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

Abstract

It is shown that a time-dependent external four-vector potential ${\mathit{a}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{ext}}$(r\ensuremath{\rightarrow}t) and a conserved external four-current ${\mathit{j}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{ext}}$(r\ensuremath{\rightarrow}t), with a given fixed initial state of the combined condensed-matter--electromagnetic system, lead, respectively, to a conserved self-consistent four-current density ${\mathit{J}}_{\mathrm{\ensuremath{\mu}}}$(r\ensuremath{\rightarrow}t) and a self-consistent four-vector field ${\mathit{A}}_{\mathrm{\ensuremath{\mu}}}$(r\ensuremath{\rightarrow}t), such that a map (${\mathit{a}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{ext}}$,${\mathit{j}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{ext}}$)\ensuremath{\rightarrow}(${\mathit{J}}_{\mathrm{\ensuremath{\mu}}}$,${\mathit{A}}_{\mathrm{\ensuremath{\mu}}}$) 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.