= Contact-interaction Bose-gas field integral
{title2=$S_E=\int[\bar\psi(\hbar\partial_\tau-\hbar^2\nabla^2/(2m)-\mu)\psi+g|\psi|^4/2]$}
For a uniform dilute repulsive gas, the periodic complex-field integral has uniform saddle $n_0=\mu/g$ for $\mu>0$. The paired Gaussian fluctuation kernel has <determinant> $(\hbar\omega_n)^2+\varepsilon_k(\varepsilon_k+2gn_0)$, giving the <Bogoliubov spectrum>. In three dimensions the physical low-energy coupling is $g=4\pi\hbar^2a_s/m$ and the weak-depletion parameter is $na_s^3\ll1$. Contact coupling beyond mean field needs ultraviolet matching to the <scattering length>.
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