Finding a quantum theory of gravity

One of the hardest problem in fundamental physics is to find a consistent theory of Quantum Gravity (QG) that matches with current experimental data. Gravity can be quantized as an effective field theory, but we know that this description breaks down at the Planck scale.

Many people try to tackle QG and a common mistake that many of them make is that they fail to understand the problem. For some reason there is a misconception that we do not have a theory of QG. This misconception wrongly motivates people to come up with their own version of QG.

But the problem of QG is exactly the opposite. There is no shortage of QG theories. On the contrary, we have too many of these, infinitely too many.

Let me explain. QG is a non-renormalizable Quantum Field Theory (QFT). This means that we can define it as an effective field theory at low energies, but as we try to extend the theory to high energies, new interactions would arise in the theory. Each such interaction adds a counter term to the theory, which adds a new free parameter to the theory. There are infinitely many counter terms, generating infinitely many free parameters.

Therefore, we have a theory of QG with infinitely many parameters that have to be determined by experiment. The fact that there is no experimental data to determine these parameters does not make our lives any easier.

I hope that this didn’t scare you off from working on QG, because this was the easy part about quantizing gravity. A bigger issue is that exactly at the Planck scale, where QG becomes non-renormalizable, it also becomes strongly coupled. But our effective field theory approach to quantization is a perturbative approach. Once the coupling is strong, of the order of 1, the perturbative approach fails.

But all is not lost. There is still a workaround. We can cut off the theory at high energies, above the energies reached by any current experiment, yet bellow the Planck scale. This is referred to as a UV cut-off. Adding a cut-off is actually a standard procedure in QFT called regularization (in practice, dimensional regularization is usually used). Without regularization, infinities show up in QFT calculations. By adding a regularization, all intermediate calculations become finite. Then, at the end of the calculation we can remove the regularization by pushing the UV cut-off to infinity. If the QFT is well behaved, all the infinities in the intermediate calculations would cancel out and the final results would be finite.

What happens if the QFT is not well behaved, like in the case of QG? Well, nobody said that we have to take the UV cut-off to infinity. Sure, taking the UV cut-off to infinity makes our life easy, as it means that we do not need to worry about the details of the UV cut-off effecting our results.

But who said that life is easy? Maybe there is a real UV cut-off in nature. So far, there is no experimental data that supports a finite UV cut-off. On the contrary, a UV cut-off would break Lorentz invariance and there is tons of experimental data showing that there are very tight constraints on Lorentz Invariance Violation (LIV). For example, according to this paper, the lower limit for LIV is actually above the Planck scale. Therefore, if someone is interested in going this route, they need to come up with a UV cut-off that is consistent with existing limits on LIV and also results in a consistent QFT.

Just a little word of warning — this endeavor might be a bit more complicated than I made it look. You see, there are two approaches, perturbative and non-perturbative.

The perturbative approach has the advantage that it starts at low energies, so you can choose your starting point as the current standard model for particle physics together with general relativity. The downside is that you will have to show that the perturbative expansion gives some usable results. Notice that I am not asking for a consistent theory. My bar is much lower. For example, setting the UV cut-off to 1% of the Planck scale might have been good enough, if it wasn’t ruled out by experimental data constraining LIV.

In the non-perturbative approach, you need not worry about a consistent expansion. Instead you need to worry about reproducing matter and interactions at low energies. For example, if the theory is defined on the lattice, its content is defined at the lattice scale. Yet from renormalization group flow we know that the low energy theory could look completely different. Again, like in the perturbative case, if we did not have to worry about LIV, we could have set the lattice scale to 1% of the Planck scale and everything might have worked out. We would have started with the standard model and general relativity at the lattice scale and probably we would have been able to fine tune the lattice couplings to reproduce low energy physics.

Unfortunately, the lattice scale is constrained to be much shorter than the Planck scale. We have no idea which matter fields and what interactions at this scale have a chance of flowing to the known low energy physics. We could try guessing, but there are infinitely many options with infinitely many free parameters. And even for a single guess with specific parameters, calculating how the theory would look like at low energies would be a monumental task, probably beyond anything that mankind has ever achieved.

To summarize, there are infinitely many theories of QG that match current experimental data, and there are infinitely many theories of QG that are self-consistent. But there are none that we know of that satisfy both conditions.

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