= Solution
For conventional <superconductivity>, an attractive interaction pairs opposite-spin <fermions>. A complex pairing <Hubbard–Stratonovich transformation> introduces a <Cooper pair> field $\Delta$. The fermionic fields then appear quadratically, giving the <pair-field effective action of a superconductor>
$$
S_{\rm eff}[\Delta,A]=\int\frac{|\Delta|^2}{g}-\operatorname{Tr}\ln G^{-1}[\Delta,A]+S_{\rm EM},\qquad g>0.
$$
The trace convention counts each independent spin-pair block once. The uniform saddle is the <BCS gap equation>; the fermionic excitations have $E_k=\sqrt{\xi_k^2+|\Delta|^2}$ in the fully gapped case. A slow-field expansion near the transition yields <Ginzburg-Landau theory of superconductivity>. Normalize its pair field $\Psi$ so $|\Psi|^2$ is an effective pair density, with pair mass $m_*$ and signed charge $q=-2e$:
$$
F=\int d^3x\left[a|\Psi|^2+\frac b2|\Psi|^4+\frac1{2m_*}|(-i\hbar\nabla-q\mathbf A)\Psi|^2+\frac{\mathbf B^2}{2\mu_0}\right],\qquad b>0.
$$
For $a<0$, the uniform minimum has $\rho_0=|\Psi|^2=-a/b$, with arbitrary constant phase. Writing $\Psi=(\sqrt{\rho_0}+h)e^{i\theta}$ separates the <amplitude mode> $h$ from the phase. The local potential has curvature term $2|a|h^2$ and coherence scale $\xi=\hbar/\sqrt{2m_*|a|}$. A neutral condensate has a gapless phase <Goldstone mode> when suitable dynamics is included.
The charged theory is invariant under
$$
\theta\mapsto\theta+q\chi/\hbar,\quad\mathbf A\mapsto\mathbf A+\nabla\chi,\quad\phi\mapsto\phi-\partial_t\chi.
$$
The <gauge-invariant condensate momentum> is $\hbar\nabla\theta-q\mathbf A$. Only $\hbar\nabla\theta-q\mathbf A$ and $\hbar\dot\theta+q\phi$ enter physical phase dynamics. At nearly constant amplitude, varying $F$ gives
$$
\boxed{\mathbf j=\frac{q\rho_0}{m_*}(\hbar\nabla\theta-q\mathbf A),\qquad
\lambda^{-2}=\frac{\mu_0q^2\rho_0}{m_*}.}
$$
For a smooth phase, taking the curl and using <Ampère's law> gives $\nabla^2\mathbf B=\mathbf B/\lambda^2$: the <Meissner effect> screens a static bulk <magnetic field> over the <London penetration depth> $\lambda$. In a gauge with $\theta=0$, the phase stiffness is a quadratic term in $\mathbf A$, the superconducting version of the <Higgs mechanism>.
To identify the dynamical mode, a simple lossless long-wave <superconducting phase-density action> is
$$
\mathcal L=\frac{\chi_p}{2}(\hbar\dot\theta+q\phi)^2-\frac{\rho_0}{2m_*}(\hbar\nabla\theta-q\mathbf A)^2+\frac{\epsilon_0\mathbf E^2}{2}-\frac{\mathbf B^2}{2\mu_0},\qquad \chi_p>0.
$$
Here $\chi_p$ is the pair compressibility coefficient. Eliminating the longitudinal electric potential gives
$$
\boxed{\omega^2=\omega_p^2+c_s^2k^2,\qquad \omega_p^2=\frac{q^2\rho_0}{\epsilon_0m_*},\qquad c_s^2=\frac{\rho_0}{m_*\chi_p}.}
$$
Thus the neutral sound mode becomes a gapped longitudinal <plasma oscillation>; transverse electromagnetic waves acquire $\omega^2=c_{\rm vac}^2k^2+c_{\rm vac}^2/\lambda^2$ in this simple bulk model. This redistribution of the phase degree of freedom into electromagnetic collective modes is the <Anderson-Higgs mechanism in a superconductor>. Local <gauge symmetry> is a redundancy, not an independently observable symmetry whose breaking creates another massless particle. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§2.3, superconductors, superfluids and gauged phase dynamics]{ref}
The static functional alone does not determine a frequency or damping rate. The displayed mode dispersions require the added dynamical assumptions; real screening, normal carriers, boundaries and dissipation modify them. The amplitude fluctuation is distinct from the phase mode and is not removed by the gauge choice.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2013/paper-1-qcm2013-phase-modes.png]
{title=Illustrative long-wave phase modes: neutral sound and the charged condensate plasma gap}
{source=paper-1-qcm2013-phase-modes.py}
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