Past exam of the theoretical physics I course of the University of Cambridge / 2019 / 3 / e / Solution

At long wavelength,This is the sound-like Goldstone mode of the broken global $U(1)$ continuous symmetry. A spatially uniform infinitesimal phase change is $\delta \psi =i{\psi }_0\delta \theta$, entirely in the ${\chi }_2$ direction, and costs no potential energy. Slow spatial variations acquire gradient energy and propagate as sound. Thus the dispersion relation realizes Goldstone's theorem through a gapless excitation; it is not a massive phase oscillation.

The amplitude and phase are canonically paired by the first-order time derivative in the quadratic condensate fluctuation Lagrangian. There is one positive-frequency branch, rather than an additional independent amplitude branch of a relativistic second-order-time theory. At high momentum,so the same branch becomes particle-like. The healing length $\xi =\hslash /\sqrt{2M\mu }$ sets the crossover scale.