Transverse plasma polaritons and longitudinal plasmons describe the
propagation of light-matter modes in an isotropic metal. However, in a layered
metal the anisotropy of the bare electromagnetic response mixes the
longitudinal and transverse excitations, making the distinction between
polariton and plasmon blurred at small wavevectors, where retardation effects
of the electromagnetic interactions become quantitatively relevant. In the
usual Kubo approach for the linear response, this effect appears as a mixing
between the density and the transverse current fluctuations, that requires to
revise the standard RPA approach for density correlations where only the
instantaneous Coulomb potential is included. In this paper we derive the
general expression for the density and current correlation functions at long
wavelength in a layered metal, showing that below a crossover scale set by the
anisotropy of the plasma frequencies retardation effects make the dispersion of
the generalized plasma modes different from the standard RPA result. In
addition, the mixed longitudinal and transverse nature of these excitations
reflects in a double-peak structure for the density response, that can be
eventually accessed by means of high-momentum resolution electron-energy-loss
or X-rays spectroscopies.Comment: 22 pages, 3 figure