Time-dependent functional theory of coupled electron and electromagnetic fields in condensed-matter systems

It is shown that a time-dependent external four-vector potential a_μᵉˣᵗ(r{→}t) and a conserved external four-current j_μᵉˣᵗ(r{→}t), with a given fixed initial state of the combined condensed-matter--electromagnetic system, lead, respectively, to a conserved self-consistent four-current density J_μ(r{→}t) and a self-consistent four-vector field A_μ(r{→}t), such that a map (a_μᵉˣᵗ,j_μᵉˣᵗ){→}(J_μ,A_μ) is invertible under certain simple conditions. With this as a basis, the stationary action principle is used to derive the self-consistent equations for the electron and the electromagnetic fields appropriate for a description of the combined condensed-matter and electrodynamic systems, where real or virtual pair creation is ignored.