Coherent Excited States in the Theory of Superconductivity: Gauge Invariance and the Meissner Effect

We discuss the coherent states generated in the Bardeen, Cooper, and Schrieffer theory of superconductivity by the momentum displacement operator ${\rho }_{\mathrm{Q}}=\Sigma n^{}\exp \left(i\mathrm{Q}·{\mathrm{r}}_n\right)$. Without taking into account plasma effects, these states are like bound Cooper pairs with momentum $\hslash \mathrm{Q}$ and energies lying in the gap, and they play a central role in the explanation of the gauge invariance of the Meissner effect. Long-range Coulomb forces recombine them into plasmons with equations of motion unaffected by the gap. Central to the argument is the proof that the non-gauge-invariant terms in the Hamiltonian of Bardeen, Cooper, and Schrieffer have an effect on these states which vanishes in the weak-coupling limit.