# Does quantizing metric fields mean quantum gravity?

• A
• Haorong Wu
The divergence of the spectrum of the field amplitudes is an issue that string theory must solve in order to be a complete theory.In summary, the papers discuss canonical quantization for gravitational perturbation. This canonical quantization belongs to another quantum gravity theory, which is not quantum gravity.

#### Haorong Wu

TL;DR Summary
In some papers, the gravitational field is canonically quantized. Does this lead to quantum gravity?
(I am not sure which forum this post belongs to. Hope someone kindly helps me move it to a proper forum.)

In papers, for example, here, here, and here, the authors start from the Lagrangian for matters and gravitational fields, then Dirac's constrained canonical quantization is used. They achieve annihilation and creation operators for gravitational perturbation.

I am not familiar with quantum gravity theories. I know there are several competing candidates, including String theory and LQG. These theories face different problems. Does this kind of canonical quantization belong to some of them, or it does not correspond to quantum gravity?

Thanks.

This is not quantum gravity, in those papers metric field is not quantized. This is decoherence of quantum matter caused by classical gravitational interaction.

• vanhees71
@Demystifier. As in the first paper, Eqs. (23) and (24) introduce creation and annihilation operators ##b## and ##b^\dagger##. Well, I will copy from the paper

We express the quantized TT perturbations in terms of creation and annihilation operators $$\hat h_{ij}(x)=\sqrt {\frac 2 \kappa} \sum_r \int \frac {d^3k}{(2\pi)^3 \sqrt{2 \omega_k}} L^r_{ij}(k)(\hat b_r(k)e^{ikx}+\hat b^\dagger_r(k)e^{-ikx} )$$

I am confused because I only see creation and annihilation operators in quantum theory. And here, ##b## and ##b^\dagger## are operators for the gravitational field. The corresponding operators for the matter field are labeled by ##a## and ##a^\dagger##.

So these operators can be used on classical fields?

• vanhees71 and Demystifier
You are actually right, in this paper it is quantum gravity. Sorry!

• vanhees71
Thanks, @Demystifier. No need to apologize.

I have gone through the papers again. It seems they are mainly considering an effective field theory in a low-energy limit. If I am correct, the non-renormalizable problem in String theory happens because of the ultraviolet divergence. I am not sure what problems are faced by the LQG theory. Some say that this theory will face a non-renormalizable problem as well when it takes classical limits.

Therefore, is this true that different quantum gravity theories coincide in the low-energy limit, and we can study quantum gravity in an effective field theory?

Haorong Wu said:
is this true that different quantum gravity theories coincide in the low-energy limit, and we can study quantum gravity in an effective field theory?
I'm not sure we even know what the low energy limit is of any quantum gravity theory, in the sense of actually being able to derive it. Most physicists appear to believe that any valid quantum gravity theory will have as its low energy limit the known quantum field theory of a massless spin-2 field, which was studied in the 1960s and 1970s and whose classical limit is known to be General Relativity. (This theory is also known to be non-renormalizable, but that is not an issue if we view it, as most physicists do, as an effective field theory that describes the low energy degrees of freedom.) However, I think that belief is based on physical intuition rather than any actual derivation from a quantum gravity theory such as string theory or LQG.

• vanhees71 and Haorong Wu
Haorong Wu said:
If I am correct, the non-renormalizable problem in String theory happens because of the ultraviolet divergence.
No, in string theory there is no ultraviolet divergence. The non-renormalizability due to ultraviolet divergences is a feature of the field theory of gravity.

• vanhees71
Demystifier said:
No, in string theory there is no ultraviolet divergence. The non-renormalizability due to ultraviolet divergences is a feature of the field theory of gravity.
Oh, sorry. I must misunderstand an introductory talk about quantum gravity given by Hermann Nicolai.