  We derive the full linear-response theory for non-relativistic quantum
electrodynamics in the long wavelength limit, show quantum modifications of the
well-known Maxwell's equation in matter and provide a practical framework to
solve the resulting equations by using quantum-electrodynamical
density-functional theory. We highlight how the coupling between quantized
light and matter changes the usual response functions and introduces new types
of cross-correlated light-matter response functions. These cross-correlation
responses lead to measurable changes in Maxwell's equations due to the
quantum-matter-mediated photon-photon interactions. Key features of treating
the combined matter-photon response are that natural lifetimes of excitations
become directly accessible from first principles, changes in the electronic
structure due to strong light-matter coupling are treated fully
non-perturbatively, and for the first time self-consistent solutions of the
back-reaction of matter onto the photon vacuum and vice versa are accounted
for. By introducing a straightforward extension of the random-phase
approximation for the coupled matter-photon problem, we calculate the first
ab-initio spectra for a real molecular system that is coupled to the quantized
electromagnetic field. Our approach can be solved numerically very efficiently.
The presented framework leads to a shift in paradigm by highlighting how
electronically excited states arise as a modification of the photon field and
that experimentally observed effects are always due to a complex interplay
between light and matter. At the same time the findings provide a new route to
analyze as well as propose experiments at the interface between quantum
chemistry, nanoplasmonics and quantum optics.
