The Keldysh formalism applied to time-dependent current-density-functional theory

In this work we demonstrate how to derive the Kohn-Sham equations of time-dependent current-density functional theory from a generating action functional defined on a Keldysh time contour. These Kohn-Sham equations contain an exchange-correlation contribution to the vector potential. For this quantity we derive an integral equation. We further derive an integral equation for its functional derivative, the exchange-correlation kernel, which plays an essential role in response theory. The exchange-only limits of the latter equation is studied in detail for the electron gas and future applications are discussed.