 Contact-interaction Bose-gas field integral

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 $(\hslash {\omega }_n{)}^2+{\epsilon }_k({\epsilon }_k+2g{n}_0)$, giving the Bogoliubov spectrum. In three dimensions the physical low-energy coupling is $g=4\pi {\hslash }^2{a}_s/m$ and the weak-depletion parameter is $n{a}_s^3\ll 1$. Contact coupling beyond mean field needs ultraviolet matching to the scattering length.