I encountered this equation when I was reading the article "Black Hole Superradiance Signatures of Ultralight Vectors"
$$A_\mu=\frac{1}{\sqrt{2m}}\Big(\psi_\mu (\vec{r})\exp(-i\omega t)+\mathrm{c.c.}\Big),$$ where $A$ is a vector field in QFT, and $m$ is the mass of the vector boson. This is a non-relativistic expansion. $A$ itself satisfies the Proca equation of motion and Lorenz gauge, $$D_\mu F^{\mu\nu}=mA^{\nu}$$ $$D_\mu A^\mu=0.$$ Where $D$ is the covariant derivative in the Kerr geometry. But to the leading order, it is just the derivative in the flat spacetime.
The above expansion will result in $\psi$ satisfying a Schrödinger-like equation. My question is that how is the normalization of $\psi$ determined in this case? Should each component of $\psi$ integrate to 1? Or should $|\psi_x|^2+|\psi_y|^2+|\psi_z|^2$ integrate to 1?
