Generalized P,Q and G conditions

The N-Representability Problem entails characterizing the set of second order reduced states that are contractions of N-electron states of the Fermion Fock algebra. This problem is formulated in the form of finding the conditions that a positive linear functional defined on a subspace of this algebra must satisfy in order to be extended to the whole algebra. As this algebra is a w*-algebra one can utilize a theorem by Kadison that shows it is sufficient to consider the values of linear functionals on projectors contained in the subspace in order to determine whether they have positive extensions. Thus we find the form of projectors belonging to the subspace of one and two particle operators and subsequently show that the extension conditions needed in the N-Representability Problem correspond to generalized P, Q and G conditions plus the additional constraints that the functionals be dispersion free on the number operator and their values on one particle operators determined by their values on two particle operators.