Conserving approximations in cavity quantum electrodynamics: Implications for density functional theory of electron-photon systems

By analyzing the many-body problem for nonrelativistic electrons strongly coupled to photon modes of a microcavity I derive the exact momentum/force balance equation for cavity quantum electrodynamics. Implications of this equation for the electron self-energy and the exchange-correlation potential of the quantum electrodynamic time-dependent density functional (QED-TDDFT) are discussed. In particular I generalize the concept of Φ derivability to construct approximations which ensure the correct momentum balance. It is shown that a recently proposed optimized effective potential approximation for QED-TDDFT is conserving, and its possible improvements are discussed.