I am a little confused about a few papers I read on the Einstein-Klein-Gordon (EKG) equations.
From what I understood one takes the energy-stress-tensor of the scalar field:
$$T_{\mu\nu } = −\partial_\mu \varphi ∂_\nu \varphi − \frac12 g_{\mu\nu}∂_\alpha\varphi∂_\alpha\varphi − V(φ )$$
$$V(φ) = −\frac12 (mφ)^2 + \fracκ4φ^4$$
Where $κ$ is the usual self-interaction coupling constant .
Then this stress-tensor is plugged into Einstein's equation and solved usually with the Schwarzschild or other convenient metrics.
Now from what I know about QFT, isn't $φ$ an operator that maps the Hilbert space $H$ of particle states to $H$ itself? Doesn't that make the components of the energy-stress-tensor observables and thus operators as well? If so, then how can one equate the components of the Einstein tensor (which are purely geometric tensor fields) to operators?
I once read that people (as of today) usually plug in < $T_{μν} $ > in Einstein's equation, but in the papers I read they directly used the operator itself and not it's expectation value.
I also wondered how people plug the Maxwell stress tensor into Einstein's equations in a similar way? I get that the EM and scalar field are real valued fields, but shouldn't we use the expectation values in Einstein's equation?
What am I missing? Here is a link to one such paper: http://arxiv.org/abs/0805.3211