 Quadratic condensate fluctuation Lagrangian

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 $i\hslash \dot{\chi }=-{\hslash }^2{\nabla }^2\chi /(2M)+\mu (\chi +{\chi }^{\ast })$ 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, $\hslash \omega =\sqrt{{ϵ}_k({ϵ}_k+2\mu )}$ with ${ϵ}_k={\hslash }^2{k}^2/(2M)$. 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.