An easy, well written discussion of the casual causal dynamical triangulations approach to quantum gravity: hepth/0509010: “The Universe from Scratch” by J. Ambjørn, J. Jurkiewicz. and R. Loll. If I were young (wait a second), interested in quantum gravity, and this article didn’t get me interested in this technique, I think I would check my pulse (ducks shoe thrown by Lubos.)
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Actually it’s causal dynamical triangulations. But I suspect that casual would have been a better name for it.
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Just to be provocative, let me propose an analogy: Loop quantum gravity and causal dynamical triangulations are to string theory as biological computation is to quantum computation.
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What would you say to someone who insistently offered biological computation as a competitor to quantum computation?
Some of the LQG people seem to me to be on a Naderite quest to compete with string theory. In the sense that Nader has helped and could help in many ways, and that even his criticisms of the Democrats can be helpful; but that trying to compete with the Democrats is not helpful. Moreover, that the Naderite standard of success is not winning, but getting recognized as a competitor.
I do not know whether dynamical triangulations is another such Naderite quest, or perhaps really the same quest, or neither. It starts to look that way. There are semirigorous results (from renormalization theory) that bare 4dimensional quantum gravity is incomplete, like chemistry without quantum mechanics. Jacques Distler has a related discussion) According to Jacques, the minority quantum gravity factions are gambling that these results are rigorously wrong.
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Their claim goes beyond saying that you should add matter to have good physics. They think that you must add matter just for mathematical consistency. They also argue that if by a fluke some consistent theory looks like bare 4D gravity, there won’t be a consistent way to add matter to it. But we can take that fallback argument as beside the point here.
Of course there are a variety of ways to simulate charge and matter, going back to the original KaluzaKlein theory. If any such “ideas” turn out to be inevitable artifacts of quantum gravity, then it establishes what the string theorists claim. Indeed in string theory matter is an artifact of the core gravitylike theory; it isn’t added ad hoc.
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Doh! Thanks Greg for catching my dopeheaded mistake.
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I think it is important to separate loop quantum gravity from causal dynamical triangulations. Sure, both are nonperturbative attempts to build quantum gravity, but the nature of the two seems to me to be very distinct.
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I do not mean renormalization in the narrow sense of adding counterterms to Feynman diagram expansions, rather I mean it in the stronger sense of the renormalization semigroup, scaling, and universality classes. That isn’t just a framework for doing physics, it’s a body of nonrigorous mathematics.
When I say “nonrigorous mathematics”, I mean that some of it can be expressed as rigorous mathematical conjectures; maybe much of it will eventually be rigorous mathematics. In any case it applies to artificial models (like the Ising model, hex percolation, etc.) just as well as to realistic ones (like the Standard Model).
I can try to apply the point more directly. (As best as I understand it. Really I am repeating what various physicists say.) If you think that “causal dynamical triangulations” is a theory of quantum gravity, then in particular you should conjecture that classical GR is a valid macroscopic limit of it, right? Well, the string theorists, and even some nonstring theorists, bet that it isn’t. Never mind whether or not it’s true, they don’t think that the model is macroscopically gravity at all. Indeed, their arguments for that predate string theory itself.
So your aphorism about a lack of imagination is fine, except that (if Jacques is really explaining these issues properly) it could be an argument in favor of string theory and against the Naderite alternatives. Because string theory has clear properties that allow it to be an exception to the prior nonexistence arguments; the alternatives don’t.
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Well I agree that putting matter content into these models is a very important question and one which may doom all these endeavors. That being said, however, I still think the work in CDT has demonstrated something very useful. In particular it has succeeded where Euclidean methods of dynamical triangulation have failed. It has pointed out that to work one needs to work with path integrals in which topology change is not allowed and that the Lorentzian signature of the space is very important. If nothing else, it points out the shortcomings of Euclidean quantum gravity (the only “sensible” way to do quantum gravity, as Stephen Hawking would say )
I would also point out that there is a possible way out of this problem, and that is that it might be possible to use ideas along the lines of Wheeler’s mass without mass and charge without charge ideas. Of course there are problems already with most of Wheeler’s constructions (geons are unstable, for example), but I think the general idea is very attractive. In this type of approach, for instance, it could be that matter is represented in a theory of quantum gravity by topological defects. Of course, this is all a fairly tale, because I really don’t know what I’m talking about.
I do firmly believe that there is a lot more wiggle room in the relationship between gravity and quantum theory than many would have us believe. “What is proved by impossibility theorems is a lack of imagination.”
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“They think that you must add matter just for mathematical consistency.” Which is exactly why I don’t think this is the end of the discusion! Mathematical consistency within a certain framework (namely renormalization theory.) Are we SO certain that this is the only framework which will work for physics? To me this would be like saying to the guys who invented quantum theory: “But look! You’re theory doesn’t obey classical physics!” I think that position is too dogmatic at this point(I am a memember of the Church of Renormalization, but only the part which works for our standard model?)
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Of course, there is also an alternative to all of this. It could be that a suitable generalization of general relativity actually gives rise to quantum theory. The arguments I’ve seen as to why one HAS to quantize gravity all work toward showing that if you don’t quantize gravity, then quantum theory has to be modified (made nonlinear, for example.) And this is bad, as far as we know. But there doesn’t seem to be a reason why quantum theory cannot arise from a deeper theory which, suitably interpreted also yields general relativity. I’ve seen some attempts to do this, but all of them are really flakey. If I had infinite time, I would work on this. Instead I am stuck with this damn 24 hour day!
Murray GellMann likes to call such deeper theories “field general theories”, a name a kind of like.
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Boy, I scan the web for interesting content, but all everyone ever talks about is quantum gravity…
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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I took a closer look at hepth/0509010; it’s also helpful to look at an expanded, more technical version, hepth/0505154.
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 nonshellable 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 noninvariance.
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 hepth/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 nonstring theorists too. Before considering the important question of what is or is not physically realistic, it is their mathematical conjecture that 4dimensional gravity is not reachable as the infinite scaling limit of a microscopically 4dimensional, 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 hepth/0505154 is evidence against.
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Thanks Moshe for the comment. I wish I understood the arguments behind (1) better. I have a question, however, about (2). It seems to me that while it is true that “the number of degrees of freedoms decreases along RNG flow,” I’m not sure that this is a direct attack on the results presented in this argument. This is because, while it is true that the “dimensionality” goes from two to four, these are only the degrees of freedom associated with the order parameter of the dimensionality. Thus it could be that the two dimensional phase they are running from has other degrees of freedom, not corresponding to the dimensionality, right? Or am I missing something?
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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Hi Dave,
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 falloff 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 welldefined (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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Hi guys,
There is a point about universality and lattice definitions that perhaps is not wellappreciated. 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 welldefined process of continuum limit which “forgets” about the details of the original discretization and concentrate on small set of “universal” quantities. Those are the welldefined 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 xdimensional 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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I am an 8th grader. Could you sum up CDT for me?
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I am very interested in string theory, and when i read an article about CDT, I knew I had to learn more. I briefly skimmed through the report that Cornell university published about CDT, and it was obviously hard for me to understand, having not taken calculus yet. Could you possibly sum it up in words that a brilliant 8th grader could understand?
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I have been lokking for an update. Is this forum closed, if so why can I still post?
I would really like an answer.
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Please excuse my error. Its looking, not lokking.
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