Quantum Energies of GR: Applying Bohr-Sommerfeld

[Karlisbad]

If we apply the Bohr-Sommerfeld quantization to GR (semiclassical)

$$\oint_{S} \pi _{ab} dg_{ab}=\hbar (n+1/2)$$

In this case if "Energies" (or whatever you call energy since in Quantum GR H=0 for the "Hamiltonian constraint" ) then using Einstein equation we see that the "curvature" (quantum version) can't be arbitrary (curvature of the surface is quantizied) and that the WKB wave function would be:

$$\Psi=e^{iS/\hbar}$$ of course the question there is if we can get the action S from the HIlbert-Einstein Lagrangian, or if the WKB method for energies and wavefunctions applied here.

 

[Demystifier]

A technical question:
What is S in your first equation?
(Obviously, not the same as S in the second one.)

 

[Karlisbad]

Oh..sory "Demystifier"..i forgot to change the letter.. one "S" is the action the other is just to indicate that the integral is performed over a close Hyper-surface on R-4 space (in a similar fashion ot usual WKB formula) i will change it.