Is there really no paper proposing a unified equation for quantum gravity?

Question

Modified from https://physics.stackexchange.com/questions/751787 (Completely rewritten.) 

Obviously, there is no equation yet for quantum gravity. But there are also surprisingly few papers that claim that such an equation must exist at all. A rumor I heard recently was: 

> "No paper really claims that a (yet unknown) equation for quantum gravity must exist"

Google scholar found only three candidates. I know about a few (popular) books that make the claim that an equation exists, but I found only 3 papers that state that an equation for quantum gravity must exist. (Even string theory is unclear on the matter...)

Are researchers in quantum gravity looking for an equation of motion at all? Or did researchers stop believing that such an equation is possible?

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Attempt one was the WdW equation, with its story told in https://arxiv.org/abs/1506.00927. It was falsified.

The second is "Time and a Physical Hamiltonian for Quantum Gravity", Viqar Husain and Tomasz Pawłowski, Phys. Rev. Lett. 108, 141301 (2012). It did not work out, but I admire their courage. None of the 185 papers that cited it continues to explore evolution equations.

The third is "Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation", Palemkota Maithresh and Tejinder P. Singh, https://arxiv.org/abs/1908.04309. This is rather new, and the equation they propose has yet to be explored and tested by other people. Again, I admire their courage.

Comments

In most textbooks on QFT you also don't find equations of motions for QED, the most accurate theory we have. The reason is that these equations do not make sense without renormalization, and renormalization is at present (except in toy models in lower dimensions) never done on the level of equations of motion but only on the level of derived information such as correlation functions, S-matrix elements, etc..

Thus it is unreasonable to expect that in quantum gravity the situation would be different.

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In my university, we do write the equations of QED on the blackboard. They are simple and short. A quick search shows them everywhere, e.g. just before section 8.3.3. in https://www.theoretical-physics.net/dev/quantum/qed.html

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The link in your comment goes to 404. Not found.

Equations purported to describe QED in fact do not describe it. It is known since 1928 that they are inconsistent because of lack of renormalization, and, as Vladimir Kalitvianski remarked, because they do not take into account the infrared problems of real QED. The latter requires treating the electrons as infraparticles.
 

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@Klaus: You don't know what you are talking about. Wikipedia is not always correct in its interpretation - it doesn't say that the equation derived are only semiclassical equations, not QED! I augmented my answer below to address your reference to Wikipedia in more detail.

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@Klaus: Your revised link still only gets classical field equations! It is said explicitly that 'The first equation is the Dirac equation in the electromagnetic field and the second equation is a set of Maxwell equations () with a source , which is a 4-current comming from the Dirac equation.' Note that the Maxwell equations are equations for the classical electromagnetic foield, and the Dirac equation is an equation for a single particle or for a classical field, but not for a qauantum field! The corresponding equations for field operators are known to be inconsistent - the reason why renormalization is needed!

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@Klaus: What is the use of these equations if they are inconsistent, if they do not give physical solutions without additional terms (counter-terms) and resummations of soft diagrams? They alone are not QED equations.

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Anybody questioning the consistency of Maxwell's equations and of Dirac's equation does not know what he is talking about - except if he is called Dirac or Maxwell. 

Answer 1

Writing down equations of motions is unreasonable for interacting quantum field theories in 4-dimensional spacetimes.

Perhaps you meant writing down Lagrangians rather than equations of motion?

For QED, the Lagrangian (unlike valid equations of motion for the fields) can be written down, and is probably found in every textbook on quantum field theory.

But this can also be done for canonical gravity; see, e.g., equation (28) on p.22 of

• Burgess, C. P. (2004). Quantum gravity in everyday life: General relativity as an effective field theory. Living Reviews in Relativity, 7, 1-56.

By formal variation, one can obtain classical equations of motion, but for the QED Lagrangian, these (given, e.g., in Wikipedia, without any caveat about their meaning) describe the dynamics of a single Dirac particle coupled to a classical electromagnetic field. To obtain QED one would have to quantize these equations, but this introduces ultraviolet and infrared divergences that show that the operator versions of the classical equations are inconsistent. The necessary renormalization destroys their validity even perturbatively, where QED is very successful.

Similarly, the Lagrangian for classical gravity (or variations of it) produce equations of motion for the classical gravitational field, not for its quantum version. Their quantum interpretation (and indeed all of quantum gravity beyond its semiclassical approximation) is fraught with difficulties. See Chapter B8 (''Quantum gravity'') of my Theoretical Physics FAQ.

There is also a Lagrangian of classical gravity coupled to the Dirac and the electromagnetic field. Its variation produces the Einstein–Maxwell–Dirac equations. They describe the dynamics of a single Dirac particle coupled to classical gravity and a classical electromagnetic field. To obtain quantum gravity one would have to quantize these equations. Nobody knows how to do it. The same problems as in QED arise, but the nonrenormalizability (according to power counting) poses additional problems.