Is general relativity required from quantum mechanics?

One of the paradoxes against quantum mechanics was claimed by Einstein during one of the Solvay conferences: a paradox called the Einstein box also explained in this question. The solution proposed by Bohr contemplates the use of the gravitational redshift predicted by general relativity. In retrospect, does the confirmation of quantum mechanics imply the correctness of general relativity (as far as redshift predictions are concerned)? Or more generally: does quantum mechanics require general relativity for its consistency?

• The discussion under section "Uniform Gravitational field and acceleration" in en.wikipedia.org/wiki/Gravitational_redshift#History states that the redshift for uniform gravitational field follows only from the equivalence principle (which is also manifested in Newtonian gravity) and doesn't involve any mathematical apparatus of GR. Commented Sep 26, 2021 at 4:21
• I read that section you referred to about Einstein's box, and I don't really get why Bohr's answer is correct. He says GR introduces uncertainty in the clock time via gravitational redshift, but why should that be the case? The redshift equation is exact. Is it relying on the precise height of the box being subject to quantum positional uncertainty? Commented Sep 26, 2021 at 15:42
• Here there is a related question that explain the Bohr answer physics.stackexchange.com/questions/8295/…
– Pipe
Commented Sep 26, 2021 at 16:36

Let's work with these summarises of Einstein's argument and Bohr's response, the latter repeating the former's assumptions about relativity and gravity. The response notes$$\color{red}{\Delta E}\color{blue}{\Delta t}=\color{red}{c^2\Delta m}\color{blue}{c^{-2}gt\Delta q}=\color{orange}{gt\Delta m}\Delta q\ge\color{orange}{\Delta p}\Delta q,$$where the red quantities are equal by special relativity, the blue quantities are equal by general relativity, and the orange quantities are equal by the acceleration $$g$$ Galileo knew about.

Does this thought experiment prove QM implies the coloured parts? No.

In an "I can get less $$\Delta E\Delta t$$ than Heisenberg said" argument, the explanation brings in extraneous-to-QM physics (in this case, the coloured parts). A counterargument uses that extra physics, which isn't a consequence of QM in either argument's view. If Einstein's argument works, QM is incompatible with the physics cited; if it doesn't, they may be compatible, but one needn't imply the other. If a third physicist doesn't grant these other ideas, that doesn't contradict QM; it just means they can't use them to work out what happens in the experiment.

• Nice point. So the question is: a posteriori, knowing that QM works and red and orange quantities too, the non understanding of the experiment briggs the necessity of GR? Because if I assume that it is not possible to get $\Delta E \Delta t$ arbitrary small and I also assume that the orange and yellow parts are true, then is necessary to have the blue part?
– Pipe
Commented Sep 26, 2021 at 20:23
• @Pipe Yellow? (Advise me of any colour changes I should try to address colour blindness; unfortunately, some LaTeX colours are very pale.) To answer your question, that's a good example of "physicist 3": if they don't know the blue fact, all they know is$$\frac{\Delta E\Delta t}{\Delta p\Delta q}\ge\frac{\Delta t}{c^{-2}gt\Delta q},$$but it's unknowable how small that can get.
– J.G.
Commented Sep 26, 2021 at 20:38
• Sorry for the color confusion: not yellow but red.
– Pipe
Commented Sep 26, 2021 at 20:45

There is a calculation of the vacuum energy in quantum mechanics which mismatches that of general relativity by roughly a hundred orders. Nevertheless, it predicts something qualitatively that is theorised in GR.

Moreover, Feynman in his lectures on gravitation attempts to derive GR from quantum mechanics. He actually gets quite far.

Finally, there is string theory - whichbis a quantum theory - and where as Witten points postdicts GR. Except thay this is not quite correct. It postdicts GR with higher curvature corrections.

All this is not quite finding that quantum theory requires gravity for its consistency. But one expects that in quantum gravity, that this will be the case. It's in my opinion, one of the hallmarks of a unified theory.

• Enjoyed reading this answer. One of the many reasons physics is such an exciting field. Commented Sep 26, 2021 at 15:57
• @Michael Burt: Thanks, I enjoyed writing it. It's comments like these that draws me back to SE. Commented Sep 30, 2021 at 7:08