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The Bose-gas phase-only action gives a thermal infrared integral $\int d^{d}q/q^2$ and a zero-temperature equal-time integral $\int d^{d}q/|q|$. Thus an infinite uniform short-range interacting gas has no true finite-temperature condensate in $d\le 2$, and the one-dimensional ground state also lacks true condensate long-range order. This does not exclude local density coherence, a finite-size condensate description or two-dimensional algebraic order and superfluidity.^[ref]

Put $\rho ={\rho }_0+\delta \rho$ and retain quadratic fluctuations in the smooth zero-winding sector. The quadratic density-phase action isAt wavelengths long compared with the healing length, drop the density-gradient term. Completing the square givesThe real Gaussian density integral, or its equivalent contour shift, contributes only a field-independent determinant. Absorb that normalization and the uniform saddle action into $S_0$. To this quadratic long-wave accuracy,The Gaussian extension of $\delta \rho$ to the full real line is a fluctuation approximation around positive ${\rho }_0$; it is not an exact replacement of the global density constraint. The compact field's vortex/winding sectors also lie beyond this smooth-phonon integral.

The Bose-gas phase-only action is a continuum harmonic chain with Euclidean inverse propagator ${\omega }_n^2/g+{\rho }_0{k}^2/m$. Continuing to real frequency givesRestoring $\hslash$ gives $E_k\sim \hslash c_s|k|$ for a wavenumber $k$. This is the phonon branch of the Bogoliubov spectrum. Keeping the omitted density-gradient term replaces $g$ by $g+k^2/(4m{\rho }_0)$ and yields $E_k^2=c_s^2{k}^2+k^4/(4m^2)$ in the adopted units, so the full quadratic spectrum is $\sqrt{{ϵ}_k({ϵ}_k+2g{\rho }_0)}$ with ${ϵ}_k=k^2/(2m)$.

The phase-only action also tests the condensate assumption. At zero temperature its equal-time phase variance has an infrared contribution $\int d^{d}k/|k|$, logarithmically divergent in one dimension; at positive temperature the zero Matsubara mode gives $\int d^{d}k/k^2$, divergent in dimensions at or below two. Thus this same low-energy theory exposes the regimes where the mean-field condensate cannot describe true thermodynamic long-range order. It permits a finite infrared fluctuation at zero temperature for $d\ge 2$ and at positive temperature for $d>2$, within the remaining weak-coupling assumptions.

At wavelengths with $q^2/(4m{\rho }_0)\ll g$, neglect the density-gradient term in the quadratic density-phase action. Complete the square in its remaining density terms:The convergent Gaussian integral can be performed by a contour shift or by its Fourier-transform identity. Its determinant is independent of $\phi$ at this order. Combining that normalization with the saddle action into a constant $S_0$ givesThe saddle contribution to $S_0$ is $-\beta V{\mu }^2/(2g)$; the additional measure-dependent constant need not be assigned an invented numerical value. The density Gaussian extends the fluctuation integration beyond its exact positivity domain, which is justified only while fluctuations around ${\rho }_0>0$ are small. The smooth zero-winding assumption is also required as discussed above. A constant global phase zero mode gives a finite compact-group volume, handled separately from nonzero phonon modes.

Fourier transforming the Bose-gas phase-only action gives a kernel ${\omega }_n^2/g+({\rho }_0/m)q^2$. Analytically continuing ${\omega }_n\mapsto -i(\omega +i{0}^{+})$ gives its real-frequency poles,In the present units the excitation energy equals $\omega$. Each real Fourier coordinate is a harmonic oscillator with inertia coefficient $1/g$ and spring coefficient ${\rho }_0{q}^2/m$, hence frequency equal to the square root of their ratio. This is the direct relation to a quantum harmonic chain. The mode is a gapless sound phonon; its finite speed reflects both the compressibility $\partial \rho /\partial \mu =1/g$ and the phase stiffness ${\rho }_0/m$.

Retaining the density-gradient term provides a useful check. Its integration instead gives a temporal coefficient $1/A_q$, soThis is the Bogoliubov spectrum. Its small-$q$ limit is the derived sound branch; at large $q$ within the contact model it crosses towards particle-like behaviour. The crossover in the squared dispersion is $q\sim 2\sqrt{mg{\rho }_0}$, precisely the scale beyond which the discarded density gradient is not small.

The same action checks the dimensional restriction in the saddle analysis. At nonzero temperature its zero Matsubara mode gives phase fluctuations proportional to $\int d^{d}q/q^2$; at zero temperature integrating the frequency gives $\int d^{d}q/|q|$. Their infrared divergences invalidate uniform true long-range order for $d\le 2$ at nonzero temperature and for $d=1$ at zero temperature. The long-wave collective description can remain useful for correlations even where a global condensate saddle is not a literal symmetry-broken state. The contact coupling here is an effective low-energy parameter; this calculation does not determine ultraviolet renormalization or trap-specific surface modes.

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 $g=4\pi {\hslash }^2{a}_s/m_b$ and gas parameter $n{a}_s^3\ll 1$. These are symbolic model definitions, not supplied numerical parameters. The grand-canonical coherent-state integral has periodic fields and actionThe contact term is the normal symbol of $g{\hat{\psi }}^{†2}{\hat{\psi }}^2/2$. 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 isWith ${\epsilon }_k={\hslash }^2{k}^2/(2m_b)$ and the Fourier convention $e^{i\mathbf{k}\cdot \mathbf{x}-i{\omega }_n\tau }$, the paired quadratic kernel isThe one-half in the Nambu spinor action removes pair double counting. Analytic continuation of the inverse kernel gives the Bogoliubov dispersionAt high momentum it approaches the free-particle kinetic energy plus its mean-field shift. The crossover scale is the healing length $\xi =\hslash /\sqrt{2m_{b}g{n}_0}$. 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 $\frac{1}{2}[E_k-{\epsilon }_k-\mu +{\mu }^2/(2{\epsilon }_k)]$; the subtraction follows from renormalization of a three-dimensional dilute-gas contact coupling, not discarding an infinite answer arbitrarily. The thermal contribution is $k_{B}T\int d^{3}k/(2\pi {)}^3\log (1-e^{-\beta E_k})$. 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 containsnumber-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 actionThis has the same sound pole. A term $i\hslash n_0{\partial }_{\tau }\theta$ 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.^[ref]

A phase gradient carries superflow ${\mathbf{v}}_s=(\hslash /m_b)\nabla \theta$. 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 $v_k^2=[({\epsilon }_k+g{n}_0)/E_k-1]/2$ and a small zero-temperature depletion of relative order $\sqrt{n{a}_s^3}$ 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 $v_L={\inf }_{k>0}{E}_k/(\hslash k)=c_s$ for this homogeneous dispersion. Thus 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.