= Superconducting phase-density action
{title2=$\mathcal L_\theta=\chi_p(\hbar\dot\theta+q\phi)^2/2-\rho_0(\hbar\nabla\theta-q\mathbf A)^2/(2m_*)$}
A lossless low-energy charged-condensate <action> has a positive compressibility coefficient $\chi_p$, effective pair density $\rho_0$, mass $m_*$ 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 $\omega^2=q^2\rho_0/(\epsilon_0m_*)+[\rho_0/(m_*\chi_p)]k^2$ 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. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§2.3, phase effective theory and coupling to electromagnetism]{ref}
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