# General Relativity - Einstein field equation and quantum field theory

Einstein field equation has many solutions. Out of them, is there any solution that is incompatible with quantum field theory?

Also, what solutions of Einstein field equation would be incompatible with some variants of string theory? (let us restrict the scope of string theory into superstring theory.)

Thanks.

• Can you be specific about what you mean by 'compatible'? GR and QFT are written in incompatible languages. Do you mean to ask if there is an obstruction to using a solution of GR as a fixed background for a QFT? May 3, 2012 at 5:57
• @BebopButUnsteady yes. I was meaning an obstruction to using a solution of GR as a fixed background for a QFT (and string theory). May 3, 2012 at 6:01
• @user27515: Our QFTs work on a flat spacetime, and flat spacetime is a solution of the Einstein equations. Then again, to put those puzzle pieces together in a detailed model, you would only be allowed such fields, which produce an energy, which corresponds to the generated gravitational field (flat metric). There lies the problem. If by "with quantum field theory" you mean the standard model, then there aren't too many possibilities you can go, I believe. May 3, 2012 at 12:06
• To further clarify: by "obstruction" do you mean that it's impossible to construct any quantum field theory? Or would you be interested in cases where the curved space time causes difficulties? Examples of the latter are well known. May 5, 2012 at 9:27
• @twistor59 the latter case. May 7, 2012 at 14:10

I'll mention just one example of the complexity that curved space introduces into the quantization process.

Consider Minkowski space quantization of the free Klein Gordon Field, which satisfies

$$(\Box+m^2)\phi=0$$

A fundamental step in the procedure is performing the mode expansion

$$\phi(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}}(a_{\bf{k}}e^{-ikx}+a^*_{\bf{k}}e^{ikx})d^{3}\bf{k}$$

Here we have a splitting into a negative frequency part

$$\phi^-(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}a_{\bf{k}}e^{-ikx}}d^{3}\bf{k}$$ and a positive frequency part

$$\phi^+(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}a^*_{\bf{k}}e^{ikx}}d^{3}\bf{k}$$

Upon quantization, the $a^*_{\bf{k}}$ in the positive frequency parts become creation operators and the $a_{\bf{k}}$ in the negative frequency parts annihilation operators. The splitting is covariant - the exponentials contain Lorentz scalars.

Now if we try to do the same thing in curved space to the (covariant form of) the Klein Gordon equation, we can find spacetimes for which there's no clear way to perform this splitting because in general, there's no "natural" time coordinate. In the Minkowski case, we had the action of the Poincare group to allow us to deal with the different possible time coordinates - we even had a Poincare invariant vacuum state, but here there's no equivalent of the Poincare group action.

The particle content of the theory depends upon this splitting, and the ambiguity we have in the curved case (or even in the flat case if we allow non inertial frames) is the origin of the Hawking and Unruh effects.

That was just a single example of a problem that crops up right from the word "go" in curved space quantization. There has been a lot of effort expended over the last few decades studying quantization on curved spacetimes. For a review, see here.