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From the "Theory of Everything" Wikipedia article (first paragraph):

Over the past few centuries, two theoretical frameworks have been developed that, together, most closely resemble a TOE. These two theories upon which all modern physics rests are general relativity and quantum mechanics.

Can someone elaborate on what is meant here exactly by taking general relativity and quantum mechanics "together"?

I have no idea what is meant by this other than something along the lines of: "Use GR and forget about QM when the situation is appropriate, and vice versa". Is this notion of "togetherness" this trivial?

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...something along the lines of: "Use GR and forget about QM when the situation is appropriate, and vice versa".

Yes, that's pretty much what it means. In slightly more detail:

  • We can use quantum theory to show that classical theory is (usually) a good approximation on macroscopic scales, and then we can use that approximation to account for matter and electromagnetism within general relativity.

  • After using that approximation to determine the spacetime metric appropriate for a given macroscopic situation, we can use that metric as a prescribed background (possibly time-dependent, but prescribed) in quantum field theory, ignoring the back-reaction of the quantum fields on the metric. This is a good enough approximation for deriving Hawking radiation (that's how Hawking did it), but it's not a good enough approximation for resolving the black hole information paradox.

This disappointingly trivial version of "together" seems to be good enough for just about any experiment we're likely to be able to do in the forseeable future. That's a great outlook for engineers, but it's terrible outlook for physicists. As physicists, we would love to have a larger supply of paradoxical experimental results to help guide the development of a real theory of everything — one that isn't just a usually-peaceful coexistence of two fundamentally incompatible theories. We do have some clues, but they are very indirect: the black hole information paradox and the cosmological constant problem are among the few precious clues we have. That's why they get so much attention in quantum-gravity research.

By the way, we know that both GR and the Standard Model of particle physics are merely approximations, but we usually assume that the basic structure of quantum theory will continue to be a sufficient framework for a theory that unifies them. We have (mathematical) evidence from string theory that it is sufficient in asymptotically flat spacetime or asymptotically anti-de Sitter (AdS) spacetime, but the real world is more accurately modeled by asymptotically de Sitter (dS) spacetime. Several experts have questioned the suitability of the conventional structure of quantum theory in the de Sitter context, but I don't think the dust has settled yet. We'll see.

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  • $\begingroup$ +1 Btw, as a physicist, what is your preferred theory of QG? $\endgroup$
    – KP99
    Sep 26 at 2:21
  • $\begingroup$ Did Einstein have "a larger supply of paradoxical experimental results" when he develop general relativity? $\endgroup$ Sep 26 at 3:18
  • $\begingroup$ @KP99 I don't even want to guess what the correct theory might be. I've learned more from the AdS/CFT correspondence (an incarnation of string theory) than I have from other approaches to QG, but the real world doesn't seem to be asymptotically AdS, so the degree to which those mathematical insights apply to the real world is still a big open question. $\endgroup$ Sep 26 at 13:34
  • $\begingroup$ @flippiefanus Good point! $\endgroup$ Sep 26 at 13:34
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Well, it's really about special relativity. General relativity is Newton's gravity incorporating the principle of special relativity. This waa Einsteins achoevrment. He accomplished it by showing that gravity was a field theory just like electromagnetism but where the field is the metric and spacetime the medium.

It's also understood how to incorporate special relativity into quantum theory. This was first accomplished by Feynman in his theory of QED. This was the quantum theory for the theory of electromagnetism. Later, similarctheories were established for the weak and strong force after Yang & Mills had written down their eponynous equations where they generalised the structure group of QED from U(1) to non-abelian groups. Later, Abdus Salam and Stephen Weinberg unified electromagnetism and the weak force into the electroweak force.

The main entry point into QED for calculations is through perturbative calculations (pQFT). Often it is claimed that QFT is perturbatively ill-defined. However, the mathematical technology has advanced far enough so that the perturbative framework is well-defined and rigorous. This is of course, as it should be, as pQFT was coming out with the correct results.

What remains is unifying QFTs with gravity. This means incorporating the relativised quantum theory with the relativised theoey of gravity. This is a long, uphill struggle. The main two contenders are Loop quantum gravity, which takes a conservative view and String Theory, which takes a radical view. And I mean radical. There are many objrcts in this theory which have not been observed - not only strings, but DBranes, higher dimensions and a tower of heavy states. Although it is often said, string theory has not given any testable predictions, there is actually one: higher curvature corrections to General Relativity. These have not, so far, been observed - or so I assume - otherwise the shouts of delight from the string theory community would have overwhelmed all of us!

The main entry point into quantum gravity is the semi-classical calculation of Hawking validating the Bekenstein-Hawking entropy formula for black holes. This has been derived in both the Loop Quantum Gravity frameworks by counting quantum geometry states and in String Theory, albeit in higher dimensions and with a higher number of charges.

This semiclassical calculation by Hawking uses the theory of QFT on curved spaces. It assumes that the dynamically curved spacetime changes slowly enough that it can be taken to be static. There is a well developed theory of how to do QFT on such backgrounds.

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