The equation of motion for the Green function is combined with the Bethe-Salpeter equation for the scattering amplitude yielding a concise and formally closed system of three equations that encapsulates the essence of Green function theory. Two of the three equations formally resemble a Dyson-like relation. We prove that this formally simple set is exactly equivalent to Hedin's equations. Our derivation therefore constitutes an alternative to Hedin's derivation which is based on functional derivatives. Furthermore, we briefly discuss how approximations can be introduced as a hierarchy of approximations to the four-point Green function.

In this article, we put forward a new approach to electrodynamics of materials. Based on the identification of induced electromagnetic fields as the microscopic counterparts of polarization and magnetization, we systematically employ the mutual functional dependencies of induced, external and total field quantities. This allows for a unified, relativistic description of the electromagnetic response without assuming the material to be composed of electric or magnetic dipoles. Using this approach, we derive universal (material-independent) relations between electromagnetic response functions such as the dielectric tensor, the magnetic susceptibility and the microscopic conductivity tensor. Our formulae can be reduced to well-known identities in special cases, but more generally include the effects of inhomogeneity, anisotropy, magnetoelectric coupling and relativistic retardation. If combined with the Kubo formalism, they would also lend themselves to the ab initio calculation of all linear electromagnetic response functions.

We examine the refractive index from the viewpoint of modern first-principles materials physics. We first argue that the standard formula, n 2 = ε r µ r , is generally in conflict with fundamental principles on the microscopic level. Instead, it turns out that an allegedly approximate relation, n 2 = ε r , which is already being used for most practical purposes, can be justified theoretically at optical wavelengths. More generally, starting from the fundamental, Lorentz-covariant electromagnetic wave equation in materials as used in plasma physics, we rederive a well-known, three-dimensional form of the wave equation in materials and thereby clarify the connection between the covariant fundamental response tensor and the various cartesian tensors used to describe optical properties. Finally, we prove a general theorem by which the fundamental, covariant wave equation can be reformulated concisely in terms of the microscopic dielectric tensor.

We argue that the amazing progress of first-principles materials physics necessitates a revision of the Standard Approach to electrodynamics of media. We hence subject this Standard Approach to a thorough critique, which shows both its inherent conceptual problems and its practical inapplicability to modern ab initio calculations. We then go on to show that the common practice in ab initio materials physics has overcome these difficulties by taking a different, microscopic approach to electrodynamics of media, only superficially resembling the Standard Approach whose validity is restricted to the macroscopic domain. As this paradigm shift went largely unnoticed outside and partly even within the ab initio community, the present article aims at a systematic development and paradigmatic discussion of this new, microscopic first-principles approach to electrodynamics of media, for which we propose the name Functional Approach.