= Solution
The <density-phase action of a Bose gas> follows directly from $\psi=\sqrt\rho e^{i\phi}$:
$$
\bar\psi\partial_\tau\psi=\tfrac12\partial_\tau\rho+i\rho\partial_\tau\phi,\qquad
|\nabla\psi|^2=\frac{|\nabla\rho|^2}{4\rho}+\rho|\nabla\phi|^2.
$$
The total <derivative> integrates to zero for periodic density. Hence
$$
S=\int\left[i\rho\partial_\tau\phi+\frac{|\nabla\rho|^2}{8m\rho}+\frac\rho{2m}|\nabla\phi|^2+\frac g2(\rho-\rho_0)^2-\frac{\mu^2}{2g}\right].
$$
A uniform density fluctuation has nonzero quadratic cost $g(\delta\rho)^2/2$, whereas a uniform phase change has no energy cost. Thus the radial fluctuation is massive in the static quadratic-action sense, while the phase is massless. The <number-phase conjugacy> term $i\delta\rho\,\partial_\tau\phi$ couples their dynamics: this does not imply an additional independent gapped <quasiparticle> branch in the nonrelativistic Bose gas.
The <compact phase winding term> $i\rho_0\int\partial_\tau\phi$ vanishes for the smooth zero-winding <phonon> sector. Globally the phase can wind by $2\pi$ while the complex field remains periodic, so that term should not be discarded across all topological sectors without a further argument.
Back to article page
