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There is a point about universality and lattice definitions that perhaps is not well-appreciated. Suppose you discretize a model, then put it on a computer, probably use some random sampling of all possible configurations, and then just calculate. Most of the numbers that come out are completely meaningless lattice artifacts. If you run your simulation again those numbers will change. There is a well-defined process of continuum limit which “forgets” about the details of the original discretization and concentrate on small set of “universal” quantities. Those are the well-defined ones, independent of details of the calculation.

The point of that is that by construction all properties of the discretization, the fact that it is geometrical and has nice causality properties , looks x-dimensional and what not- those are exactly the things that are to be “forgotten” in a very technical sense. If one wants to claim anything about physics it should involve one of the universal quantities.

I would not be at all surprised if with enough fiddling one can find *some* 4dim continuum limit, that is not at all the interesting point. Getting gravity this way (as defined by universal Newtonian potential) looks like a very tall order.

best,

Moshe

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An obvious possible combinatorial model for gravity is a sum over triangulations. This has been considered in depth in the Euclidean case. It works really, really well in two dimensions, even if you just mindlessly sum over triangulations giving them all the same weight. In general in physics the weight in a weighted sum is called the “action”. In higher dimensions, the experience so far is that the simulation looks terrible no matter what action you try.

Ambjorn, Jurkiewicz, and Loll take a somewhat ad hoc, somewhat geometric action due to Regge (I think), but restrict the allowed triangulations. The triangulations that they allow are those that are called shellable in geometric topology. These are the triangulations that are consistent with a discrete time coordinate (which in topology would be called a Morse function or a foliation) without any topology changes in the spatial slices (no Morse tranisitions). They gloss over two subtle points: (1) Even a spacetime with no topological evolution has many non-shellable triangulations. (2) One triangulation can have more than one shelling. It is not clear if they mean shellable or shelled triangulations. Probably the latter, because the former is would be computationally difficult. I am not sure whether this is a fundamental point, although it does open the door to Lorentz non-invariance.

Their main result is that the spectral dimension (which means inferring the dimension from particle diffusion) seems to asymptote to 4. Actually, their curve fit gives 4.02, which they optimistically interpret as approximately 4. But look carefully at Figure 5, which shows the curve fit. Zoom into a lot, like 1500%. The black fitting curve aims too high in most of the middle and aims too low at the right. So 4.02 looks like an underestimate for their result, not an overestimate. They say in hep-th/0505154 that they outright assumed a fit with a horizontal asymptote. But it also seems possible that there is no horizontal asymptote, that instead the value slowly drifts to infinity.

This is in keeping with a message from many string theorists, and some non-string theorists too. Before considering the important question of what is or is not physically realistic, it is their mathematical conjecture that 4-dimensional gravity is not reachable as the infinite scaling limit of a microscopically 4-dimensional, statistical model. (Four dimensions meaning either 4+0 or 3+1; statistical meaning either quantumly or classically.) They have evidence for this conjecture. I do not see that hep-th/0505154 is evidence against.

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Jacques has a recent post on what exactly one assumes when trying to quantize GR as a field theory. In a nutshell this assmues the existence of UV fixed point, which is necessarily strongly coupled, and then Einstein GR is the IR limit of some perturbation of this fixed point. The validity of this picture depends on every detail of the physics between the Planck scale and the IR, thus attempting to just quantize pure GR gives you no insight whether this picture is the correct one.

About point 2, what I meant is not really contact with experiment, but contact with diagnostic tools we are familiar with. My definition of dimension will use for example the fall-off of the gravitational potential. If you want to invent some new diagnostic tools, such as some properties of random walks on the lattice, I would ideally have liked to see that they are well-defined (universal and gauge invariant), and will be curious what do they give for the zillions of theories that are already known. For example if using this tool QCD will turn out to be 17 -dimensional, I would not call the number they calculate a dimension in any decent sense.

best,

Moshe

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I do agree wholeheartly with you that making contact with physics we already know is something I would love to see in ALL forms of speculative physics.

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Let me jump in with my two cents about CDT:

1. It may be that some modifications to usual wisdom is needed, but CDT is not trying to do that at all. Rather, it is good old fashioned lattice definition, and the only way you have to avoid getting (literally) different answers in different runs is to use RNG and universality. If they do that, based on completely general arguments they will find themselves in some 4dim fixed point. Alas there is not a single indication that Einstein gravity comes from a 4dim fixed point, so some miracle is needed. It would be nice to have motivation for this miracle- before plunging into years of research on technical details.

2. Again, based on completely general argument, the number of degrees of freedoms decreases along RNG flow , so any claimed flow going from two to four dimensions is highly suspicious. The easiest explanation for their results is that they are completely formal, there is no indication that properties of random walks on the lattice are measurable or physical (gauge invariant). One wonders for example what happens in conventional lattice QCD. I have to admit to a certain amount of frustration with these formal mathematical arguments that do not even attempt to make any connection with the physics we already know.

best,

Moshe

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Murray Gell-Mann likes to call such deeper theories “field general theories”, a name a kind of like.

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