Did anybody ever claim that a unified Lagrangian exists?

Question

Continuing the issue raised by Klaus 1 month ago, I have a question. Is there a unified Lagrangian that combines particle physics and general relativity? The answer probably is: we do not know.

So here is the next question: Did anybody ever state in writing that such a unified Lagrangian must exist? 

Is there an argument in favor (or against)?

There are so many people working on string theory, on loop quantum gravity and on other ideas.

Did anybody working on any idea for unification ever claim that unification is achieved using a Lagrangian?

What are the expectations?

Comments

One can write down putative Lagrnagians, but one doesn't know how to solve the corresponding quantum field theory without introducing infinitely many renormalization parameters. That's why it remains an open problem.

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@Arnold I'd like to read about these putative Lagrangians!

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@Arnold I like the Christian part of your website.

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@Jean: The variation of the Lagrangian of classical gravity coupled to the Dirac and the electromagnetic field 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. Coupling to the standard model rather than electrodynamics is analogous; I don't know whether anyone has written it down explicitly, but there is no difficulty doing so.

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@Arnold Ok, so we know at least one person who believes that there is a unified Lagrangian! (Even though your link goes to a strange wikipedia page.)  Could you maybe provide another link with the Einstein-Maxwell-Dirac Lagrangian? Thank you in advance for all your help and patience.

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@Jean: See, e.g., equation (14) in the paper

Jakubiec, A., & Kijowski, J. (1985). On interaction of the unified Maxwell-Einstein field with spinorial matter. letters in mathematical physics, 9, 1-11.

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@Arnold Thank you. In fact, equation (8) is an even simpler version. Interesting!

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The opposite seems to apply.

The entropy of black holes suggests that they are composed of distinct components. These components are shared by both space and particles, as black holes can be described as either curved space or compressed matter. Based on black hole thermodynamics, these shared components are estimated to be of Planck size. Their behaviour is crucial in defining the unified theory - which is still unknown.

Combining general relativity with quantum theory suggests that the Planck scale represents the smallest measurable length in nature. This limitation indicates that the common constituents, regardless of their specific details, cannot be described using a Lagrangian. Consequently, the unified theory of relativistic quantum gravity cannot be described by a Lagrangian either, independent of the specific type of common constituents. 

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Mere suggestions are not answers.Moreover, ChatGTP is well-known for producing spurious comments to even simple scinetific questions.