On Hugenholtz-Pines' Theory about Ground State Energy of Interacting Bosons

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Abstract
In their recently published paper Hugenholtz and Pines have developed a new method to evaluate the ground state energy of a system of interacting bosons, the merit of which consists in the elimination of the zero-momentum state and accordingly the applicability of the linked cluster expansion. Their theory, however, does not give correct results because of the undue treatment of the zero-momentum state; this will be proved in a simple example. Up to third order in coupling parameter of the ground state energy we compare the result of the usual Rayleigh-Schrödinger perturbation method with that obtained according to Hugenholtz-Pines' theory and find that the latter does not agree with the former.

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