= Density-phase action of a Bose gas
{title2=$\psi=\sqrt\rho e^{i\phi}$}
In units $\hbar=1$, polar fields rewrite the coherent-state action as
$$
S=\int\left[i\rho\partial_\tau\phi+\frac{|\nabla\rho|^2}{8m\rho}+\frac\rho{2m}|\nabla\phi|^2-\mu\rho+\frac g2\rho^2\right],
$$
after removing the periodic total density <derivative>. The potential has positive-density minimum $\rho_0=\mu/g$ for $\mu>0$. Density curvature gives a finite static amplitude cost, while phase changes cost only <derivatives>. <Number-phase conjugacy> couples their dynamics; the two fields do not automatically represent two independent excitation branches.
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