# Quantum Energies of GR: Applying Bohr-Sommerfeld

In summary, the conversation is discussing the application of the Bohr-Sommerfeld quantization to semiclassical General Relativity. It is suggested that if the energies (or equivalent) are applied using Einstein's equation, the quantum version of curvature cannot be arbitrary and the WKB wave function would be of the form e^(iS/h), where S is the action and the integral is performed over a closed hypersurface in R-4 space. The question of whether the action S can be obtained from the Hilbert-Einstein Lagrangian or if the WKB method can be applied to energies and wavefunctions is also raised.
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.

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

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.

## 1. What is the Bohr-Sommerfeld quantization rule?

The Bohr-Sommerfeld quantization rule is a mathematical formula that describes the allowed energy levels of a quantum system. It states that the action of a system must be equal to a multiple of Planck's constant, h, divided by 2π.

## 2. How does the Bohr-Sommerfeld quantization rule relate to quantum energies?

The Bohr-Sommerfeld quantization rule is used to calculate the quantized energy levels of a quantum system. It takes into account the wave-like nature of particles and the quantization of angular momentum.

## 3. What is the significance of applying the Bohr-Sommerfeld quantization rule to General Relativity?

Applying the Bohr-Sommerfeld quantization rule to General Relativity allows for a better understanding of the quantum properties of gravity. It helps to explain how gravity behaves on a microscale and how it interacts with other fundamental forces.

## 4. Can the Bohr-Sommerfeld quantization rule be applied to all quantum systems?

No, the Bohr-Sommerfeld quantization rule is only applicable to systems that have a well-defined classical limit. This means that it cannot be applied to systems that do not have a clear classical counterpart, such as black holes.

## 5. How does the Bohr-Sommerfeld quantization rule relate to the uncertainty principle?

The Bohr-Sommerfeld quantization rule is related to the uncertainty principle in that it sets limits on the precision with which certain physical quantities can be measured. The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty, and the Bohr-Sommerfeld quantization rule helps to explain this limitation in terms of quantized energy levels.

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