Grassmann products, cumulants, and two-electron reduced density matrices

Some properties of Grassmann products of one-electron operators and of cumulants and related quantities for two-electron reduced density matrices (2-RDMs) are reported. The results suggest that any physical significance of the Grassmann product plus cumulant decomposition must be based on the physics of the system described, not just on the mathematical structure. The Grassmann product plus cumulant decomposition of the 2-RDM is resolved into components based on unitary invariants. Simple examples using mathematically acceptable but physically unlikely wave functions show that a Grassmann product differing from the usual one may better approximate the 2-RDM, and that the usual Grassmann product, cumulant, and 2-RDM can be mutually proportional.