  We investigate the microscopic description of exciton-polaritons that
involves electrons, holes and photons within a two-dimensional microcavity. We
show that in order to recover the simplified exciton-photon model that is
typically used to describe polaritons, one must correctly define the
exciton-photon detuning and exciton-photon (Rabi) coupling in terms of the bare
microscopic parameters. For the case of unscreened Coulomb interactions, we
find that the exciton-photon detuning is strongly shifted from its bare value
in a manner akin to renormalization in quantum electrodynamics. Within the
renormalized theory, we exactly solve the problem of a single exciton-polariton
for the first time and obtain the full spectral response of the microcavity. In
particular, we find that the electron-hole wave function of the polariton can
be significantly modified by the very strong Rabi couplings achieved in current
experiments. Our microscopic approach furthermore allows us to obtain the
effective interaction between identical polaritons for any light-matter
coupling. Crucially, we show that the standard treatment of polariton-polariton
interactions in the very strong coupling regime is incorrect, since it neglects
the light-induced modification of the exciton size and thus greatly
overestimates the effect of Pauli exclusion on the Rabi coupling, i.e., the
saturation of exciton oscillator strength. Our findings thus provide the
foundations for understanding and characterizing exciton-polariton systems
across the whole range of polariton densities.
