Representability Conditions by Grassmann Integration

TL;DR

This paper introduces a novel framework using Grassmann integrals to formulate and derive key representability conditions for fermion systems' density matrices, connecting quantum chemistry conditions with Grassmann algebra.

Contribution

It presents a new Grassmann integral approach to derive and unify various fermionic density matrix conditions, including G-, P-, Q-, T1-, and T2-Conditions, and characterizes quasifree Grassmann states.

Findings

Positivity condition for Grassmann integrals established

G-, P-, Q-, T1-, T2-Conditions derived from Grassmann integrals

Every bounded generalized one-particle density matrix corresponds to a unique quasifree Grassmann state

Abstract

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, in turn, induces the well-known G-, P- and Q-Conditions of quantum chemistry by an appropriate choice of the integrand. Similarly, the T1- and T2-Conditions are derived. Furthermore, quasifree Grassmann states are introduced and it is shown that every so-called generalized one-particle density matrix which is bounded between 0 and 1 corresponds to a unique quasifree Grassmann state.