Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the ”ǫ and μ ” description

The derivation of macroscopic Maxwell equations (M-eqs.) from microscopic basis is made in a general, model-independent way. Starting from a general Lagrangian of interacting matter-EM field as a most reliable basis, we first set up simultaneous equations for microscopic ”EM field and induced current density”, where a single (microscopic) susceptibility tensor is enough to describe the whole response. Macroscopic averaging is done by applying long wavelength approximation (LWA) to the microscopic response, which results in a more general macroscopic scheme than the traditional one in terms of a new macroscopic susceptibility containing the effects of both electric and magnetic polarizations in a generally inseparable way. This procedure does not increase the number of field variables, neither the number of consitutive equation. It is demonstrated that the electric and magnetic susceptibilities correspond to the first and second order terms, respectively, of the LWA expansion of the microscopic susceptibility in the high symmetry cases where no mixing of electric and magnetic dipole transitions occurs. Plane wave dispersion equation takes a different form, which is reduced to the well known form in high symmetry cases.