  Strong light-matter coupling provides a promising path for the control of
quantum matter where the latter is routinely described from first-principles.
However, combining the quantized nature of light with this ab initio tool set
is challenging and merely developing, as the coupled light-matter Hilbert space
is conceptually different and computational cost quickly becomes overwhelming.
In this work, we provide a non-perturbative photon-free formulation of quantum
electrodynamics (QED) in the long-wavelength limit, which is formulated solely
on the matter Hilbert space and can serve as an accurate starting point for
such ab initio methods. The present formulation is an extension of quantum
mechanics that recovers the exact results of QED for the zero- and
infinite-coupling limit, the infinite-frequency as well as the homogeneous
limit and we can constructively increase its accuracy. We show how this
formulation can be used to devise approximations for quantum-electrodynamical
density-functional theory (QEDFT), which in turn also allows to extend the
ansatz to the full minimal-coupling problem and to non-adiabatic situations.
Finally, we provide a simple local-density-type functional that takes the
strong coupling to the transverse photon-degrees of freedom into account and
includes the correct frequency and polarization dependence. This is the first
QEDFT functional that accounts for the quantized nature of light while
remaining computationally simple enough to allow its application to a large
range of systems. All approximations allow the seamless application to periodic
systems.
