The N-representability conditions on the reduced second-order reduced density matrix (2-RDM), impose restrictions not only in the context of reduced density matrix theory (RDMT), but also on functionals advanced in one-matrix theory such as natural orbital functional theory (NOFT), and on functionals depending on the one-electron density such as those of density functional theory (DFT). We review some aspects of the applications of these N-representability conditions in these theories and present some conclusions.
Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs)
ni. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.
We formulate the density matrix of many-body systems as functional integrals in the holomorphic representation, similar to the S-matrix formulation in field theory. Symmetry and antisymmetry requirements are built into the integration variables: c-numbers for bosons and Grassmann numbers for fermions. The integrals have well-defined normalization. They yield first-order differential equations for semiclassical extremal configurations and corrections to it, and do not require periodicity conditions on the individual integration paths, unlike the formulations found in the literature. We present some applications to simple boson and fermion systems.