Dispersive Corrections to Electromagnetic Energy and Power Flux Are Imprints of Eliminated Degrees of Freedom

Abstract

In a spatially dispersive medium the electromagnetic energy and flux are given by the frequency and wave-vector derivatives of the constitutive parameters, a result known as the Agranovich--Ginzburg (AG) formulas. Here, we clarify the physical meaning of the derivative terms in these relations and show that they measure the energy and flux carried by the degrees of freedom of the internal field-matter dynamics that the Maxwellian constitutive description hides. We construct an extended state-space model of such dynamics for a wide class of temporally and spatially dispersive media, from which the Maxwellian constitutive relations follow upon elimination of the internal variables, the effective material operator resulting from the Schur complement of the eliminated internal block. A congruence identity relates this complement to the extended state-space description, which allows us to extend the AG formulas to general bianisotropic media and to establish the conditions of their validity in the presence of dissipation. The AG dispersive corrections are thus the imprints of the eliminated degrees of freedom rather than independent constitutive postulates. They hold for fields driven at arbitrary real frequency and wave vector, and not only for eigenmodes, provided that the sources act on the fields alone and that any dissipation is carried by the electromagnetic subsystem. We verify the construction on the Lorentz, Drude, magnetized-plasma, and hydrodynamic models, the last of which describes a nonlocal metal. Additionally, a capacitively loaded wire medium is treated in the Supplemental Material.