Representability conditions by Grassmann integration (Q300266)

A key to understanding fermions is provided by the insight that differential and integral calculus for anticommuting variables is needed in this particular domain of field theory and so one has to cope in textbooks with the problem of dealing with this new repertoire of modern mathematical physics, i.e. real Grassmann variables. The present paper uses Grassmann integrals to formulate conditions for the \textit{representability} on one- and two-particle fermionic density matrices by Grassmann integration. Before the authors give a definition of what they call \textit{reduced density matrices}. A positivity condition is needed for some kind of Grassmann integral. Furthermore, theorems are given that imply G-, P-, and Q-conditions, well known in quantum chemistry. Quasifree Grassmann states are also considered. However, the structure of fermionic density matrices in general is still the main subject of this article and all necessary material is developed within six sections.