  Cavity modification of material properties and phenomena is a novel research
field largely motivated by the advances in strong light-matter interactions.
Despite this progress, exact solutions for extended systems strongly coupled to
the photon field are not available, and both theory and experiments rely mainly
on finite-system models. Therefore a paradigmatic example of an exactly
solvable extended system in a cavity becomes highly desireable. To fill this
gap we revisit Sommerfeld's theory of the free electron gas in cavity quantum
electrodynamics (QED). We solve this system analytically in the long-wavelength
limit for an arbitrary number of non-interacting electrons, and we demonstrate
that the electron-photon ground state is a Fermi liquid which contains virtual
photons. In contrast to models of finite systems, no ground state exists if the
diamagentic $\textbf{A}^2$ term is omitted. Further, by performing linear
response we show that the cavity field induces plasmon-polariton excitations
and modifies the optical and the DC conductivity of the electron gas. Our exact
solution allows us to consider the thermodynamic limit for both electrons and
photons by constructing an effective quantum field theory. The continuum of
modes leads to a many-body renormalization of the electron mass, which modifies
the fermionic quasiparticle excitations of the Fermi liquid and the
Wigner-Seitz radius of the interacting electron gas. Lastly, we show how the
matter-modified photon field leads to a repulsive Casimir force and how the
continuum of modes introduces dissipation into the light-matter system. Several
of the presented findings should be experimentally accessible.
