Model-independent derivation of macroscopic Maxwell equations from microscopic basis: Beyond the "\epsilon and \mu" description

Pointing out the incompleteness of conventional macroscopic Maxwell equations (M-eqs.), we propose a new form derived from the long wavelength approximation (LWA) of microscopic nonlocal response. From the general Hamilonian of matter and matter-EM field interaction (containing spin dependent terms due to relativistic correction), we first set up the simultaneous equations for microscopic "vector potential $\Vec{A}$ and induced current density $\Vec{I}$", and then extract the macroscopic components by applying LWA. This leads to new macroscopic M-eqs. with a single macroscopic susceptibility $\chi_{\rm em}(\Vec{k}, \omega)$ between $\Vec{I}$ and $\Vec{A}$, which describes both electric and magnetic polarizations and their mutual interference in its fully quantum mechanical expression. In the absence of chirality and under the condition to use magnetic susceptibility defined with respect to $\Vec{B}$, this scheme is shown to be equivalent to the conventional "$\epsilon$ and $\mu$ scheme". In the case of chiral symmetry, the phenomenological constitutive equations by Drude, Born and Fedorov cannot be justified by this microscopic approach. As a single susceptibility scheme of macroscopic M-eqs., this result is on an advanced level by its fully quantum mechanical description of the whole set of $O(k^0)$, $O(k^1)$ and $O(k^2)$ terms of susceptibility providing a consistent picture of such a scheme.