= Quadratic condensate fluctuation Lagrangian
{title2=$\mathcal L_2=\frac{i\hbar}{2}(\chi^*\dot\chi-\chi\dot\chi^*)-\frac{\hbar^2}{2M}|\nabla\chi|^2-\frac\mu2(\chi+\chi^*)^2$}
Expand a repulsive <Bose-Einstein condensate> around the real uniform saddle $\psi_0=\sqrt{\mu/g}$ as $\psi=\psi_0+\chi$. After removing a constant and a temporal <boundary term>, the quadratic <Lagrangian density> is the displayed expression. Its equation $i\hbar\dot\chi=-\hbar^2\nabla^2\chi/(2M)+\mu(\chi+\chi^*)$ mixes a fluctuation and its <complex conjugate>. Writing $\chi=\chi_1+i\chi_2$ gives canonically paired amplitude and phase fluctuations and one positive <Bogoliubov quasiparticle dispersion> branch, $\hbar\omega=\sqrt{\epsilon_k(\epsilon_k+2\mu)}$ with $\epsilon_k=\hbar^2k^2/(2M)$. The phase direction is a <Goldstone mode>; its zero uniform restoring force does not produce a second independent positive-frequency amplitude branch in this first-order-in-time theory.
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