  We develop a quantum theory of plasmon polaritons in chains of metallic
nanoparticles, describing both near- and far-field interparticle distances, by
including plasmon-photon Umklapp processes. Taking into account the retardation
effects of the long-range dipole-dipole interaction between the nanoparticles,
which are induced by the coupling of the plasmonic degrees of freedom to the
photonic continuum, we reveal the polaritonic nature of the normal modes of the
system. We compute the dispersion relation and radiative linewidth, as well as
the group velocities of the eigenmodes, and compare our numerical results to
classical electrodynamic calculations within the point-dipole approximation.
Interestingly, the group velocities of the polaritonic excitations present an
almost periodic sign change and are found to be highly tunable by modifying the
spacing between the nanoparticles. We show that, away from the intersection of
the plasmonic eigenfrequencies with the free photon dispersion, an analytical
perturbative treatment of the light-matter interaction is in excellent
agreement with our fully retarded numerical calculations. We further study
quantitatively the hybridization of light and matter excitations, through an
analysis of Hopfield's coefficients. Finally, we consider the limit of
infinitely spaced nanoparticles and discuss some recent results on single
nanoparticles that can be found in the literature.
