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Assume one can write the metric field on a lattice, so on each lattice point one has a value of $g^{\mu\nu}$. Similar to the way lattice QCD is formulated. Then later taking the distance between lattice points to go to zero.

Why is not as good as techniques such as dynamical triangulation or loop quantum gravity? The latter have the distance between vertices defining a minimum fixed length such the Planck length but the graph itself is dynamic.

So my question is what advantage does one gain from having a dynamic lattice rather than a fixed lattice as in lattice QCD?

Is it because we don't want to take the limit to continuous space as we add more lattice points? Then I guess we'd have to sum over different types of lattice in order to maintain Lorenz symmetries.

Is lattice quantum gravity doomed to failure?

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  • $\begingroup$ In what way do you think putting GR on a lattice circumvents non-normalizability, which is usually cited as the reason standard QFT-techniques don't work for GR? Your comparison with QCD is a red herring - QCD is normalizable, it just has a large coupling, making it unsuited for perturbation theory with the coupling as the parameter. $\endgroup$ – ACuriousMind Jan 20 at 14:51
  • $\begingroup$ Well maybe that's the answer to the question. If you can contrast that with the other approaches. $\endgroup$ – zooby Jan 20 at 14:58
  • $\begingroup$ Recall that gravity (quantum or classical) features dynamical spacetime. A "fixed lattice as in lattice QCD" instead involves a fixed flat spacetime, so no gravity is going to come out of that. You also need to be careful to realize diffeomorphism invariance. Let me see if I can ping @R.G.J who actually works on "lattice quantum gravity" in the form of dynamical triangulations (which he obviously doesn't consider doomed to failure) and can tell you more than I can. $\endgroup$ – David Schaich Jan 25 at 12:44
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    $\begingroup$ Agree with @DavidSchaich above. If you want to have a consistent theory of gravity on the lattice, the lattice must itself be dynamical. The preferred choice is called dynamical triangulations(DT) because first it was used for quantum gravity in 2d where the 2-simplex are triangles (hence, the name).DT had great success in 2d & it was shown that continuum limit corresponds to QG2(Quantum gravity in 2d). Next was to do QG4 and discretize S^4 (4-sphere) with 4-simplices. Note that simplicial complex like these is among the simplest tessellation of S^4. This remains work in progress till date. $\endgroup$ – R.G.J Jan 26 at 1:22
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    $\begingroup$ 2/2: Defining the continuum limit in QG4 is the main deal. Lattice gravity models assume Weinberg’s conjecture of interacting fixed point in the UV with a finite number of parameters to be tuned, and this is known as -“asymptotic safety”. But, it was found that for DT in 4d there was no critical point corresponding to 2nd order phase transition, so the hope of taking a continuum limit (CL) was dashed with the simple Regge like discretization. Recently, there are hints that by having a more complicated action, one can alter the phase structure & have a critical point where CL can be taken. $\endgroup$ – R.G.J Jan 26 at 2:39

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