= Solution
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:
$$
\frac g2(\delta\rho)^2+i\delta\rho\partial_\tau\varphi
=\frac g2\left(\delta\rho+\frac i g\partial_\tau\varphi\right)^2
+\frac1{2g}(\partial_\tau\varphi)^2.
$$
The convergent <Gaussian integral> can be performed by a contour shift or by its Fourier-transform identity. Its <determinant> is independent of $\varphi$ at this order. Combining that normalization with the saddle <action> into a constant $S_0$ gives
$$
\boxed{Z=e^{-S_0}\int\mathcal D\varphi\,e^{-S_{\rm eff}[\varphi]},\qquad
S_{\rm eff}=\frac12\int_0^\beta d\tau\int_Vd^dr
\left[\frac{(\partial_\tau\varphi)^2}{g}+\frac{\rho_0}{m}|\nabla\varphi|^2\right].}
$$
The 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 $\omega_n\mapsto-i(\omega+i0^+)$ gives its real-frequency poles,
$$
\boxed{\omega(q)=c|q|,\qquad c=\sqrt{g\rho_0/m}.}
$$
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 $\rho_0q^2/m$, 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 $1/A_q$, so
$$
\omega^2(q)=\frac{\rho_0 q^2}{m}\left(g+\frac{q^2}{4m\rho_0}\right)
=c^2q^2+\frac{q^4}{4m^2},
\qquad\omega(q)=\sqrt{\epsilon_q(\epsilon_q+2g\rho_0)},\quad\epsilon_q=\frac{q^2}{2m}.
$$
This 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 $q\sim2\sqrt{mg\rho_0}$, 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 $\int d^dq/q^2$; at zero <temperature> integrating the <frequency> gives $\int d^dq/|q|$. Their <infrared divergences> invalidate uniform true <long-range order> for $d\le2$ 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.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2015/paper-1-bose-spectrum.png]
{title=Original dimensionless Bogoliubov dispersion and its sound and particle limits}
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