The two-body reduced density matrix of the high-density electron gas

For the first time, the cumulant 2-body reduced density matrix (= 2-matrix) of the spin-unpolarized homogeneous electron gas (HEG) is considered. This $\gamma_{\rm c}$ proves to be the common source for both the momentum distribution $n(k)$ and the static structure factor $S(q)$. Within many-body perturbation theory, this $\gamma_{\rm c}$ is given by only {\it linked} diagrams (with 2 open particle-hole lines as well as with closed loops and interaction lines). Here it is worked out in detail, how the 1-body quantity $n(k)$ follows from the 2-body quantity $\gamma_{\rm c}$ - through a certain contraction procedure, cf Eqs.(2.25)-(2.28). In particular, this $\gamma_{\rm c}$ is developed for the high-density HEG. Its correctness is checked by deriving from it $n(k)$ and $S(q)$, known from the random-phase approximation (RPA). This study opens the way to a more sophisticated HEG description in terms of cumulant geminals or/and variational methods. Besides, the cumulant structure factor (CSF) of the exchange in lowest order is explicitly given and sum rules for the CSFs and their small- and large-$q$ behavior (beyond RPA) are systematically summarized within the plasmon sum rule, coalescing theorems, and the inflexion-point trajectory.