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I was trying to reason about how could quantum mechanics be related to the space-time curvature, and I have ended up in an apparent contradiction, which puzzles me. It would be nice if someone could point out if I am mistaken.

Let's say one wants to determine a distance, for instance, the position of a particle, with high precision. Then, according to the uncertainty principle, one has to sacrifice accuracy on how well the momentum of the particle can be known, so trying to resolve a distance more precisely involves an increase in momentum uncertainty. On the other hand, according to general relativity, the curvature of spacetime is related to the energy and momentum of whatever matter present, so, if curvature is dependent on momentum, increasing momentum uncertainty should lead to increasing "curvature uncertainty" (although I don't think I have ever heard this term used). This is where I'm having trouble: since the distance between two points depends of the curvature between them (that is, the intrinsic curvature of a surface depends of the distance relationships that hold within it), then an increase in curvature uncertainty would imply an increase in distance (hence position) uncertainty, which seems in contradiction with the initial assumption.

I suppose the apparent paradox comes from introducing this "curvature uncertainty" in my argument, which does seem a bit odd. Is the whole argument incorrect, or could it be used as a way to show that infinitely small distances can't be measured?

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    $\begingroup$ I don't know whether it's correct or not, but it probably could be made a bit more formal. The energy and momentum are those of the matter present in spacetime, while the distances relevant to curvature are distances between events in spacetime, not the positions of particles, though obviously they are related $\endgroup$
    – Javier
    Oct 11 '16 at 1:10
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Good question. That's no surprise. We still have not reconciled GR (general relativity) with quantum theory (QT). We don't know how to do what you ask.

The measurement of a particle's position can be done exactly in QT, as you said, and it means that you've picked a position eignestate of it. The momentum variable then has infinite variance, in that state, thus confirming the uncertainty principle. This is true for a quantum particle in a FIXED spacetime background, i.e., in a specific curvature. We know this because we have worked out how to deal with quantum particles or quantum fileds Ina fixed background.

But you then go a step further, as you should, and say, well, that momentum, then must make the source of the gravitational field a quantum entity, and the field then must be quantum, and thus some aspect of it must be in a quantum state. If it's a small effect, we can treat it perturbatively, and calculate that effect. But then there are perturbations again on the position, and so on, where we have to do an infinite number of perturbations. We don't know how to calculate that without the infinities it involves; quantum gravity, done that way is not renormalizable, it's been proven to be so. If you wanted the first perturbation, and it was small, like one particle changed, it would actually converge easily, but in fact all particles, all sources of the gravitational filed enter in, and we have to account for all of them. Because everything couples to a gravitational field that's been a problem.

The good news is that gravity couples weakly to matter and anything else, and we can do a 1 step approximation and in fields which are not too strong get close enough to an answer. In fact, mostly we simply treat the particles affected as very few, and treat them quantically, and treat the source of the gravitational field (I.e, curvature) classically. That's what was done by Hawking and others in treating how a quantum effect causes radiation form a Black Hole.

The current work on quantum gravity goes along multiple lines of research, with string theory and loop quantum gravity being two of the most popular. An equivalence between gravity in a (specific, AntideSiter) spacetime and a conformal QT in its boundaries is another approach, called AdS/CFT correspondence. Till we figure that out we have no answer for your question.

The popular conceptual view of what gravity may look in the quantum realm, which would make itself manifest at the Planck scales of $10^{-33}$ cms is a bubbling and rapidly changing foam that represents the precursor to spacetime at larger normal distances.

Google Quantum Gravity and see as an intro the wiki article at https://en.m.wikipedia.org/wiki/Quantum_gravity

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  • $\begingroup$ Thank you for your answer! So, if I understood correctly, theorists are looking for a background-independent quantum theory? $\endgroup$
    – dahemar
    Oct 11 '16 at 9:38
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    $\begingroup$ @David Herrero Marti. Background independence is not fully well defined. See the wiki page on it. Anyway, LQG is manifestly background independent, String Theory is not manifestly so but has some features of background independence. Remains a point of contention $\endgroup$
    – Bob Bee
    Oct 11 '16 at 17:56
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Read chapter 21 in Misner-Thorne-Wheelers Gravitation. In it you will see the derivation of the Arnowitt-Deser-Misner (ADM) approach to relativity. I will not reproduce that here, but sketch a bit of this. The Riemann curvature results in a Hamiltonian constraint $H~=~G_{ijkl}\pi^{ij}\pi^{kl}$. Here $\pi_{ij}$ is the momentum metric conjugate to the metric $g_{ij}$ for a spatial surface and $G_{ijkl}$ is a superspace metric. We have then the natural quantization condition $$ \hat\pi^{ij}~=~-i\frac{\delta}{g_{ij}}, $$ where it is clear there is the commutator $$ [\hat\pi^{ij},~g_{kl}]~=~-i\delta^i_k\delta^j_l. $$ This leads to an uncertainty relationship between the momentum metric operator and the spatial metric. The uncertainty relationship can be shown in a standard way.

The momentum metric is constructed from the extrinsic curvature, and so this is a form of uncertainty between curvature and the metric. This can be carried further with the Hamiltonian constraint. A good candidate to look at would be a product of metric components $g_{ij}g_{kl}$.

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  • $\begingroup$ See more on ADM quantization of General Relativity at en.m.wikipedia.org/wiki/Canonical_quantum_gravity. It is basically called the canonical quantization of gravity. It did not work. It points out that it had irremediably problems that got partially resolved by the Ashtekar-Barberos treatment, which still had serious issues. The survivor of all that is basically the current Loop Quantum Gravity. Not sure where things stand on it. $\endgroup$
    – Bob Bee
    Oct 11 '16 at 7:03
  • $\begingroup$ Sure, there are problems with the Wheeler-De Witt equation. Variants of this with Ashtekar variables and LQG are a sort of low energy quantum gravity at best. Of course the loops or struts have problems with a "hard UV" cut-off, and there are other issues with LQG. I think in some WKB-ish or semi-classical way the WDW equation and LQG has some bearing quantum gravity. There are also potential deep reasons why LQG fails as a fundamental quantum gravity theory. $\endgroup$ Oct 11 '16 at 10:11

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