Both the Kohn-Sham theorem and the Gell-Mann & Low theorem provide a way to map an interacting condensed matter system on a non-interacting system, each in their own language. The Kohn-Sham approach does so for density functional theory, calculating the properties of a system from the one-electron density. On the other hand, the Gell-Mann & Low approach is used in second quantized solid-state physics where the one-electron Green function is the important object. Despite the difference in language, is there some deeper connection between the two approaches? And what determines which one of the two languages is the one that is used?