Gauge-coupling sign in Ginzburg-Landau superconductivity ($D_{\eta }=\nabla -i\eta qA/\hslash$)

For energy with kinetic term ${\hslash }^2|D_{\eta }\psi {|}^2/(2m)$, $\eta =\pm 1$, variation gives $-{\hslash }^2{D}_{\eta }^2\psi /(2m)+(\alpha +\beta |\psi {|}^2)\psi =0$ and current $j=\eta q\hslash \mathrm{Im}({\psi }^{\ast }\nabla \psi )/m-q^2|\psi {|}^{2}A/m$. Thus changing the sign of the covariant derivative changes the phase-gradient term of the current. The displayed energy, field equation and current must use the same convention.

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.