Advances in Theoretical and Mathematical Physics

Volume 19 (2015)

Number 6

Representability conditions by Grassmann integration

Pages: 1141 – 1182



Volker Bach (Institut für Analysis und Algebra, Technische Universität Braunschweig, Braunschweig, Germany)

Hans Konrad Knörr (Aalborg Universitet, Institut for Matematiske Fag, Aalborg, Denmark)

Edmund Menge (Technische Universität Braunschweig, Institut für Analysis und Algebra, Braunschweig, Germany)


Representability conditions on the one- and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals. A positivity condition for a certain kind of Grassmann integral is established which by an appropriate choice of the integrand, in turn, induces the well-known $\mathrm{G}$-, $\mathrm{P}$- and $\mathrm{Q}$-Conditions of quantum chemistry. Similarly, the $\mathrm{T_1}$- and $\mathrm{T_2}$-Conditions are derived. Furthermore, quasifree Grassmann states are introduced and, for every operator $\widetilde{\gamma} \in \mathcal{H} \oplus \mathcal{H}$ with $0 \leq \widetilde{\gamma} \leq \mathbb{1}$, the existence of a unique quasifree Grassmann state whose one-particle density matrix is $\widetilde{\gamma}$ is shown.

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