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The notion that "we have no reliable quantum theory of gravitation" seems to be widely accepted (see example PSE discussion here). But is it really so?

According to the modern Effective Field Theory (EFT), at experimentally accessible energies, we DO have exceptionally reliable quantum theory of gravitation (see e.g. here, which gives a very accurate formula eq. 20 for the quantum corrections to the Newton potential between two masses at low energies). We just don't have reliable quantum theory of gravitation at Planck scale energies. However, note that the standard model (SM) is also merely an effective field theory of a more fundamental UV complete theory at Planck scale. Therefore, we don't have a reliable quantum theory of any force at Planck scale. In this regard, can we say that the status of quantum theory of gravitation (QG) is not different from e.g. QED?

One may disagree with the above comparison of QG with QED. The rationale is that QED is renormalizable and QG is not, since gravity coupling has negative mass dimension. However, in the modern EFT/Wilsonian RG point of view, coupling with negative dimension is perfectly allowable. QG is renormalizable as well: you just have to carefully absorb divergences into higher order Lagrangian terms. For more details of QG renormalization, see Section 4 of the paper referenced above. The paper says that:

The renormalization of divergences is also not that big of a deal, although it was the focus of this subject for many years. The divergences themselves come from the high energy end of the theory, which we know is not reliable. The ultimate UV completion will eventually tell us the correct way to treat this domain, and will predict the value of the coefficients. So renormalization is a necessary step, but one without much content...The real power of the effective field theory is that it shifts our attention from the UV (where we do not know the physics) to the IR (where we do). There, EFT techniques allow one to make real predictions. This is because we know the light degrees of freedom active there and we know their interactions.

So, is there really a reliable quantum theory of gravitation?

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  • $\begingroup$ Is this a bit like saying, "Newtonian physics is extremely accurate on the surface of the earth, so prior to the invention of the telescope we had a reliable general-relativistic theory of motion"? We had a reliable theory of motion, sure, but it wasn't general-relativistic. Likewise, unless a theory of gravity addresses the Planck scale then it is not (in the parlance of physicists) a quantum theory, and unless we can test it (which we don't expect to be able to do before the Planck scale since as you say the effects are not practically measurable) its reliability is not known. $\endgroup$ Commented Nov 24, 2022 at 9:19
  • $\begingroup$ Or look at it another way: imagine I have two theories of Quantum Gravity, QG1 and QG2. Both of them diverge from general relativity by a certain amount at testable scales, well below what we can measure, and in fact QG2 diverges by twice as much as QG1. Does that mean they are both "reliable theories of quantum gravity", despite the fact they disagree? For that matter, what about QG0, which says "there is no quantum gravity effect at all at any scale: just use GR". Is that a reliable theory of quantum gravity? $\endgroup$ Commented Nov 24, 2022 at 9:22
  • $\begingroup$ "don't expect to be able to do" - directly, I mean. Testing a theory less directly, e.g. because it makes predictions in cosmology, would be nice. And also obviously we didn't actually have Newtonian mechanics prior to the invention of the telescope, but if we had it, and perhaps if we didn't have Mercury so that Newtonian mechanics wasn't obviously incorrect even without the aid of a telescope: then would the fact we couldn't observe any general-relativistic motion mean that pretty much anything would have been a reliable theory of general-relativistic motion? ;-) $\endgroup$ Commented Nov 24, 2022 at 9:37

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We don't expect quantum gravity effects to become observable until we approach the Planck energy, so the effective field theories for quantum gravity work at energies where they don't predict anything observable, and they don't work at energies where the effects would be observable. This is a strange way to define a reliable theory.

You are quite correct that the Standard Model is also (probably) an effective field theory, but it does make predictions in regimes where we can make experimental observations. This is the big difference from quantum gravity.

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  • $\begingroup$ The equation 20 of the paper I cited actually gives a very accurate formula for the quantum corrections to the Newton potential between two masses at low energies. It's just that we don't have any accurate instrument to measure these tiny effects. Note that these are low energy, non-Planck-scale effects. $\endgroup$
    – MadMax
    Commented Nov 23, 2022 at 16:46
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    $\begingroup$ The Planck scale is where we expect full non- perturbation quantum gravity to become relevant. Perturbation quantum gravity effects can become observable long before that. (E.g. Hawking radiation/ black hole evaporation) $\endgroup$
    – TimRias
    Commented Nov 23, 2022 at 16:57
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    $\begingroup$ @TimRias Hawking radiation/black hole evaporation is not a quantum gravity calculation. It is a Standard Model calculation done on a fixed curved spacetime. $\endgroup$ Commented Nov 24, 2022 at 6:53
  • $\begingroup$ The Planck energy is about 2 Gigajoules. It is not really that much energy. The Planck mass is about 22 nanograms, which is also not a whole lot. I too do not expect quantum gravitational effects to become observable before we reach such levels, but I also don't expect them to become observable at that point, or for long after that point. What exactly do you mean? $\endgroup$
    – Arthur
    Commented Nov 24, 2022 at 9:59
  • $\begingroup$ Calculating the energy loss into Hawking radiation of gravitons very much is a pertubative quantum gravity calculation and its effects would be visible in the rate of evaporation if we were ever to observe the evaporation of a black hole. The calculation of the backreaction of Hawking radiation on a black hole also is a pertubative quantum gravity calculation. $\endgroup$
    – TimRias
    Commented Nov 24, 2022 at 11:15
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Is there a reliable quantum theory of gravitation?

It depends. Reliable for what? If as philosophers of science we wear our realist cap, any theory we know isn't 100% true gets a thumbs down. But if we wear our instrumentalist cap, a slightly wrong theory is useful for some things and not others, and the former utility comes from reliable predictions in a suitable context, called a domain of applicability or names along those lines.

the standard model (SM) is also merely an effective field theory of a more fundamental UV complete theory at Planck scale. Therefore, we don't have a reliable quantum theory of any force at Planck scale. In this regard, can we say that the status of quantum theory of gravitation (QG) is not different from e.g. QED?

There's an important difference: "The very-high energy spectrum of any $d$-dimensional quantum field theory is that of a $d$-dimensional conformal field theory. This is not true for gravity." In other words, the entropy-energy relation of black holes precludes a standard quantum cum conformal characterization of gravity, so we have a clearer idea of other interactions' trans-Planckian behaviours.

in the modern EFT/Wilsonian RG point of view, coupling with negative dimension is perfectly allowable

This comes down to where a non-renormalizable theory, which has infinitely many parameters, is applicable/reliable; long story short, it's at energy scales where a few terms are enough for precision.

The ultimate UV completion will eventually tell us the correct way to treat this domain, and will predict the value of the coefficients

Whatever that completion looks like, it's not the familiar conformal kind other fundamental interactions get. But the more we learn about gravity's quantization, the more broadly applicable our understanding will be. For a taste of what we know already, see e.g. this and Eqs. (333) and (334) here.

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The notion that "we have no reliable quantum theory of gravitation" seems to be widely accepted (see example PSE discussion here).

I disagree with this statement, especially under the light of the linked question. Searching for the word "reliable" on that post leads you to a single comment in one of the answers. The answer itself however states, and I quote,

... we are not sure that general relativity works at such small scales. It is possible that the laws of gravitation breakdown at the quantum level. That is to say, we have no consistent quantum theory of gravitation at all energies and scales.

I believe John Donoghue completely agrees with this phrasing. Gravity as an EFT is, at least in my opinion, certainly reliable at small energies. After all, it is the quantized version of general relativity, which is one of the most successful theories in the history of physics. Furthermore, GR is arguably the most reliable of all EFTs, exactly because its UV cutoff is so high. As far as we know, you need to go to extremely high energy scales for the EFT to actually break down, scales way larger than what we could expect for the SM. If I recall correctly, this is argued very well in Percacci's An Introduction to Covariant Quantum Gravity and Asymptotic Safety. In this specific sense, we have an extremely reliable quantum theory of gravitation.

The problem with the question, however, is that "reliable" isn't really that much of a well-defined concept, and different people might read your question in different ways. GR is definitely unreliable near the Planck scale. Does that mean it is fair to say "we have no reliable quantum theory of gravitation"? It depends on what exactly you mean by those words.

In summary, the answer to

So, is there really a reliable quantum theory of gravitation?

is "It depends". It depends on what you mean by the word "reliable". If you want to calculate the first quantum corrections to classical GR at low energies, we do have a reliable theory, at least from a theoretical perspective (do remember we still lack have direct evidence that gravity is actually quantized). If you want to ask what happens near a black hole's singularity, then we certainly have no reliable theory.

In this regard, can we say that the status of quantum theory of gravitation (QG) is not different from e.g. QED?

I'd say it is different. The reason being that our understanding of QED is theoretically admissible to all scales in which we can expect it to work. For example, while we do understand the SM to be an effective field theory, there is no inconsistency (that I know of) in assuming it to be valid up to the Planck scale. While other particles might exist in the scales between the electroweak scale and the Planck scale, there are no inconsistencies in assuming the SM to hold. You'd only reach an inconsistency at the Landau pole, which is way beyond the Planck scale. Hence, in principle, the limitations of the SM are either due to experiment (i.e., there are other particles we don't know about) or due to the Planck scale, which is a quantum gravity problem. The problem due to the Planck scale is because we can no longer reliably use a QFT developed in Minkowski spacetime, since gravitational effects are expected to be relevant at such scales. Hence, QED is sort of more reliable than gravity, because it doesn't have a theoretical breakdown until very after gravity spoils it.

Remark: it might be that the effective field theory of gravity is actually consistent and reliable to all scales, as long as it is considered non-perturbatively. This is the core of the asymptotic safety program, discussed in the Percacci book I mentioned above.

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