Gauge-invariant condensate momentum (${\mathbf{p}}_s=\hslash \nabla \theta -q\mathbf{A}$)

A pair field transforms as $\Psi \mapsto e^{iq\chi /\hslash }\Psi$ when $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$. Its phase gradient alone is gauge dependent; ${\mathbf{p}}_s=\hslash \nabla \theta -q\mathbf{A}$ is invariant. With $\varphi \mapsto \varphi -{\partial }_t\chi$, the temporal combination $\hslash \dot{\theta }+q\varphi$ is also invariant. The supercurrent is $\mathbf{j}=q{\rho }_s{\mathbf{p}}_s/m_{\ast }$. A negative electronic pair charge is consistent with either sign of current once this convention is retained throughout.