Linear absorptive dielectrics

Starting from Maxwell's equations for a linear, nonconducting, absorptive, and dispersive medium, characterized by the constitutive equations $D(x,t)={{ε}}₁(x)E(x,t)+{{∫}}_{{-}{∞}}ᵗds{χ}(x,t{-}s)E(x,s)$ and $H(x,t)=B(x,t)$, a unitary time evolution and canonical formalism is obtained. Given the complex, coordinate, and frequency-dependent, electric permeability ${ε}(x,{ω})$, 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 {C} {C}{}erenkov and transition radiation as well as atomic decay. The last case suggests a useful generalization of the density of states to the absorptive situation.