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We analyze the original proof by Luttinger and Ward of the Luttinger theorem,according to which for uniform ground states of systems of (interacting)fermions, which may be metallic or insulating, the number of k pointscorresponding to non-negative values of G_s(k;mu) is equal to the total numberof particles with spin index s in these ground states. Here G_s(k;mu) is thesingle-particle Green function of particles with spin index s at the chemicalpotential mu. For the cases where the two-body interaction potential isshort-range, and in particular for lattice models, we explicitly demonstratethat this theorem is unconditionally valid, irrespective of the strength of thebare interaction potential. We arrive at this conclusion by amongst otherthings demonstrating that the perturbation series expansion for self-energy interms of skeleton diagrams, as encountered in the proof of the Luttinger-Wardidentity, is uniformly convergent for almost all momenta and energies. Wefurther investigate the mechanisms underlying some reported instances offailure of the Luttinger theorem. With one exception, for all the casesconsidered in this paper, we show that the apparent failures of the Luttingertheorem can be attributed either to shortcomings of the employedsingle-particle Green functions or to misapplication of this theorem. The oneexceptional case brings to light the possibility of a genuine failure of theLuttinger theorem for insulating ground states, which we show to be broughtabout by a false limit that in principle can be reached on taking thezero-temperature limit without the value of mu coinciding with thezero-temperature limit of the chemical potential satisfying the equation ofstate at finite temperatures; no such ambiguity can arise for metallic states.

On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions