 Solution

For a dilute three-dimensional weakly interacting Bose gas with repulsive short-range scattering, use a contact coupling $g=4\pi {\hslash }^2{a}_s/m>0$ and $n{a}_s^3\ll 1$. In imaginary time ${\tau }_E$, the grand-canonical action isThe uniform saddle has $n_0=\mu /g$. Its real-time equation is the Gross–Pitaevskii equation; quantum depletion means condensate and total density need not coincide exactly.

Writing $\psi =\sqrt{n_0}+\delta \psi$ and retaining quadratic fluctuations gives a Nambu kernel with diagonal ${ϵ}_k+g{n}_0$, anomalous coupling $g{n}_0$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. The Bogoliubov quasiparticle dispersion isPhase and density fluctuations form the gapless sound mode associated with broken global $U(1)$ symmetry. At large momentum the spectrum crosses towards particle behaviour; the healing length is $\xi =\hslash /\sqrt{2mg{n}_0}$. The depletion integral $\int d^{3}k{v}_k^2/(2\pi {)}^3$ is finite in three dimensions and gives $(n-n_0)/n\simeq 8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$ to leading order. The contact coupling denotes a scattering-length-renormalized interaction; naive ultraviolet zero-point integrals require renormalization.

The condensate phase gives superfluid velocity ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$ and quantized circulation $\oint {\mathbf{v}}_s\cdot d\mathbf{l}=2\pi \hslash \ell /m$. The Landau criterion, $v_c={\min }_k{E}_k/(\hslash k)$, gives $c_s$ for this homogeneous Bogoliubov spectrum. Boundaries and vortex nucleation can cause dissipation below that value, so it is not a universal measured critical speed. Condensation and superfluid stiffness are related but distinct; dimensional fluctuations can remove true condensate order while retaining superfluid responses.