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 $\mathrm{\Phi }$ 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.

• Received 27 September 2018
• Revised 8 November 2018

DOI:https://doi.org/10.1103/PhysRevB.98.235123