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The density-phase action of a Bose gas follows directly from $\psi =\sqrt{\rho }{e}^{i\varphi }$:The total derivative integrates to zero for periodic density. HenceA 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 $i\delta \rho {\partial }_{\tau }\varphi$ couples their dynamics: this does not imply an additional independent gapped quasiparticle branch in the nonrelativistic Bose gas.

The compact phase winding term $i{\rho }_0\int {\partial }_{\tau }\varphi$ 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.

On the physical density-phase contour put $\bar{\psi }=\sqrt{\rho }{e}^{-i\varphi }$. Direct differentiation givesIntegrating the Laplacian by parts with the stated bulk boundary convention, and dropping the periodic total derivative of $\rho$, produces the density-phase action of a Bose gas,Write $\rho ={\rho }_0+\delta \rho$ and $\varphi ={\varphi }_0+\phi$, where ${\varphi }_0$ is constant. To quadratic order,The uniform static density fluctuation has positive curvature $g$. Its inverse static susceptibility at wavevector $q$ is $A_q=g+q^2/(4m{\rho }_0)$, which remains nonzero at $q=0$. This is the sense in which the amplitude or density fluctuation is massive. The phase has no term proportional to ${\phi }^2$ without derivatives; a uniform phase rotation costs no action. It is the massless direction required by global $U(1)$ symmetry.

The term $i\delta \rho {\partial }_{\tau }\phi$ shows density-phase conjugacy. It is important to distinguish static density stiffness versus an excitation gap: the density and phase do not form two independent normal-mode branches with one gapped and one gapless particle. In this nonrelativistic first-order theory they supply the conjugate variables of one positive Bogoliubov branch, derived below. The omitted cubic terms include $\delta \rho |\nabla \phi {|}^2/(2m)$ and corrections from the density-gradient denominator.

There is also a background term $i{\rho }_0\int d\tau d^{d}r{\partial }_{\tau }\phi$. It vanishes for smooth fluctuations with zero temporal winding. Periodicity of $\psi$, however, permits $\phi (\beta )-\phi (0)=2\pi w$. The compact phase winding term must be retained when such sectors are included; it is not generally the same constant in all sectors. The local Gaussian phonon calculation concerns the smooth zero-winding sector and does not describe vortices or points where the density vanishes.