Static density stiffness versus an excitation gap ($S_2\supset g(\delta \rho {)}^2/2+i\delta \rho {\partial }_{\tau }\varphi$)

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.