Insert “string” pun here

Over at Peter Woit’s blog there’s a lively discussion about the differences between string theory and intelligent design. There are a few obvious ones: one is based Fields Medal caliber math and the other on elementary mistakes in probability; one is studied at an Institute and the other at an “Institute”. But arguably, neither theory has yet made a clear prediction or explained what it sets out to in a non-circular way. String theorists explain the muon mass by invoking an infinite set of Calabi-Yau manifolds, some of which presumably yield the right value; ID’ers explain the complicated dance of bees by invoking a yet more complicated designer.

Of course, an important difference is that most string theorists admit the situation sucks. Many are searching for some deeper principle that would pick out a preferred vacuum (or set of vacua, or probability distribution over vacua) non-anthropically. Based on what little I know, it doesn’t sound like an enviable task. Today I had lunch with Frederik Denef, a string theorist who’s interested in the computational complexity of finding a minimum-energy vacuum, given a collection of scalar fields. He’s formulated some toy problems, all of which are provably NP-hard (or as hard as unique-SVP under a uniqueness assumption). I was impressed by Denef’s knowledge of complexity, and by his willingness to state precise problems that I could understand. But his work suggests an obvious conundrum: if finding an “optimal” Calabi-Yau is so hard, then how did Nature do it in the first place? (If the string theorists ever succeed, will a voice in the sky boom “Thanks, dudes!” just before space as we know it disappears?)

In short, if the ID’ers are armed squatters in the apartment building of science, openly scorning the materialistic concept of rent, then the string theorists are model tenants who often drop by the landlord’s office to say good afternoon, and by the way, that check from 20 years ago should clear any day. (In their defense, the other quantum gravity theorists’ checks haven’t cleared either.) To me, this raises an interesting question: does science need a notion of “resource-bounded falsifiability,” which is to Popper’s original notion as complexity is to computability?

58 Responses to “Insert “string” pun here”

  1. David Molnar Says:

    What kind of resource makes sense in this context? Scientific person-years?

  2. Greg Kuperberg Says:

    It is interesting the extent to which quantum computation is in the same bind as string theory. Computer scientists should study real computers, just like physicists should make predictions. It’s been more than ten years already, so where are the quantum computers? Ten years of hype and promises. No quantum computers, no real progress. Tick, tock.

    The convenience of this argument is that you can make it without reading papers in the field. Certainly you don’t have to understand papers in the field, even if you do glance at them. String theorists are working on a very hard problem, as are quantum computation researchers. That inevitably leads to the attack on either group that it makes no progress.

    On the other hand, a useful critique of either field can be phrased in a pessimistic style that sounds like an “admission” of no progress. It’s the old “smoking gun”.

    To turn to slightly more technical matters: Denef’s problem, as you phrase it, is not the same problem as finding a Calabi-Yau manifold with any particular properties. I don’t know that anyone has proved that any relevant problem there is NP-hard. In any case, one reason that it is hard is simply that the real world is a quantum system, but people are classical computers and they have only built classical computers.

    The 6 small dimensions of spacetime could be a quantum superposition of different Calabi-Yau manifolds, and even that is a crude description of what string theorists think is really going on. What they really think is that any Calabi-Yau manifold is at best an approximation to some new microscopic structure which has not yet been defined. They know that the 5 healthy-looking string theories are 5 different extreme limits of this new thing, which they call M-theory.

    Beyond that, there is wisdom in the comments by Ygorff in the “lively discussion” that you mention. As best I can tell, anyway.

    As for the blog “Not Even Wrong”, I can make an analogy: I think that Peter Woit is to string theory as Leonid Levin is to quantum computation. And that’s not counting Levin’s established work in classical computation.

  3. Scott Says:

    “What kind of resource makes sense in this context? Scientific person-years?”

    Just like in complexity, one could consider different resources: person-years are analogous to circuit size, years themselves are analogous to circuit depth… :)

  4. Scott Says:

    Greg: I was going to say something about quantum computing in my post, but then I figured I’d just wait for the inevitable comment and respond to that. :)

    I’m interested in the limits of efficient computation in nature. Right now, that means I have to be interested in BQP, because for 80 years there hasn’t been a serious alternative to linear QM with tensor product structure. (Whatever other biases I have, I don’t think one can accuse me of not actively looking for such alternatives.) This interest, and not the prospect of actual quantum computers, has always been my motivation for working on quantum complexity.

    By contrast, if I were interested in quantum gravity, I’d need to consider several alternatives to string theory: LQG, spin foams, etc. Even if I didn’t like these alternatives, their very existence would tend to decrease my confidence in the assumptions underlying string theory. I might work on string theory anyway, but I certainly wouldn’t claim it as the “only real idea there is for quantum gravity,” as Witten did in a talk at IAS.

    I suspect that this sort of premature triumphalism — and not the continued pursuit of string theory as one idea among others — is the reason why scientists from other fields have started to “tick, tock” more loudly.

  5. David Molnar Says:

    So what’s the scientific instantiation of the diagonalization method for showing problems require a certain number of person-years? Once we have that, can we formulate a hierarchy theorem? :)

    I suspect if we push on this, we’ll end up with either incoherence or scientific problems that are contrived enough to seem obviously absurd. I could be wrong…

  6. Jacques Distler Says:

    “By contrast, if I were interested in quantum gravity, I’d need to consider several alternatives to string theory: LQG, spin foams, etc. Even if I didn’t like these alternatives, their very existence would tend to decrease my confidence in the assumptions underlying string theory. I might work on string theory anyway, but I certainly wouldn’t claim it as the ‘only real idea there is for quantum gravity,’ as Witten did in a talk at IAS.”

    It’s not a matter of liking or disliking the alternatives. It’s a matter of whether they are viable candidates. As I’ve tried to explain, they are not.

    That doesn’t mean I don’t think they’re worth thinking about. There’s a long and noble history of “wrong” ideas having useful (and, sometimes, unexpected) spinoffs. The same may well turn out to be true of String Theory, which continues to have a rich set of spinoffs for Quantum Field Theory and for Mathematics but which could, ultimately, turn out to be wrong.

  7. Greg Kuperberg Says:

    Scott: You’re right, there isn’t a serious alternative to quantum probability. There are alternatives, but they aren’t serious. If I were clueless, I could accuse you of ignoring alternatives like biological computation. But the truth is that biological computation isn’t a real alternative to quantum computation, moreover not all quantum computation experts ignore it.

    Loop quantum gravity is to string theory as biological computation is to quantum computation. (At least, that is my very strong outside impression.) This is not a perfect analogy, because the LQG leaders have spent much more time playing up their effort as an alternative. Moreover Witten and some others have not ignored it. They have read the main LQG papers, and they have talked to the LQG leaders. They say that LQG isn’t credible as quantum gravity (although maybe some of it is credible as something else), and it sounds to me like an informed judgment.

    It isn’t triumphalism. Modulo some very likely conjectures, string theory is a consistent perturbative model of quantum gravity. (Perturbative means that all calculations are formal power series in h-bar.) Consistency of LQG is an unlikely conjecture. (One may also conjecture that biological computation is a new complexity class because we don’t understand biology, but this is also unlikely.) When Witten says “the only real idea”, he means the only consistent mathematics known to be relevant to the question.

    It’s a mistake to study inconsistent mathematics. Also to study consistent but irrelevant mathematics. This wisdom is not any kind of false claim about what string theory has accomplished.

    It’s also important to remember that the Internet decomposes into intellectual echo chambers, one of them being the echo chamber of skeptics of string theory.

    A better idea is to look at good review articles and textbooks. For example, Polchinski’s textbook is to string theory what Nielsen and Chuang is to quantum computation. But I concede that it is challenging material.

  8. Bram Says:

    The difference between string theory and quantum computation is that string theory is an attempt to build a new theory to explain something, while quantum computation takes an existing theory and engages in a thought experiment to demonstrate an inconsistency, or at least an absurdity. We may not know a way of creating a physical experiment to test quantum computation today, but there’s no way for us to philosophically justify just forgetting about it until either the theory is modified to avoid the problem or a way is figured out of actually building a quantum computer and conducting the experiment.

  9. Greg Kuperberg Says:

    String theory isn’t quite exactly the same as quantum computation; it’s an analogy. But it’s not as if string theorists have nothing to go on.

    Quantum probability is true, and general relativity is true. In their present form, they contradict each other. Therefore there should be a reconciliation. But that’s not so easy, because both theories are pretty rigid. The string theorists have found one famous perturbative reconciliation. (Technically, five, but there is some strong evidence that they are non-perturbatively eqiuvalent.) It only works in 9+1 dimensions, but hey, it’s a start. If it’s the only consistent reconciliation that you can think of, you should develop it. Maybe the 6 extra dimensions mean something.

    It’s also too bad that it is only perturbative. That is the regrettable but long-standing compromise of quantum field theory. Maybe you can move past it with hard work.

    The alternative quantum gravity theorists seem much less concerned about the rigidity of general relativity in 3+1 dimensions. The approach is to assume a microscopic semblance of general relativity, then to conjecture that that implies macroscopic agreement. There are actually many microscopic semblances of general relativity — it is very easy to be creative here — but, according to Jacques and others, macroscopic convergence is a devastating restriction which is not ameliorated by microscopic resemblance. If we can believe people like Jacques on this point, then this is not a viable alternative.

  10. Drew Arrowood Says:

    Popper was working in a tradition of analytic philosophy. Russell and Whitehead had looked at the assertions of Kant and Hegel (who said such things as it takes time to go from 7, adding 5, and getting 12) that mathematical knowledge was synthetic, and tried to make such knowledge analytic and atemporal. Philosophers of science have a long tradition of not thinking about the cost of inference.

  11. Scott Says:

    Drew: I completely agree with you. Turing deliciously ridiculed that tradition in Computing Machinery and Intelligence:

    “The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles.”

  12. Scott Says:

    Greg: Well, obviously I’m not a quantum gravity expert, but I was influenced not only by the “Internet echo chamber” but by reading the papers of Lee Smolin, Carlo Rovelli, John Baez, and others, as well as Roger Penrose’s new book. Unlike Leonid Levin and the ID’ers, these people don’t simply criticize; they develop alternatives whose strengths and weaknesses seem complementary to those of string theory. The easy response would be that, if it’s these people’s word against the string theorists’, then a non-expert might be forgiven for withholding judgment.

    But let me try to give a better response. Because of black hole entropy arguments, all parties seem to agree that a bounded region of spacetime has a finite-dimensional Hilbert space. For me, this is an absolutely fundamental fact about the universe, and one that any quantum theory of gravity ought to make manifest. In LQG and spin foam models, it’s obvious how this happens, since (basically) spacetime itself is a superposition over edge-labeled graphs. In string theory it’s less clear, since you start with a fixed background spacetime which is continuous, and then do perturbations around that. How discreteness arises in such a setting is one thing I wish I understood better.

    Smolin likes to argue that LQG is grappling now with issues like background-independence and the discreteness of spacetime, which any quantum gravity theory (including string theory) will have to confront eventually. I find this argument just as persuasive as the argument that any quantum gravity theory (including LQG) will eventually have to recover a smooth spacetime on large scales.

    Regarding consistency, my understanding was that the amplitudes in string theory are given by an infinite sum, and that not only could that sum diverge, but only the first two terms have been shown to be finite! Maybe this is less serious than the analogous divergence problems in LQG, but if that’s the case, I would want to understand why.

  13. Jacques Distler Says:

    >Because of black hole entropy arguments, all parties seem to agree that a bounded region of spacetime has a finite-dimensional Hilbert space.

    Finite entropy ≠ finite-dimensional Hilbert space, but I’ll let that slide. The important point is that the entropy of a blackhole scales like the area, not like the volume. Equally important is that the coefficient (1/4, in Planck units) is precisely known.

    >For me, this is an absolutely fundamental fact about the universe, and one that any quantum theory of gravity ought to make manifest.

    See Strominger and Vafa’s tour-de-force computation of the entropy of extremal (BPS) blackholes. This was originally done in 5D, but the result has, subsequently, been extended to other dimensions and to the near-extremal case.

    >In LQG and spin foam models, it’s obvious how this happens, since (basically) spacetime itself is a superposition over edge-labeled graphs.

    The LQG attempt to reproduce this “fundamental” result

    a) gets the coefficient of proportionality between the area and the entropy wrong.
    b) tacitly assumes that the result is proportional to the area from the outset.

    >Smolin likes to argue that LQG is grappling now with issues like background-independence and the discreteness of spacetime, which any quantum gravity theory (including string theory) will have to confront eventually. I find this argument just as persuasive as the argument that any quantum gravity theory (including LQG) will eventually have to recover a smooth spacetime on large scales.

    I strongly disagree with Lee’s characterization of the situation of String Theory vis-a-vis background independence. But, since I have written extensive responses elsewhere, I won’t do that here.

    As to the discreteness of spacetime, I am shocked to hear that you find that argument “just as persuasive”. It may or may not be true that spacetime is fundamentally discrete at short distances. We have not a shred of evidence either way. On the other hand, we do have abundant evidence that spacetime is smooth at long distances.

    >Regarding consistency, my understanding was that the amplitudes in string theory are given by an infinite sum, and that not only could that sum diverge,…

    In any physically-interesting QFT, perturbation theory is always an asymptotic expansion, with zero radius of convergence.

    >… but only the first two terms have been shown to be finite!

    This is false. The proof of finiteness, to all orders, is in quite solid shape. Explicit formulæ are currently known only up to 3-loop order, and the methods used to write down those formulæ clearly don’t generalize beyond 3 loops.

    What’s certainly not clear (since you asked a very technical question, you will forgive me if my response is rather technical) is that, beyond 3 loops, the superstring measure over supermoduli space can be “pushed forward” to a measure over the moduli space of ordinary Riemann surfaces. It was a nontrivial (and, to many of us, somewhat surprising) result of d’Hoker and Phong that this does hold true at genus-2 and -3.

    >Maybe this is less serious than the analogous divergence problems in LQG …

    The problems of LQG are not even remotely comparable.

  14. Greg Kuperberg Says:

    I didn’t compare Lee Smolin to Leonid Levin. I agree that Smolin is in a rather different position in that he is trying to construct an alternative. I also agree that it would be valid to withhold judgment. But the right way to do that is to say, “I have no idea who is right”, and not to make the positive statement that, for example, Witten was presumptuous in his comments. Nor to ask leadingly if the string theorists are running out of time.

    The problem with the spin foam and and LQG calculations of entropy is that they are ad hoc calculations that are not part of a larger consistent framework.

    Now it is true that the terms of power series expansion of string theory, say for the vacuum expectation, have not been proven to be finite except for genus 0, 1, and 2 (which I would count as three terms rather than two, but never mind that). But unproven is not the same thing as unlikely. That all of the other terms are finite is a rigorous and very likely conjecture. Smolin likes to point out that this conjecture is open, but I don’t think that he has any heuristic argument that the conjecture is false. It’s a very good idea to try to prove this conjecture. It seems like a misreading to turn away because the conjecture is open. It’s like saying, “the Riemann hypothesis is open, therefore number theory needs an alternative”.

    On the other hand, string theorists, and some non-string theorists too, have strong heuristic arguments that the main conjecture of LQG, that it converges to general relativity, is false.

    It is true that the main conjecture of LQG is also much less rigorous, even as a conjecture. Nonetheless the heuristic argument against it is quite general and would require a very good new idea to circumvent. Without any kind of convergence to gravity, you’re giving the name “quantum gravity” to something that you have no reason to believe is gravity.

    So I don’t see this as a case of complementary strengths and weaknesses. Rather, LQG looks more like a Naderite quest to compete with string theory. (I.e., Nader’s struggle is to be an alternative to the Democrats, not to actually win elections.)

  15. Greg Kuperberg Says:

    Sorry, it seems that some of my comments about the perturbative finiteness of string theory are too conservative, since Jacques is saying that finiteness has been proven to all orders. Although now I have a question: Is it also known that the string measure on the supermoduli space of supercurves can be integrated over that moduli space? Or is the result that you describe only that the measure on supermoduli space is finite?

  16. Jacques Distler Says:

    The statement is just about the “pointwise” construction, not about its integral.

    One also needs to be careful to distinguish infrared divergences from UV divergences. The former are divergences of the measure as one approaches the boundary of the moduli space (in string theory, we work with the Deligne-Mumford compactification of the moduli space, or its super generalization). When present, they do, indeed, cause the integral to diverge.

    But IR divergences are physical. They are supposed to be there, and it would be a mistake to wish them away.

    UV divergences would (in the context of string perturbation theory) correspond to divergences of the measure in the interior of the moduli space. That’s a different matter. Those — were they to exist— would not correspond to any physically desirable divergence of the integral.

    Moreover, they could not be removed by the usual procedure of renormalization, because string theory, having no adjustable coupling constants, does not admit any counterterms.

  17. Greg Kuperberg Says:

    Then it sounds like there is a sense in which Smolin’s “criticism” is true, if more as a critique than a criticism. The relevant integrals over the supermoduli space of curves are conjectured to be finite for every genus, but this is only known in genus 0, 1, 2, and 3. (I am not sure of the role of marked points in the status of this question, which can represent, among other things, the external tubes of a stringy “Feynman diagram”.)

    But I gather that it’s an extremely likely conjecture for every genus, supported by a variety of consistency checks.

  18. Anonymous Says:

    “The problem with the spin foam and and LQG calculations of entropy is that they are ad hoc calculations that are not part of a larger consistent framework.”

    Ahhh! This is the crunch, right? I agree perfectly about the adhocness of said calculations … but it sounds like you are dismissive of anything that is not couched in a nest on a great edifice the size of String theory. In other words, EVERYTHING gets dismissed … because waiting around for a theory to develop to that extent is surely like waiting … forever.

  19. Greg Kuperberg Says:

    But the definition of string theory is not a great edifice. It’s a very short definition, which moreover has no free parameters. The topology of compactified dimensions gives you some choices for some questions, but not for black hole entropy calculations.

    The results of string theory really are a great edifice. So far it is an edifice of consistency and mathematical ideas rather than predictive physics. Although it also has hints of predictive physics.

    As Jacques said, the LQG calculation of black hole entropy is so ad hoc that it has no merit as a consistency check. If it were merely somewhat ad hoc, so that it had some check left in it, then it would still be interesting.

  20. Anonymous Says:

    “But the definition of string theory is not a great edifice. It’s a very short definition.”

    We agree about the LQG stuff, by the way. So, hit us with the short definition … and then maybe we can discuss alternative short definitions?

  21. Jacques Distler Says:

    >The relevant integrals over the supermoduli space of curves are conjectured to be finite for every genus …

    Depending on what we are calculating, the integrals are allowed, even expected to diverge (these are the IR divergences I spoke of before).

    The absence of UV divergences is, as I said, well-established.

    I think you are conflating the absence of UV divergences with the absence of a certain (unwanted) IR divergence. Namely, if supersymmetry (in flat space) is broken, the genus-g vacuum amplitude is nonvanishing. This leads to an IR divergence of the genus-2g vacuum amplitude.

    The proof that this latter effect does not happen is rather less appealing than that of the absence of UV divergences. I think the proof is correct, but ugly.

    Since we now strongly believe the result to be true, not only to all orders in perturbation theory, but nonperturbatively as well (certainly, in theories with 32 supercharges, like the Type-II string in flat space), a more direct proof seems desirable.

  22. Greg Kuperberg Says:

    Jacques: I agree that what you are quoting seems muddled and I’m not sure exactly what statement I really mean. My recollection is that somewhere in the comment section of Lubos Motl’s blog, Smolin got Lubos to concede that some natural perturbative finiteness conjecture is still open. Unfortunately, these comments appear to have been purged from Lubos’ site.

    In any case it’s a very small point in this present discussion. Even though Lubos conceded some such point, he insisted that it was proven “at the physical level of rigor”, as is said about many QFT results.

    Anonymous: Bosonic string theory is defined as follows. It is a perturbative expansion in which 2-dimensional surfaces (strings moving through time and interacting by making pairs of pants) replace Feynman diagrams. There is only one “diagram” for each genus of surface. The value of this diagram is given by integrating an action over all positions of the surface in spacetime. The action that you integrate is exp(i*alpha*A), where alpha is the string coupling constant and A is the area of the surface (as it is embedded in spacetime).

    So string theory is a non-perturbative quantum field theory on surfaces, reinterpreted as a perturbative tubular Feynman diagram expansion. The QFT itself makes sense for any spacetime, but the reinterpretation is only consistent when (1) spacetime satisfies the GR equations, and (2) spacetime is 25+1-dimensional.

    Superstrings are the same idea, except that the string’s worldsheet (i.e., its evolution tube) and spacetime are both supermanifolds (of a restricted kind). The area Lagrangian of the base QFT is replaced by a natural super-analogue. The “tubular Feynman diagram” reinterpretation is now only consistent in 9+1 dimensions, again with the GR equations.

    That is really everything in the definitions of strings and superstrings, except that I glossed over some details of the “super” part, and I didn’t mention heterotic strings. (They are a cross between bosonic strings and superstrings.) But as I said, even though the definition is short, its analysis is not.

  23. Jacques Distler Says:

    >In any case it’s a very small point in this present discussion.

    Of course it’s a minor point.

    But the so-called “String Sceptics” can’t seem to be bothered to get straight what statement they’re questioning. Then people like Scott (no offense) read what they’ve written, and repeat it in even more muddled form.

    Pretty soon, we have what sounds like a major crisis at the foundations of String Theory on our hands.

  24. Scott Says:

    “Then people like Scott (no offense) read what they’ve written, and repeat it in even more muddled form.”

    Of course. No offense taken. ;)

  25. Greg Kuperberg Says:

    One more thought this morning: It is completely reasonable to look for models of quantum gravity, or reality otherwise, that doesn’t look like string theory. Two examples that come to mind are Matrix theory and 11-dimensional supergravity. Both of these arise as strong-coupling limits of string theory. But their definitions are completely different, and it is one reason that people believe that eventual definition of M-theory will also be completely different.

    So the point is not to stay loyal forever to one narrow definition, although the old definition of superstring theory from 20 years ago has been monumentally useful. The point is that the string theory community is a big tent that will accept any viable fundamental model. Maybe not every string theorist is so accepting, but enough of them are. Witten, in particular.

    The string theorists’ intuition is that viability should imply a direct connection to string theory. That may seem unfair, but it could be well-founded.

    But part of the Naderite mindset is that anything accepted into the big tent belongs to the other side, and therefore doesn’t count as open-mindedness. It’s an eternally divisive quest to compete with string theorists (or in the case of the real Nader, with Democrats). Merely writing papers in loop quantum gravity or whatever is just fine and it would be unfair to criticize Smolin or Rovelli or anyone else just for that. Perpetually offering it as an alternative to string theory seems closed-minded and self-defeating.

  26. Anonymous Says:

    “The string theorists’ intuition is that viability should imply a direct connection to string theory. That may seem unfair, but it could be well-founded.”

    I don’t doubt for a second that mathematically speaking you are right. The alternatives that I would consider ‘viable’ could easily be brought into the BIG TENT.

    Physically, however, I find it extremely alarming that ANY viable alternative must be evaluated in the context of the tent. Surely the only way to understand an emerging picture is with the ‘right’ physical intuition, which may or MAY NOT have something to do with strings.

  27. Greg Kuperberg Says:

    The string theorists are physicists and this is their intuition. Do you want physical intuition or not?

    Okay, Smolin is also a physicist and his intuition is radically different from that of the strings theorists. So who is right? It’s just my feeling that Smolin, Rovelli, and a few others, are biased; while Witten, Jacques Distler (our guest string theorist), and many others aren’t. I can point to a variety of clues for this, but ultimately the only path to wisdom is to delve more deeply into the material yourself.

    On that note, I think that Polchinski’s two-volume set, “String Theory”, is an outstanding textbook. It is challenging material and I personally haven’t learned all that much of it, but it’s still great stuff.

  28. Anonymous Says:

    All right. Perhaps we could discuss some alternative short definitions. In my favourite one Feynman diagrams can also turn into pants etc. BUT IT ISN’T STRING THEORY and the only large intersection between it and Strings historically is TFTs.

    Definition 1: The necessity of imposing Gray’s categorical comprehension as a physical principle for measurement forces the consideration of dimension raising products and categorical cohomologies.

    Perhaps you might delve a little more deeply into these ideas.
    Kea

  29. Peter Woit Says:

    “As for the blog “Not Even Wrong”, I can make an analogy: I think that Peter Woit is to string theory as Leonid Levin is to quantum computation. And that’s not counting Levin’s established work in classical computation.”

    This is pretty funny, coming from Greg Kuperberg. In his comments here and on other blogs, he loves to go on about the glories of string theory, while not once making an accurate statement about the subject. I’m resisting the temptation to quote some of Kuperberg’s claims from this thread about string theory and explain why they are wrong (Distler does explain what’s wrong with at least one of them). Based on past experience, it would just be an utter waste of time. When challenged about any of his completely incorrect statements about string theory, he admits that he doesn’t know much about the subject. What is it about string theory that leads people to engage in this kind of behavior?

    Unlike Kuperberg, Jacques Distler generally knows what he is talking about, although he approaches things with the fanatical true-believer’s perspective. Every problem with string theory is denied or minimized, whereas problems with any alternate approach are made much of.

    Distler’s comment that “The proof of finiteness, to all orders, is in quite solid shape” is seriously misleading. I assume he chose the terminology “in quite solid shape” carefully, instead of something less ambiguous like “there is a rigorous proof”. The fact of the matter is that there is no complete proof of multi-loop finiteness in the literature. The closest thing to such an argument is due to Berkovits, but in his published papers he acknowledges gaps in his argument. I am not an expert in this subject, but I have discussed it extensively with people who are much more experts on this than Distler is, and they don’t believe that a complete proof of finiteness now exists.

    While there is no complete proof, it is certainly reasonable to expect that these amplitudes are finite. But the series they form seems to be undoubtedly divergent. Distler is quite right when he says this is also true of physically important QFTs. But in the QFT case we have non-perturbative versions of the theory, and at least for asymptotically free theories there is extensive evidence that these theories are non-perturbatively finite and consistent.

    When string theorists tell you that they have a “finite theory of quantum gravity”, they’re engaging in a bit of deception. What they actually have is (assuming multi-loop finiteness) a divergent series which they hope is an asymptotic series for some still unknown non-perturbative version of string theory.

  30. Jacques Distler Says:

    “Unlike Kuperberg, Jacques Distler generally knows what he is talking about, although he approaches things with the fanatical true-believer’s perspective. … Distler’s comment that “The proof of finiteness, to all orders, is in quite solid shape” is seriously misleading. … I am not an expert in this subject, but I have discussed it extensively with people who are much more experts on this than Distler is…”

    I have learned, from experience, that it is best to avoid responding publicly to Peter Woit. Invariably, the end result is another one of his trademark paranoid tirades.

    I’d be happy to discuss the matter further, via private email, if you’re interested.

    I’m sorry, however, to have to bow out of this public discussion.

  31. Greg Kuperberg Says:

    Kea: If the idea to replace edges in Feynman diagrams by tubes and vertices by pants, then I don’t see how you could call it anything other than string theory. At least not if the the diagrams retain their perturbative interpretation.

    (NB, I see that your nickname appears in sci.physics.strings and is associated with the e-mail address of Marni D. Sheppeard. Coincidence?)

    Jacques: It’s completely reasonable not to respond to Peter Woit if you have nothing to say in response (or nothing new to say, etc.). But if you mean that we won’t see you in Aaron’s blog any more, I think that you’re overreacting. You have to respect Aaron, because he’s willing to get his hands dirty. (Except for the gloves, I concede.)

  32. Greg Kuperberg Says:

    Did I say Aaron? Good grief, I meant Scott, son of Aaron.

    It’s too bad that only the blog host can edit the blog ex post facto.

  33. Jacques Distler Says:

    “I think that you’re overreacting.”

    OK, for your and Scott’s sake…

    When most people (at least, most quantum field theorists) use the term “finiteness,” they are referring to UV finiteness.

    There are no cavils about UV finiteness of String perturbation theory. However, as I emphasized above, there’s a quite different notion of finiteness that people do still worry about (though only a little).

    Say you start off in flat space, with zero tree-level cosmological constant. Imagine that, at g-loops, you induce a nonzero cosmological constant. At this point, flat space is no longer a solution to the quantum-corrected equations of motion and, at (2g)-loops, you discover an IR-divergence of the vacuum amplitude.

    This is not a phenomenon that has any analogue in non-gravitational QFTs. However, it is something that crops up in any quantum theory of gravity. It is, in no sense, peculiar to String Theory.

    The technical proof that this divergence doesn’t occur in String Theory is due to Atick, Moore and Sen. Many people (include both myself and Woit’s unnamed “experts”) are dissatisfied by their proof.

    But, conceptually, you know why it “has” to be true. Type IIB supergravity in 10D does not admit a cosmological constant term — that’s just incompatible with local IIB supersymmetry. So there’s no way loop corrections could induce one.

    Type IIA admits a cosmological constant (a theory called “massive Type IIA”), but its value is quantized, so again, you can’t induce it by radiative corrections if it vanishes at tree-level. [Alternatively, IIA and IIB String Theories differ only in the sign of the GSO projection in the left-moving Ramond Sector. This choice cannot affect calculation of the divergence in question.]

  34. Anonymous Says:

    “At least not if the the diagrams retain their perturbative interpretation.”

    Which they don’t.

  35. Anonymous Says:

    Thanks for spelling my name correctly.

  36. Greg Kuperberg Says:

    It is an amazing fact that the perturbative expansion/definition of any non-trivial quantum field theory (for some definition of “non-trivial”) has zero radius of convergence; and string theory shares this feature. It means that if you expect either flavor of theory to exist non-perturbatively, then it needs a separate definition. String theorists know this, which is why much of their attention lately is devoted to the unknown “M-theory”.

    But zero radius of convergence does not mean that you can’t make inferences. For example, suppose that you take an interest in some function f(x) whose power series is sum_{n=0}^infty 2^(n^2) x^n. Its radius of convergence is zero, and technically speaking the power series doesn’t say anything about any finite value. Nonetheless the power series does satisfy the functional equation f(x) = 2x f(4x)+1. You can suppose, since this is ultimately intuitive work, that the real f(x) satisfies the same relation. Then you might try to use this as a recurrence, e.g., f(1) = 1 implies f(1/4) = 3/2 implies f(1/16) = 19/16 implies … I think, although I haven’t tried to prove it, that the values that you get this way match the power series.

    I don’t know of any relevance of this example to quantum field theory or string theory. Even so, the perturbative forms of both theories provide a lot of fodder for inferences. There is even a seim-rigorous machine for making inferences known as “renormalization theory”.

    Zero radius of convergence also does not imply no predictions. The only known definition of quantum electrodynamics is perturbative. Renormalization theory even suggests that that is the only definition there ever will be, unless you add new short-distance forces. Nonetheless if you plug in the fine structure constant, the early terms converge extremely well, up to the limit and maybe past the limit of low-energy experiments.

    Unfortunately real-life quantum gravity is almost certainly strongly coupled at low energy, so this tactic is not available as a test of string theory. There may be no way to probe perturbative limits with forseeable high-energy experiments, except conceivably by examining the ultimate high-energy event, the Big Bang.

  37. Scott Says:

    “Did I say Aaron? Good grief, I meant Scott, son of Aaron.”

    Yeah, for whatever reason, a lot of people make that slip. :)

  38. Peter Woit Says:

    “Unfortunately real-life quantum gravity is almost certainly strongly coupled at low energy”

    What causes someone to go on like this? Bizarre…

    Jacques: One person’s trademark paranoid tirade is another’s forceful response to someone who deals with valid criticisms of string theory by repeatedly misquoting me and using these misquotes to try and argue that I don’t know what I’m talking about. If you’d stop using that tactic we undoubtedly could have some interesting discussions. I’m sure I’d learn something, and you might too.

    Unlike some other commenters here I do have a clear idea of what I understand and what I don’t in this field. Multi-loop amplitudes are something I’m happy to admit that I don’t understand very well, although I’ve spent quite a lot of time asking questions of people who do and trying to understand what they tell me as best I can. In that spirit, one thing such people have told me is that one can’t really distinguish IR and UV divergences. This seemed plausible given what I understand about T-duality and various facts about conformal symmetry, but quite possibly I misunderstood this or have this wrong. Is this not a problem for the distinction between UV and IR divergences that you are making?

  39. Jacques Distler Says:

    >Unlike some other commenters here I do have a clear idea of what I understand and what I don’t in this field. Multi-loop amplitudes are something I’m happy to admit that I don’t understand very well …

    And yet, you have no hesitation in calling me a liar? (“Distler’s comment that ‘The proof of finiteness, to all orders, is in quite solid shape’ is seriously misleading…”)

    OK …

    I’ll have to defer to your expertise on the subject of misleading statements about the status of String Theory.

    >Is this not a problem for the distinction between UV and IR divergences that you are making?

    No, it is not.

    IR divergences are associated to divergences of the (super)string measure as one approaches the boundary of the moduli space (in string theory, we work with the Deligne-Mumford compactification). UV divergences are associated to divergences of the string measure in the interior of the moduli space.

    IR divergences are associated to massless particles in intermediate channels going on-shell. UV divergences are an entirely different matter (as I assume you are well-aware).

    T-duality has nothing to do with the matter:

    a) because the question arises already in 10D Minkowski space and
    b) because T-duality takes massless particles into massless particles.

    That said, the earliest, naive, attempt to write down the higher-genus superstring measure (as a measure on the ordinary bosonic moduli space) did, indeed, have spurious poles in the interior of the moduli space. This was subsequently understood as a singular choice of gauge. (I’m going to skip all the technicalities, some of which I’ve posted about on my blog. Perhaps you read my posts about d’Hoker and Phong’s work, and the question of the splitness of supermoduli space?)

  40. Bram Cohen Says:

    I’m not sure if David Molnar’s suggestion in the first comment is a good one, but I believe that if we measure based on units of researcher-years work to prove, and we have a standardized researcher, then the speedup theorem applies.

  41. Anonymous Says:

    Scott,

    Since I am a string theorist, my opinions will probably not count for much, but let me offer some true statements for you to ponder:

    1. In AdS/CFT string theory has a formulation (in spaces with negative cosmological constant)
    which is as background independent as one could wish for.

    2. Most of the work on LQG in the last decade is based on Witten’s work on topological QFT, he could have been triumphant about that if he chose to. As I know him to be an independent thinker of exceptional caliber, I tend to consider his judgement seriously, but I guess that makes me part of the cult.

    3. Any continuous model in nature, e.g. fluid governed by the Navier-Stokes equation, has finite entropy for finite volume, this is not a hint for discrete sub-structure. Nevertheless I don’t think this point, of having some sort of discrete structure at short distances, is particularly controversial among string theorists. The only quesion is precisely which such structure. LQG and similar approaches choose to take the long distance geometrical strunctures and simply discretize it. In string theory one tends to get more elaborate algebraic structures, such as abstarct CFT at short distance.

    best,

    Moshe Rozali

  42. Peter Woit Says:

    Jacques:

    We still seem to have this misquoting problem. If I intended to call you a liar that’s what I would have written. I have no doubt that what you write is what you believe to be true. I just happen to disagree that characterizing the state of the proof of multi-loop finiteness as “in quite solid shape” is accurate. You acknowledge the existence of at least one remaining problem, whereas saying a proof is “in quite solid shape” generally would lead most people to believe there are no remaining problems with it.

    I’ve read with interest your postings about d’Hoker and Phong’s work, as well as discussing it directly with them. While I’m no expert on this, my understanding of the situation is that the really difficult problem they had to solve was that of coming up with a gauge slice invariant amplitude.

    Correct me if I’m wrong, but my impression is that how to handle this problem at higher loops remains, either using Phong and d’Hoker’s methods, or the quite different ones of Berkovits.

  43. Jacques Distler Says:

    >Correct me if I’m wrong, but my impression is that how to handle this problem at higher loops remains, either using Phong and d’Hoker’s methods, or the quite different ones of Berkovits.

    As I explained at some length in this comment thread before you showed up,

    1)The string measure, as a measure on supermoduli space, can be written in a fashion manifestly free of UV divergences.
    2)Beyond 3 loops, it is unclear that it can be “pushed forward” to a measure on the ordinary bosonic moduli space. (As d’Hoker and Phong, quite surprisingly showed was possible in genus 2,3).
    3)In contrast, the proof of the absence of IR divergences (of the vacuum amplitude) is probably correct, but ugly and indirect. It certainly could bear improvement.

    I explained all these things before you showed up to charge that I was being “deceptive” on the subject of finiteness.

    (And, unlike some people, I don’t rely on carefully-worded statements that seem to say one thing, whereas they actually say something else. The explanations above, which I’ve repeated here for your benefit, were clear and unambiguous statements of the status of the subject, as I understand it.)

  44. Greg Kuperberg Says:

    Jacques: Certainly your wisdom on string theory is appreciated and I’m glad that you haven’t left. But I think that it’s a mistake to publicly declare that you shouldn’t respond to someone and then respond anyway. I think that you were right the first time. If it needed to be said; I was assuming that Peter Woit’s diatribe speaks for itself.

    To expand on Moshe Rozali’s comments, certainly one reason that I am skeptical of loop quantum gravity, even as a plausible alternative, is that much of it seems reactive and derivative to string theory. (And to non-string work of string theorists such as TQFTs.) For example, the LQG community appeared to jump into black hole entropy as if it was an ante laid down by string theorists. Maybe this appearance was completely unintentional and coincidental, but it just doesn’t feel right. It doesn’t help matters that so many reviews of LQG have Coke vs Pepsi comparisons with string theory.

    Also, re Moshe’s comment about discreteness, the point is that if you want the theory of everything to be locally finite (or “UV finite” as Jacques might say), you should do it the quantum way. Instead of insisting that spacetime is a microscopic Tinkertoy set, you should be open to any reason that local Hilbert spaces are finite-dimensional, or otherwise finitely cut off. (For example local life could be bounded by a finite-entropy mixed state.) Everyone knows that an atom only has finitely many stable states within a bounded radius of its nucleus. This crucial issue, which was indeed a major crisis that motivated quantum mechanics itself, was not settled by discretizing space.

  45. Peter Woit Says:

    Jacques:

    I really wish you would make a better effort to actually quote the words I write. I didn’t write that “you were being ‘deceptive’”,
    my term was “misleading”, and I used it in the passive form of saying that something you wrote was misleading, not in the active form that you were actively trying to mislead people. Sure, you can argue that this is a distinction without a difference, but if you’re going to quote me, using quotation marks, how hard is it to use exactly the words that I wrote?

    Believe it or not, I did carefully read your earlier comments before writing anything here. Thanks for the clarification of your earlier comment. I took your earlier statement that many people were “dissatisfied” with the proof of Atick, Moore and Sen, to mean that some of these many people thought it was wrong, which is different than your clarification that it is “probably correct”, and the dissatisfaction is just about its ugliness and indirectness. This last formulation is still a bit unclear, since it’s kind of ambiguous what it means for a proof to be “probably correct”. If the steps of a proof are clear, one can tell whether it is correct or not. If they’re so unclear one can’t be sure the proof is correct, it’s not really a proof. There’s no question here that the finiteness is probably true, the question is how solid an argument for this there is.

    Anyway, thanks for your comments, I did learn something, although some things you claim seem to me inconsistent with things I’ve heard from other experts. Probably in the future I’ll learn even more about this and these inconsistencies will get resolved one way or another.

  46. Peter Woit Says:

    Greg,

    Perhaps you can explain to us why “real-life quantum gravity is almost certainly strongly coupled at low energy”. If you can’t, perhaps you could explain to us why you are writing extensive comments here about issues you don’t understand, and at the same time attacking people who actually do.

  47. who Says:

    Dear Moshe,

    I wish by asking to provide you with occasion to expand on some of this:


    …2. Most of the work on LQG in the last decade is based on Witten’s work on topological QFT, he could have been triumphant about that if he chose to. As I know him to be an independent thinker of exceptional caliber, I tend to consider his judgement seriously, but I guess that makes me part of the cult.

    3. Any continuous model in nature, e.g. fluid governed by the Navier-Stokes equation, has finite entropy for finite volume, this is not a hint for discrete sub-structure. Nevertheless I don’t think this point, of having some sort of discrete structure at short distances, is particularly controversial among string theorists. The only quesion is precisely which such structure. LQG and similar approaches choose to take the long distance geometrical strunctures and simply discretize it. In string theory one tends…

    Your talking more about point 2 might clarify the language. I think you mean that most of the 1995-2005 work in LQG has been on path integral or sum-over-histories approaches which would include spinfoams and an alternative, causal dynamical triangulations, which has gained recognition lately.

    But you could have meant something different by “Most of the work on LQG in the last decade…
    You might be understood by someone to mean the decade of the 1990s and the canonical Loop Quantum Gravity which was developed in the 1990s, encountered some problems, and seems now not so much worked on except for the symmetry-reduced version used in cosmology–almost a separate field.

    In your point 2. you mention Witten’s judgment. I am curious to know what was this judgment and what was it about? Was it about the canonical LQG of the 1990s? Or about some of the different spinfoam formalisms that have been or are being tried? Did it include CDT? It would be nice to know what he said and what he was talking about.

    In regard to your point 3 I was curious about what you meant by LQG and similar approaches choose to take the long distance geometrical strunctures and simply discretize it. In string theory one tends…

    As you may know, in CDT no evidence of spacetime discreteness, or of a minimal length, has been found—of course discreteness is not assumed. In a standard version of LQG (as in Thiemann’s lecture notes) discreteness is not assumed but some discrete spectra are derived. So I am confused by your saying that one “simply discretizes” geometrical structures.

    I hope you will expand on some of your statements.
    The bothersome question that arises when people seem to be talking about the main non-string lines of research in quantum gravity, is which recent papers are they talking about. Maybe I should get some arxiv numbers for some 2005 papers so we could look at them and ask “is this what Witten had in mind?” , “in what sense is this discrete or continuous?”, “in what way does this behave as expected at large scale?”

    best,

    Who

  48. Moshe Says:

    let me clarify my statements a bit: though I admit to be weary of chatting with a virtual person, it is Halloween and in any event I think some of my comment was opaque. I am likely to make things worse, I know…

    The reference to Witten was in reply to Scott’s comment where he cited Witten’s sentence referring to string theory as the “only real idea there is for quantum gravity”. I suggested that this could be read as a reasoned judgement by a qualified and impartial person, rather that “premature triumphism” (even though I don’t completely agree). I probably overstated the significance of TQFT in LQG, and sorry about that, but my point is valid I think.

    (and I suggest to stay away from discussing specific people, sorry I opened this can of worms)

    The discrete sub-structure is an interesting question.
    The CDT appraoch is explicitly a discretization (triangulation) of spacetime, the same spacetime that one wants to recover at long distances. In my view there are regimes in string theory where the smooth spacetime picture breaks down, and then generally one has something else altogether, abstract CFT, matrix model, gauge theory or whatever.

    The quantization of areas and volumes arise in models which are not discrete from the outset, and in that sense are more similar to what happens at short distances in string theory, but still the idea of keeping some sort of geometrical structure in short distances is something that may or may not work, and I think is the common thread in all “alternative” approaches. This, and not “background independence” or “discrete structure”, is the main difference in my mind between string theory and other approaches to quantum gravity.

    Finally, in the future I would be happy to have a chat with a real person, more fun this way.

    best,

    Moshe

  49. who Says:

    Moshe, you write The CDT approach is explicitly a discretization (triangulation) of spacetime, the same spacetime that one wants to recover at long distances.

    But as I understand CDT, it is explicitly NOT a discretization of spacetime. The scale of the triangulation is allowed to go to zero in a continuum limit. The principle authors say explicitly that no evidence of discreteness has been found nor anything to suggest the existence of a minimal length.

    See paragraph 4 of the first page of text of this recent paper by Ambjorn, Jurkiewicz, Loll
    http://arxiv.org/hep-th/0505113

    “…We have recently begun an analysis of the microscopic properties of these quantum spacetimes. As in previous work, their geometry can be probed in a rather direct manner through Monte Carlo simulations and measurements. At small scales, it exhibits neither fundamental discreteness nor indication of a minimal length scale…”

    Their underlying spacetime is a continuum which they do triangulate. but this triangulation is not fixed nor does it have a fixed scale. I do not think that temporarily triangulating something essentially compromises its continuum nature.

  50. Moshe Says:

    Good, I stand corrected. This fits well with my previous point that “alternative”approaches are distiguished from string theory mainly by assuming some sort of geometry to be valid at all distance scales. As Jacques pointed out we have no evidence for or against this.

    In any event, I fear we are already boring the locals, we can stop here.

  51. who Says:

    Dear Moshe,

    I shall be sorry if you choose not to reply, but I had another point about your post. You may appreciate knowing of a mistake.

    you wrote:

    “The quantization of areas and volumes arise in models which are not discrete from the outset, and in that sense are more similar to what happens at short distances in string theory, but still the idea of keeping some sort of geometrical structure in short distances is something that may or may not work, and I think is the common thread in all “alternative” approaches. This, and not “background independence” or “discrete structure”, is the main difference in my mind between string theory and other approaches to quantum gravity.”

    It is not true that the same geometrical structure arises or is assumed at all scales. something as simple as the dimension can be different at different scales. The same link I gave in my previous post will suffice for this. Dimension turns out to be 4 at large scale but substantially less at very small scale.

    the link given earlier, which will suffice, is
    http://arxiv.org/hep-th/0505113
    but for a more complete survey there are
    http://arxiv.org/hep-th/0505154
    and
    http://arxiv.org/hep-th/0509010

    the latter is written for non-specialists.
    Geometry (even basic things like the possibility of having a metric, and what dimension, or the possibility of having something approximately like a tangent space) changes radically with scale.

    However your post says:
    “Good, I stand corrected. This fits well with my previous point that “alternative”approaches are distiguished from string theory mainly by assuming some sort of geometry to be valid at all distance scales. As Jacques pointed out we have no evidence for or against this.”

    This worries me because it is contrary to the facts. The same geometry does NOT turn out to hold at all distance scales. A 4D geometry arises at large scale and something more like 2D arises in the small. Nor is any such thing assumed.

    A similar result has been derived in Martin Reuter’s work, using a different model altogether. Loll et al have references to Reuter’s papers so i will not give them here.

    Of course I am speaking only of these two approaches—CDT and Reuter’s QEG, both important approaches to QG alternative to string.
    In some other non-string QG approaches what you say may be perfectly true—they may well assume the same geometric structure at all scales.

    Sorry if you think this is boring, just trying to get it right.
    best wishes and respect,
    who

  52. Moshe Says:

    My comment was that in all these approaches there is *some* geometry at short distances, did not claim it is the same geometry as the long distance one, or the same one at all scales. This I contrast with various approaches to string theory where the short distance physics is entirely non-geometrical and geometry emerges only at long distances (as it should). Nothing important, just making some low level taxonomy to amuse myself.

  53. who Says:

    amusing taxonomy is almost certainly the best kind. and this already it strikes me as more interesting than some of the other distinctions you mentioned

  54. Wolfgang Says:

    who,

    > I do not think that temporarily triangulating something essentially compromises its continuum nature.

    Actually it does.
    One of the issues with dynamical triangulation is the simple fact that it has no classical continuum limit.
    Deficit angles of arbitrary (large) loops will always assume values out of a discrete set only.

  55. Wolfgang Says:

    who,

    this issue was discussed on p.9ff of the paper
    http://xxx.lanl.gov/abs/hep-lat/9505002

    I should add that this does not automatically mean that there is anything wrong with DT or CDT as a quantum theory.

  56. who Says:

    hi Wolgang, I am printing out your paper
    http://xxx.lanl.gov/abs/hep-lat/9505002
    but it is going slowly so I will reply precipitously.

    the idea of continuum limit being compromised is exciting. you say:
    One of the issues with dynamical triangulation is the simple fact that it has no classical continuum limit.
    Deficit angles of arbitrary (large) loops will always assume values out of a discrete set only.

    Now I have your paper printed out. It did not take as long as I expected at first.

    Just glancing, I see that you use equilateral triangles. Loll and coworkers, at least in anything recent I have seen, seem to make a point of not doing that. I am not sure about 2D but in 3 and 4 dimensions they make the simplices be a bit “squat”—there is an “asymmetry parameter” by which the timelike edges are shorter than the spacelike, and this “asym. param” is adjusted to help get the continuum limit. Could this have anything to do with the fact that you found no continuum limit. Am I missing something? (of course, I just looked at the paper this moment, anyway a first reaction for whatever it’s worth)

    bravo you and Bernd for being among those who have done QG computer simulations using dynamical triangulation. To an outside spectator it now seems like a profitable line of investigation, that you identified very early as of potential interest.

  57. Wolfgang Says:

    > Loll and coworkers, at least in anything recent I have seen, seem to make a point of not doing that.

    They have two different link lengths for space-like and time-like links but they are fixed; Thus the comment I made applies.

  58. Wolfgang Says:

    who,

    just to make myself clear once more:
    Dynamical Triangulation does not have a classical continuum limit. This is true for all dimensions D=2,3,4, …
    This means the sum S over all triangulations (number of simplices N, link lengths a) does not approximate the integral I over all geometries in the sense that S would contain all geometries in some sense as N goes to infinity and a to zero.

    It is relatively easy to see that this is also true for causal dynamical triangulation.

    I am afraid the discussion would require more technical details and I do not want to abuse Scott’s blog.