= Static density stiffness versus an excitation gap
{title2=$S_2\supset g(\delta\rho)^2/2+i\delta\rho\partial_\tau\phi$}
A positive density curvature $g$ makes uniform static density fluctuations costly. It does not imply an additional independent gapped excitation: the first-order term makes density and phase conjugate. For a simple nonrelativistic <weakly interacting Bose gas>, their coupled quadratic dynamics yields one positive <Bogoliubov spectrum>, which is gapless at zero <momentum>. The word massive for the density field describes its static curvature in this setting.
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