On the Stability of the Hartree-Fock Solution in Many-Body Problem

A necessary condition for the stability of the Hartree-Fock solution in many-body problem. is presented. If this condition is not satisfied, then the solution becomes no longer stable as was discussed by Overhauser in connection with the one-dimensional spin wave model. A variational method is proposed in this case to construct a stable solution which has definitely .a lower energy than the Hartree-Fock value. The workability of our method is tested in some realistic examples like the B.C.S.-Bogolyubov theory of superconductivity. Then the method is applied to field theory which seems to be inconsistent in view of the presence of "ghost states"-A canonical transformation leads to a new vacuum state with lower energy, but the high momentum part of the coupling is not damped. l. Introduction