The relation between the Gross–Pitaevskii and Bogoliubov descriptions of a dilute Bosegas

Abstract

I formulate a ‘pseudo-paradox’ in the theory of a dilute Bose gas with repulsive interactions:the standard expression for the ground state energy within the Gross–Pitaevskii (GP)approximation is lower than that in the Bogoliubov approximation, and hence, bythe standard variational argument, the former should prima facie be a betterapproximation than the latter to the true ground state—a conclusion which is of courseopposite to the established wisdom concerning this problem. It is shown that thepseudo-paradox is (unsurprisingly) resolved by a correct transcription of the two-bodyscattering theory to the many-body case; however, contrary to what appearsto be a widespread belief, the resolution has nothing to do with any spuriousultraviolet divergences which result from the replacement of the true interatomicpotential by a delta-function pseudopotential. Rather, it relates to an infrareddivergence which has the consequence that (a) the most obvious form of the GP‘approximation’ actually does not correspond to any well-defined ansatz for the many-bodywavefunction, and (b) that the ‘best shot’ at such a wavefunction always producesan energy which exceeds, or at best equals, that calculated in the Bogoliubovapproximation. In fact, the necessity of the latter may be seen as a consequence of theneed to reduce the Fock term in the energy, which is absent in the two-particleproblem but dominant in the many-body case; it does this by increasing the densitycorrelations, at distances less than or approximately equal to the correlation length , above the value extrapolated from the two-body case. As a by-product I devise analternative formulation of the Bogoliubov approximation which does not require theexplicit replacement of the true interatomic potential by a delta-function pseudopotential.