= Solution
Use the <contact-interaction Bose-gas field integral> for a homogeneous three-dimensional <weakly interacting Bose gas> with repulsive short-range interaction as the standard model. Let particle mass be $m_b$, <scattering length> $a_s>0$, physical coupling $g=4\pi\hbar^2a_s/m_b$ and gas parameter $na_s^3\ll1$. These are symbolic model definitions, not supplied numerical parameters. The grand-canonical coherent-state integral has periodic fields and action
$$
S_E=\int_0^{\beta\hbar}d\tau\int d^3x\left[\bar\psi\left(\hbar\partial_\tau-\frac{\hbar^2\nabla^2}{2m_b}-\mu\right)\psi+\frac g2(\bar\psi\psi)^2\right].
$$
The contact term is the normal symbol of $g\hat\psi^{\dagger2}\hat\psi^2/2$. In three dimensions a <contact interaction> needs a regulator and coupling matched to the measured <scattering length>; the displayed $g$ is the low-energy physical coupling. This field integral is useful because an ordered saddle and its quantum/thermal fluctuations follow from the same action.
Stationarity with respect to $\bar\psi$ gives the imaginary-time <Gross–Pitaevskii equation>. A uniform saddle exists at $\mu>0$ with $\psi_0=\sqrt{n_0}e^{i\theta_0}$ and <condensate density> $n_0=\mu/g$. Its arbitrary phase reflects global $U(1)$ <symmetry>; a selected condensate fixes a phase in the thermodynamic symmetry-breaking description. Write $\psi=\sqrt{n_0}+\eta$ after choosing that phase. Linear terms vanish, and the quadratic part is
$$
S_2=\int d\tau\,d^3x\left[\bar\eta\left(\hbar\partial_\tau-\frac{\hbar^2\nabla^2}{2m_b}+\mu\right)\eta+\frac\mu2(\eta^2+\bar\eta^2)\right].
$$
With $\varepsilon_k=\hbar^2k^2/(2m_b)$ and the Fourier convention $e^{i\mathbf k\cdot\mathbf x-i\omega_n\tau}$, the paired quadratic kernel is
$$
\mathcal M(\mathbf k,\omega_n)=\begin{pmatrix}-i\hbar\omega_n+\varepsilon_k+\mu&\mu\\\mu&i\hbar\omega_n+\varepsilon_k+\mu\end{pmatrix},\qquad \det\mathcal M=(\hbar\omega_n)^2+\varepsilon_k(\varepsilon_k+2\mu).
$$
The one-half in the <Nambu spinor> action removes pair double counting. Analytic continuation of the inverse kernel gives \b[the Bogoliubov dispersion]
$$
\boxed{E_k=\sqrt{\varepsilon_k(\varepsilon_k+2gn_0)},\qquad E_k\simeq\hbar c_s k,\qquad c_s=\sqrt{gn_0/m_b}.}
$$
At high <momentum> it approaches the free-particle <kinetic energy> plus its mean-field shift. The crossover scale is the <healing length> $\xi=\hbar/\sqrt{2m_bgn_0}$. Gaussian integration adds a fluctuation <determinant> to the saddle <free energy>. Expressed using the physical three-dimensional coupling, its zero-temperature integrand can be regularized as $\tfrac12[E_k-\varepsilon_k-\mu+\mu^2/(2\varepsilon_k)]$; the subtraction follows from <renormalization of a three-dimensional dilute-gas contact coupling>, not discarding an infinite answer arbitrarily. The thermal contribution is $k_BT\int d^3k/(2\pi)^3\log(1-e^{-\beta E_k})$. A calculation beyond mean field must keep that matching and its thermodynamic derivatives consistent.
The <density-phase action of a Bose gas> makes the physical content clear, $\psi=\sqrt{n_0+\delta n}\,e^{i\theta}$. Up to a periodic density total derivative, the long-wavelength quadratic action contains
$$
S_2=\int d\tau\,d^3x\left[i\hbar\delta n\,\partial_\tau\theta+\frac g2(\delta n)^2+\frac{\hbar^2n_0}{2m_b}(\nabla\theta)^2+\frac{\hbar^2}{8m_bn_0}(\nabla\delta n)^2\right].
$$
<number-phase conjugacy> pairs the density fluctuation and phase in the <quadratic density-phase action>. Dropping the density-gradient term only in the long-wave limit and completing the density Gaussian square gives the <Bose-gas phase-only action>
$$
S_{\theta}=\frac12\int d\tau\,d^3x\left[\frac{\hbar^2}{g}(\partial_\tau\theta)^2+\frac{\hbar^2n_0}{m_b}(\nabla\theta)^2\right].
$$
This has the same sound pole. A term $i\hbar n_0\partial_\tau\theta$ is also present before restricting to the smooth zero-winding sector; although locally a total derivative, it cannot in general be ignored in topologically nontrivial sectors. The density and phase variables describe the same coupled Bogoliubov branch, not a second independent massive particle simply because the density has a static restoring cost.<Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§2.3, printed pp.29–38: superfluid phase stiffness, amplitude fluctuations and phase dynamics]{ref}
A phase gradient carries superflow $\mathbf v_s=(\hbar/m_b)\nabla\theta$. Its energy cost defines the <superfluid number density>. At leading mean-field order the stiffness uses $n_0$, but <condensate density> and <superfluid number density> are different observables: for a Galilean-invariant continuum gas at zero <temperature> the latter is the full density even when interactions deplete the condensate. Bogoliubov coefficients give $v_k^2=[(\varepsilon_k+gn_0)/E_k-1]/2$ and a small zero-temperature depletion of relative order $\sqrt{na_s^3}$ in three dimensions. Large depletion or strong long-wavelength phase fluctuations would invalidate the small-fluctuation saddle approximation.
The <Landau criterion> for creating excitations is $v_L=\inf_{k>0}E_k/(\hbar k)=c_s$ for this homogeneous dispersion. Thus \b[repulsion produces a linear sound mode and a positive critical velocity from the <Landau criterion>], whereas an ideal quadratic gas has zero value of this criterion. Vortices, obstacles and nucleation can reduce an actual flow threshold below $c_s$; the criterion alone is not a complete theory of metastable flow. At finite <temperature> thermal <quasiparticles> reduce the <superfluid number density>. Dimensionality and finite size also matter: the uniform three-dimensional weak-depletion condensate expansion must not be transferred unchanged to a one- or two-dimensional thermodynamic system. The field integral provides systematic corrections only within its dilute, stable and sufficiently weak-fluctuation regime.
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