Superconducting phase-density action (${\mathcal{L}}_{\theta }={\chi }_p(\hslash \dot{\theta }+q\varphi {)}^2/2-{\rho }_0(\hslash \nabla \theta -q\mathbf{A}{)}^2/(2m_{\ast })$)

A lossless low-energy charged-condensate action has a positive compressibility coefficient ${\chi }_p$, effective pair density ${\rho }_0$, mass $m_{\ast }$ and signed charge $q$. Coupling its phase to the electromagnetic potentials through the displayed combinations preserves gauge symmetry. Adding the Maxwell action and eliminating the longitudinal electric potential gives ${\omega }^2=q^2{\rho }_0/({ϵ}_0{m}_{\ast })+[{\rho }_0/(m_{\ast }{\chi }_p)]k^2$ in a simple homogeneous bulk model. Removing the electromagnetic coupling leaves neutral sound. Static Ginzburg-Landau theory of superconductivity alone cannot determine these frequencies or their damping. ^[ref]