= Bose-gas phase-only action
{c}
{title2=$S_{\rm eff}=\tfrac12\int[(\partial_\tau\phi)^2/g+\rho_0|\nabla\phi|^2/m]$}
In the smooth zero-winding long-wave sector, integrate the density <Gaussian integral> by completing $g(\delta\rho+i\partial_\tau\phi/g)^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is $\omega^2=g\rho_0 k^2/m$, so the <phonon> speed is $\sqrt{g\rho_0/m}$. Keeping density <gradients> yields $\omega^2=g\rho_0k^2/m+k^4/(4m^2)$, matching the <Bogoliubov spectrum>.
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