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Recently i read about Loop Quantum Gravity in Wikipedia and found this below statement here.

Presently, no semiclassical limit recovering general relativity has been shown to exist.

But i also came across a paper in arxiv whose abstract claims that Einstein's equations arise as a result of Semi-Classical Limit from Covariant Loop Quantum Gravity.

In this paper we explain how 4-dimensional general relativity and in particular, the Einstein equation, emerge from the spinfoam amplitude in loop quantum gravity.

So is the Wikipedia article outdated or is the paper's idea falsified?

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    $\begingroup$ This exact question is hotly constested between the LQG people and the string theory people. $\endgroup$ Apr 20 '20 at 13:29
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    $\begingroup$ @JerrySchirmer only because of the lack of knowledge, or maybe it was the case 10 years ago. It is well known within the LQG community that the canonical formalism very likely does not reproduce GR in the classical limit, but the EPRL spinfoam model (aka the covariant formalism) demonstrably does, at the cost of being defined only perturbatively. $\endgroup$ Apr 24 '20 at 13:45
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The confusion arises because there are essentially two different approaches to the LQG dynamics in the literature.

The canonical approach pioneered by Thiemann is to define the matrix elements of the Hamiltonian constraint operator acting on spin networks, and then postulate the kernel of that operator to be the physical Hilbert space of LQG. It is extremely hard to make any physical predictions with this mathematical object as it is so extremely complex. However, it is possible to prove that it is mathematically well defined.

The spinfoam approach is based on the so-called EPRL spinfoam model, which is what the paper you linked talks about. It indeed reproduces General Relativity (or rather its triangulated version – the Regge theory, which becomes General Relativity when the triangulation is refined) in the $\hbar \rightarrow 0$ limit if the boundary states are taken to be the coherent (semiclassical) Livine-Speziale states. Rovelli has a nice chapter with a detailed calculation in his textbook called "Covariant Loop Quantum Gravity".

The spinfoam formulation, unlike the canonical formulation, has not been made into a mathematically well-defined and consistent formalism so far. Due to infrared divergences, it isn't clear whether the projective limit of spinfoam amplitudes exists or not, and whether, if it does exist, it specifies a projection operator onto the physical state space of quantum gravity.

This situation is a lot like the one in perturbative QFT. The theory gives meaningful physical predictions, but so far no one has been able to prove that this theory exists mathematically.

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  • $\begingroup$ That's a very helpful summary! But could you sketch the relationship (if there is one) between covariant and canonical LQG? From what you're saying I infer that they are not equivalent in a regime where LQG could reproduce GR, right? $\endgroup$
    – user21299
    Jun 6 '20 at 3:07
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    $\begingroup$ @alexarvanitakis it is afaik an open question even whether they are equivalent completely or not. It is very hard to extract anything physically useful from canonical LQG, so comparing predictions isn’t at all straightforward $\endgroup$ Jun 6 '20 at 3:08
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About a decade ago, Ashoke Sen did some calculations that show that you can use Euclidean methods compute the logarithmic corrections to the entropy of Schwarzchild black holes in terms of low-energy, classical data. He remarked that these entropies had also been calculated within loop quantum gravity, and that the results do not match the answer. This strongly suggests that LQG does not give the correct semi-classical limit.

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  • $\begingroup$ I see we're downvoting without commenting. Classy. $\endgroup$
    – user1504
    Apr 24 '20 at 17:52

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