Linear absorptive dielectrics

Starting from Maxwell’s equations for a linear, nonconducting, absorptive, and dispersive medium, characterized by the constitutive equations $\mathit{D}(\mathbf{x},t)={\varepsilon }_1\left(\mathbf{x}\right)\mathit{E}(\mathbf{x},t)+{\int }_{-\infty }^t\mathrm{ds}\chi (\mathbf{x},t-s)\mathit{E}(\mathbf{x},s)$ and $\mathit{H}(\mathbf{x},t)=\mathit{B}(\mathbf{x},t)$, a unitary time evolution and canonical formalism is obtained. Given the complex, coordinate, and frequency-dependent, electric permeability $\varepsilon (\mathbf{x},\omega )$, no further assumptions are made. The procedure leads to a proper definition of band gaps in the periodic case and a new continuity equation for energy flow. An $S$-matrix formalism for scattering from lossy objects is presented in full detail. A quantized version of the formalism is derived and applied to the generation of Čerenkov and transition radiation as well as atomic decay. The last case suggests a useful generalization of the density of states to the absorptive situation.