Two electron problem and the nonlocal equations of electrodynamics

A survey is offered for the current knowledge of nonlocal electrodynamic equations which in some cases (e.g., in solving boundary value problems in optics) can replace Maxwell's equations. The nonlocal equations are derived using the semi-classical or quantum-electrodynamic approaches. The former involves an expansion of retarded potentials in appropriate parameters and a subsequent transition, to terms of order v² /c², to quantum mechanical operators in the Lagrangian of a system of moving charges. The latter approach is to consider second- and third-order quantum electrodynamic effects for two hydrogen-like atoms arbitrarily far apart. Various nonlocal equations are derived for the propagation of photons and electromagnetic waves in spin systems, dielectrics, and metals, taking into account a variety of quantum transitions and intermediate states in the spectrum of the interacting atoms. By combining nonlocal field equations with relevant constitutive equations, a number of typical boundary-value optical problems are solved for semi-infinite media, superthin films, and for objects whose linear dimensions are much smaller than the light wavelength.