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Negative frequencies in wave propagation: a microscopic model

A change in the sign of the frequency of a wave between two inertial reference frames corresponds to a reversal of the phase velocity. Yet from the point of view of the relation $E=\hbarω$, a positive quantum of energy apparently becomes a negative energy one. This is physically distinct from a change in the sign of the wave-vector, and has been associated with various effects such as Cherenkov radiation, quantum friction, and the Hawking effect. In this work we provide a more detailed understanding of these negative frequency modes based on a simple microscopic model of a dielectric medium as a lattice of scatterers. We calculate the classical and quantum mechanical radiation damping of an oscillator moving through such a lattice and find that the modes where the frequency has changed sign contribute negatively. In terms of the lattice of scatterers we find that this negative radiation damping arises due to phase of the periodic force experienced by the oscillator due to the relative motion of the lattice.