Quantum current density in general relativity

[FunkyDwarf]

Hi All,

As shown here

http://en.wikipedia.org/wiki/Klein–Gordon_equation#Gravitational_interaction

one can modify the derivative operator in the Minkowski Klein-Gordon equation to generate the GR case for an arbitrary metric.

My question is, what would the quantum current density j be in this case? Given here
http://en.wikipedia.org/wiki/Four-current
is the classical current density in the EM case but that's not really what I'm after.

Furthermore, if i generate a radial wave equation from the GR KG equation in some metric and transform it in a Schrödinger like form, can I just use the standard tools of non-rel QM and use $\psi^* \frac{d}{dx} \psi$ for j?

Cheers!

 

[jfy4]

I think a lagrangian like
$$\mathcal{L}=\bar{\psi}\gamma^\mu \partial^\nu \psi g_{\mu\nu}$$
would get you essentially
$$\bar{\psi}\gamma^\mu \partial^\nu \psi=\theta^{\mu\nu}$$
when varied by the metric, where the current corresponds to the $0\nu$ part.

But I kinda made that up, so we should keep that tentative lol.