= Quadratic density-phase action
{title2=$S_2=\int[g(\delta\rho)^2/2+i\delta\rho\partial_\tau\phi+|\nabla\delta\rho|^2/(8m\rho_0)+\rho_0|\nabla\phi|^2/(2m)]$}
Expand the <density-phase action of a Bose gas> around $\rho_0=\mu/g>0$ and a constant phase. Density and phase are coupled by $i\delta\rho\partial_\tau\phi$, and retaining density <gradients> gives the full quadratic <Bogoliubov spectrum>. Dropping those <gradients> is a long-wavelength approximation. A small-fluctuation Gaussian extension around $\rho_0$ does not exactly remove the original nonnegative-density constraint.
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