Canonical formalism and quantization for a class of classical fields with application to radiative atomic decay in dielectrics

For the description of spectral and radiative decay properties of atoms or molecules, placed in a photonic material, the electromagnetic field in the material must be quantized and active research is taking place in this area. Here a unified account is given of such quantization procedures. Led by the Maxwell example, we consider the canonical formalism and its quantization for a class of linear evolution equations ${\partial }_{t}F=NF-G$, obeying a conservation law for $G=0\mathrm{}$. If $N$ has a nonempty null space (zero is an eigenvalue with associated nonpropagating solutions), an abstract form of the gauge concept makes its appearance and generalizations of the familiar Coulomb and Lorentz gauges are obtained. A canonical formalism is set up and quantized. The application to spatially inhomogeneous nonconducting electrodynamical systems is immediate, including the interaction with matter. Next atomic decay in a medium is considered, in particular in the presence of band gaps. For a simple two-level model with transition frequency in the gap, single-photon decay is inhibited and also a different stable eigenvalue of the combined system is found. An open problem in connection with random dielectrics, showing Anderson localization, is discussed. Finally, a mass renormalization, by means of a Kramers transformation, is presented. In general, the renormalized mass is no longer a scalar quantity.