This is a seemingly odd question, but it's come up in a few different areas related to the supercurrent from GL theory. I am happy that one can vary wrt to the vector potential to option the equation for the supercurrent, but a lot of references and literature refer to the supercurrent in an imaginary expression:
\begin{align}
\vec{J}_s = \text{Im}\left( 
\psi^* (\nabla - i \vec{A}) \psi
\right )
\end{align}
This can also be written in terms of a real component, [as is written at the bottom of the first page of this paper][1]. The confusion comes mainly from this [paper by Sadovskyy][2], where they switch between the common representation, where one can write something like
\begin{align}
\vec{J}_s = \frac{1}{2i}\left( 
\psi^* \nabla \psi
- 
\psi \nabla \psi^*
\right )
- |{\psi}|^2 \vec{A}
\end{align}

What I don't understand is how these two are related? I'm sure it's just a simple mathematical trick, but I can't see it at the moment.
Any help or guidance would be greatly appreciated. Thank you.

EDIT: The latter expression for the supercurrent may have differing factors between sources; this is just an example of a broader idea.

  [1]: https://inis.iaea.org/collection/NCLCollectionStore/_Public/27/046/27046815.pdf
  [2]: https://reader.elsevier.com/reader/sd/pii/S0021999115002284?token=0846C1D1FED130DBD72DA0E1A436FE1420F1285E72DDD8C7960F1A4D6AD2BACEB4637C9005CE83CEAA9D065C2C287F3B