The Einstein-Klein-Gordon (EKG) equations

Question

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

Answer 1

This is not how I would usually go about formulating this problem. When i think of the Einstein Klein-gordon equation, I start from an action principle:

$$S = \int d^{4}x\sqrt{|g|}\left(\frac{1}{16\pi G}R -\left[\nabla_{a}\phi\nabla^{a}\phi + V(\phi)\right]\right)$$

Which will then yield EOM:

$$R_{ab} - \frac{1}{2}Rg_{ab} = 8\pi G\left(\nabla_{a}\phi\nabla_{b}\phi -\frac{1}{2}g_{ab}\left[\nabla_{c}\phi\nabla^{c}\phi + V(\phi)\right]\right)$$

and

$$\nabla^{c}\nabla_{c}\phi - V'(\phi) = 0$$

From here, the question is what are you doing with these equations?

Are you looking at general relativity in the context of a classical Klein-Gordon source? If so, you just solve these equations.

Are you trying to do semi-classical gravity? Well, then, you set your metric to a fixed background metric, and just analyze the Klein-Gordon EOM using the appropriate $\nabla$ for this background metric, quantizing the field using a scheme like you'll find in Wald's book.

Are you looking to work through the back-reaction of semi-classical effects on the background metric? Well, then you need to write down $g_{ab} = g^{0}_{ab} + g^{1}_{ab}$ where $g^{1}_{ab} \ll g^{0}_{ab}$, assume that $\phi$ is first-order, and substitute the expectation value of your solved $\phi$ in on the right hand side, and solve for $g^{1}_{ab}$ in this limit.

Or are you trying to do something else? If you want to treat this as a fully quantum problem, you're going to need to first quantize gravity.