent and natural manner. In particular, we obtain a relativistic
coupling between the coboson's center-of-mass motion and internal structure
encoded by the mass defect. We use these results to derive modified bound-state
energies, including the motion of ions, modified scattering potentials, a
relativistic extension of the Gross-Pitaevskii equation, and the mass defect
applicable to atomic clocks or quantum clock interferometry.
