Strings 2007 is starting today, and already there seem to be a large number of different laptops connecting to this blog from wlan.uam.es. I’m hoping that some of their owners will write in here with news of how the conference is going. I’ll also try and add links here to any blog and press coverage of the conference that I see or hear about.
Witten’s new 83 page paper entitled Three-Dimensional Gravity Revisited appeared on the arXiv last night and presumably this is the detailed version of what he’ll be talking about in the first of tomorrow’s talks at the conference. It corresponds pretty much to what he talked about here in New York a few weeks ago, which was described here. I assume many of those laptops at the conference are being used to download and read copies of Witten’s paper. At his blog, Clifford Johnson notes that he won’t be at Strings 2007, and hopes that this will ensure that there will be an exciting breakthrough announced there, just like what happened when he decided not to attend Strings 1995.
As I write this, I see that string theorist Jacques Distler is there live-blogging. Here’s the first of his reports.
Update: Some of the slides of the talks are already on-line. The slides for Witten’s talk are available here.
Update: There’s an interesting posting here by Jacques Distler about the Witten talk, and, for those who enjoy such things, quite a rant from Lubos here, prompted by my comment that in this case Witten is investigating quantum gravity using non-perturbative QFT, not strings.
I’ve been reading Witten’s paper a bit more carefully, and it raises all sorts of interesting issues. He makes the point that even in 3d, we really don’t know exactly what “non-perturbative pure quantum gravity” is. He uses the Chern-Simons formulation of 3d gravity in terms of gauge theory to motivate his guess at the correct boundary CFT, and then once he has that he has something much more well-defined to study.
This is a bit reminiscent of the compact, non-gravitational situation. There Chern-Simons theory works fine perturbatively, but to understand the non-perturbative theory one connects it to a CFT on the boundary, in this case the Wess-Zumino model (this is the story that got Witten a Fields medal).
Update: B. Yen has set up a video-blog for Strings 2007, where there will be iTunes podcasts of the conference available.
Update: Latest report from Jacques Distler is that, since Witten’s one, which he got to write about:
there have been some very cool talks
(emphasis in the original), but he can’t tell us even which ones they were since his laptop is malfunctioning. Slides of the talks are available here. I’ve looked through them and, besides Witten’s, don’t see anything I would describe as “very cool”, but maybe that’s just me.
Update: All the talks are on-line now and I just looked through the last of them, and watched the summary talk by Gross. Lisa Randall discussed recent calculations of black-hole production at colliders, with the bottom line being that even in the unlikely event the gravity scale is within reach of the LHC, existing bounds already pretty much rule out the possibility of seeing the kind of dramatic effects from black hole production that have been widely advertised as something that might be seen at the LHC.
Gross noted that the conference was much less mathematical than last year’s, possibly because Yau was not involved in organizing it. He was most enthusiastic about describing the many talks on AdS/CFT, especially the Beisert talk which told about recent progress in getting an exact solution of N-4 SYM. Some talks referred to possible applications of AdS/CFT not just in QCD and heavy-ion physics, but in condensed matter physics (using the duality to get info about relevant CFTs). He told about Polchinski’s speculation that “maybe AdS/CFT will solve high Tc superconductivity”, but dismissed it with “sounds great, but seems unlikely to me.” He dealt with the landscape talks by flashing them by quickly, in a lower and less enthusiastic voice, noting that they made up at least a quarter of the talks at the conference. He dealt similarly with the cosmology/anthropic talks, describing Bousso’s as “an attempt to make the anthropic principle precise if not respectable.”
After the summary, he gave his own take on the state of string theory, saying that one had to be honest about the lack of falsifiable predictions and that now he had a slide headed “The Failures of String Theory”. He continues to feel that the main failure is because we “don’t know what string theory is”, that something is missing, some principle that would pick out not a “vacuum” but a “cosmology”, one perhaps using new ideas about what space and time are. He said he was not too upset by the landscape, because “we don’t know what the rules are” in string theory, so one can’t argue that string theory implies the landscape. He appeared to feel that he is losing the debate, complaining that this used to also be the opinion of his colleagues, but that they were going over to the other side because of the cosmological constant, saying that if another explanation of the CC was found 90% of the anthropicists would come back to his side. He tried to minimize the size of the CC problem, measuring it with respect to a supposed 1 TeV SSYM breaking scale and working in energy, not energy density units, so it is only too small by a factor 1016. He compared this to Dirac’s famous large number problem (which Dirac tried to solve not anthropically, but by time-varying constants, leading to a prediction that was falsified), which was finally “explained” by asymptotic freedom. His message to the anthropocists was “just because you don’t know an explanation doesn’t mean it doesn’t exist”.
Finally he mentioned that Strings 08 will be at CERN, Strings 09 in Rome, and no one has yet agreed to host Strings 10. He argued that the series of Strings conferences “must go on”, because they are “like the canary in the coal mine”, and if they stop that would be a very bad sign for string theory.
I’m curious to know what those in attendance thought of this; it wasn’t exactly a rousingly optimistic portrayal of the state of the subject…
Update: There’s an article in El Pais about the conference and about the state of string theory. My Spanish isn’t perfect, but as far as I can tell the piece was pretty much pure unadulterated hype, of the sort that it is one goal of the Strings XX bashes to generate.
Update: Jacques Distler finally got his computer fixed, and posted about one example of what he considered a “very cool talk” with exciting new ideas. Unfortunately, it seems the ideas are not that new, since a commenter wrote in to his blog to point to papers from four and a half years ago that do pretty much the same thing.
Hi Peter,
I wonder why three international conferences dealing with strings one way or another are being held at more or less the same time around the globe. I of course would not have dreamt of going to Strings 07, while I am indeed going to Pascos next week, which has some strings flavor but admittedly is also devoted to particles and cosmology. But die-hard stringers might have had to make a difficult choice.
By the way, I will try to report from there, but I will be there only for a couple of days so I can’t really promise anything…
Cheers,
T.
here is a sample exerpt from the talk by Witten, slides 14 – 16:
==quote==
First of all, I am only going to consider the case of negative cosmological constant.
Currently there is some suspicion that quantum gravity with Lambda > 0 doesn’t exist non-perturbatively (in any dimension) with positive cosmological constant. The reason for this is that it does not appear to be possible, with Lambda > 0, to define precise observables. This is natural if it is the case that a world with positive cosmological constant (like the one we may be living in) is always unstable.
If that is so, then a world with Lambda > 0 doesn’t really make sense as an exact theory in its own right but (like an unstable particle) must be studied as part of a larger system.
Whether that is the right interpretation or not, since I don’t know how to define any precise observables, I don’t know what it would mean to try to solve 2+1-dimensional gravity with Lambda > 0, since it isn’t clear what we’d want to compute.
==endquote==
Unfortunately this sounds like a rationalization. Perhaps it is a question of what tools one uses, as you suggested in comment to your previous post:
http://www.math.columbia.edu/~woit/wordpress/?p=570#comment-26394
If one set of tools for studying the world fails to provide observables, then before declaring the world we “may be living in” to be “like an unstable particle” it seems a sensible reaction would be to look around for better technical means. Am I missing some essential part of the message? Maybe others would comment on this passage.
__________________
Marcus,
Witten’s argument is exactly what leads to anthropic principle: if our dS universe is only a metastable vacuum, it is probably an ugly mess from a landscape of possibilities, and there is no beautiful selection principle to guide us. Declare defeat, and work on mathematically nice but physically wrong things, like 3D and AdS.
This is a unavoidable problam if all observables in QG are global, because dS space lacks the right asymptopia where the global observables can live. AFAIU, the only way around this dilemma is if there are local observables in QG after all; then we don’t have to worry about the lack of global observables.
Indeed, we know for sure that there are local observables in classical GR – Rovelli’s GPS coordinates. Now, these are only partial observables, but we could make them complete by reading a local clock. Without leaving my car (if I had one), I could use my fancy GPS equipment to read off my GPS coordinates, and read off the dials of my watch to get proper time, and get a local, complete observable. This observable is local, because I don’t have to visit a holographic screen in an AdS region of the multiverse (i.e. outside our visible dS universe) to make this observation; I can stay in my car.
Marcus,
I think this debate over dS observables is completely irrelevant to the Witten paper. All he is doing is referring to that debate and noting that the arguments in his paper involve the AdS case, and don’t apply to the dS case. He’s not making any claims in this paper of any new ideas or understanding about the dS case.
If I were still in theoretical physics I’d eagerly await next year’s Strings 2008. For it will mark the 175th anniversary of Hamilton’s article on optics and planetary motions where he invokes the spirit of Bacon:
In every physical science, we must ascend from facts to laws… and unity arises from variety: and then from unity must re-deduce variety, and force the discovered law to utter its revelations of the future.
Or if I can quote again with inlines relevant to science nearly two centuries later:
In every physical science [even string theory], we must ascend from facts to laws… and unity arises from variety: and then from unity must re-deduce variety [i.e., must reproduce the standard model], and force the discovered law to utter its revelations of the future [i.e., make novel and testable predictions of future experiments].
Hi Tommaso,
One reason there are a lot of string theory conferences (20-30 a year) is that there are a lot of string theorists. And conferences are concentrated in the summer months when people don’t have to teach. Typically there are a few smaller “satellite” conferences organized before and after the main Strings conference, at locations not too far away. I gather there are some of these going on around now. Will look forward to your reports from PASCOS!
Jim Clarage:
I don’t think that string theory should be blamed exclusively for the stagnation in modern theoretical physics. I think that the fundamental problem was best expressed by Mark Srednicki in his reply to Chris Oakley on Cosmic Variance
And I don’t agree at all with your statement that while “it would be very nice to explain SM parameters, but establishing a mathematical framework free of inconsistencies has to be done first.” Historically, progress in physics has almost never been made this way. It almost always came by somebody futzing around with some not-fully-baked theory, until something interesting popped out.
This “futzing around with some not-fully-baked theory” was very successful until 1960’s. But, apparently, it doesn’t work anymore. I think that the chance that something will “pop out” is close to zero. Maybe it is time to stop relying on random attacks and switch to a methodical and organized siege: Decide which physical principles are truly important (the principle of relativity? laws of quantum mechanics?…) and build a rigorous axiomatic mathematical framework around these principles.
Well that’s the whole problem isn’t it. That’s like saying “…to become a gazillionaire we need to decide which business idea is the right one and build a rigorous business around that…”
hmmm,
That was my point. I have an impression (please correct me if I am wrong) that there is no much interest in designing a sound plan and building the “business” from ground up. Just look at talks at String 07 or any other similar gathering. People are trying to prove theorems without even knowing what their axioms are. These are conjectures based upon previous conjectures. Using your analogy, they try to become gazillionaires by gambling in a nearby casino. Srednicki thinks this is the way to go. I respectfully disagree.
Using your analogy, they try to become gazillionaires by gambling in a nearby casino.
Well, they have to. It’s the only game in town.
So I’m afraid a lot of this is beyond my current level of understanding, but I’m trying to figure out– it’s been commented several times that Witten’s Strings 2007 paper here doesn’t seem to have much to do with string theory. However, it does seem like it has a lot to do with AdS/CFT correspondence. In fact, it seems like he’s *using* the AdS/CFT correspondence; as I understand this he’s analyzing black holes in 3-dimensional AdS spacetime by reasoning about the equivalent 2-dimensional conformal field. That’s AdS/CFT, right? So here is basically what I’m trying to figure out:
1. This paper seems to somehow be using AdS/CFT without talking about strings. I didn’t actually realize you could do that. Is there anything “special” about CFTs that correspond to spaces with strings, or is it just a matter of whether you choose to describe the space as containing strings?
2. Although the paper doesn’t explicitly talk about strings, would it make sense for the ideas in this theory to be followed up, in either a later paper by Witten or somebody else, by linking these ideas to string theory or simply adding strings to the toy model described here? If so, what would that look like? Like, would it make any sense to drop strings into the toy model here and analyze their interactions with a BTZ black hole?
Just a reminder. If you want to have general philosophical discussions about physics, please find some other forum. If you have something substantive to say specifically about the topic of this posting, String 2007, please do so.
Coin,
When people talk about “AdS/CFT” duality, they normally are talking about a relationship between string theory on five-dimensional AdS space (really AdS times a 5-sphere, the strings need 10 dimensions) and a supersymmetric gauge theory on its 4d asymptotic boundary. More generally, one hopes that this string-gauge duality works for different sorts of geometries. One could try and do this with 3d AdS space, and people have worked on that.
But this is just not what Witten is doing. He’s not looking at string theory on the AdS space, he’s looking at pure gravity, and at a Chern-Simons gauge theory, and trying to understand their relation. Pure gravity and string theory on AdS are just two very different theories. What Witten is doing just isn’t a string/gauge duality. I don’t see what you’re going to achieve by adding string theory to this story, and I suspect if Witten thought one could get anything that way, he’d certainly be mentioning it in his paper or talk at the conference. Strings are mentioned zero times in his talk
Let us leave alone for a moment the question of whether the recent paper of Witten has anything to do with String theory or not.
What it clearly implies, is that variable changes as those promoted by Ashtekar and used in LQG can not be correct, and have nothing to do with the quantum version of gravity (which is expected to have black holes to start with). Thus, in the most well understood situation of 3d gravity, where the variable change is clean and covariant (unlike 4d) Witten shows that this avriable change is menaingless in quantum level.
Should we still expect that 4d LQG tricks do makes sense ? certainly not. This is a complicated non covariant version of the 3d case, and one should not expect any conclusion besides that LQG has no meaning after quantization.
Thus, one of the conclusions of Witten’s paper is that LQG does not make any sense.
Ori,
Witten does not claim that using gauge variables and the Chern-Simons action has nothing to do with quantum gravity. He actually uses these to produce the correct central charge of the boundary CFT. Here’s what he actually does say about this:
The variables that do seem to be useless for non-perturbative quantum gravity in 3d are string variables. Witten doesn’t mention strings at all in his talk and gives no indication that he thinks they have anything to do with this problem.
Unfortunately I don’t think this tells us anything one way or another about 4d, but if one could have the kind of success there starting with Ashtekar variables that Witten seems to have starting with Chern-Simons in 3d, that would be quite spectacular.
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Peter,
I want to avoid misunderstanding and confusion. Forget for a moment how Witten derives whatever he derives, this is a matter of taste. The invariant physical conclusion is that the non perturbative complition of 3d gravity has nothing to do with the gauge theory !
Similar conclusion is expected to hold for the 4d case, where there would not be enough dof in the gauge theory to create black holes, as in the 3d case.
Most of the people here are laymen, and I want to make sure that they understand that Witten’s conclusion is that the gauge theory variables are bad. This was known to Seiberg for some time, apparently. This leaves no hope that any interesting insight can be derived from gauge theory formulation of 4d gravity, for obvious reasons. Thus, LQG seems to have been falsified.
Unrelaed comment for experts : The way Witten derives some central charges does uses gauge theory variables. This is, however, incorrect since the gauge theory variables dont make much sense. Witten admits this is the case, and wishes he had a better way to derive the central charges. He could, as well, guess the correct values by some factorization and absence of gravitational anomalies requirements. This is the second way he motivates these charges, which is much more convincing.
One should not confuse the physics conclusion (which is that the Chern-Simons formulatoins have nothing to do with gravity) with the technical details (which is an incorrect derivation of some central charges, which is as good as a guess). Note that the reason CS has nothing to do with gravity is independent of the ability of disability to derive these central charges.
Another comment is that Peter is correct that there are no stringy variables in the story. However, the construction of the paper is a very particular example of something which was well known is string theory : AdS/CFT. This also shows that the results obtained in string theory in this context are much more general, because similar techniques can be applies in non supersymmetric non stringy cases.
“Thus, LQG seems to have been falsified.”
Oh, come. It’s extremely clear that all of Witten’s [brilliant] work on this is strictly dependent on the 3D context. There is not a clue about how to extend *any* of this stuff to the enormously different 4D case. I have no time for LQG, but enthusiasts for it could quite legitimately point out that 3D gravity has been studied intensively for years, and so far it has told us exactly nothing about 4D physics. There’s no reason to suspect that this is about to change.
For all my admiration for EW, and gratitude to him for the way he has changed the subject, I have to summarize the situation as follows: he has shown that gravity with the wrong sign of the CC in the wrong dimension is wrongly described by Chern-Simons. Next talk please.
Ori,
Your claims that Witten has shown that “gauge theory variables are bad” etc. are directly contradicted by what Witten actually wrote, including the part I quoted above. You’re putting lots of highly tendentious claims in his mouth which are not in his paper and were not in his talk (or in the one I heard here in New York). I think Witten is someone who is very careful to say precisely what he means. In this case he is being very clear that the question of how to properly understand 3d non-perturbative quantum gravity is very much still up in the air. Here’s what he writes:
This is very different than claiming that he has shown that “the non perturbative complition of 3d gravity has nothing to do with the gauge theory !” and falsified the whole idea of using gauge theory variables.
The conclusion from your comment is that you have misunderstood the paper completely. You have picked some quotations out of context, and that is it.
Witten clearly write that the CS theory can not be the correct non perturbative complition, he said it explicitly in strings and it is written in the paper. This is the whole point of the paper, if you claim otherwise, you have misunderstood the paper and the talk. Let me quote for you :
“There is, however, an even more serious problem with the idea “gauge theory=gravity in 2+1 dimensions”…. ”
He has couple of such arguments against the 3d description via CS theory, which in the overall sum to what I said.
You may keep thinking whatever you want, I am the last to care, I was only willing to make sure that readers of this blog will be aware of the correct statements.
To gunpowder&noodles :
If you had read the paper, you would have understood that the arguments against the gauge theory description are completely general, and can be extended to 4d as well. Witten also says that theories with positive cosmological constant probably have no non perturbative complition (unless embedded in a more complete model), so it is not clear what is the point in trying to quantize de sitter backgrounds, without a stable ads ground state
Another way to proceed, is that u answer a question (Instead of putting more and more out of context quotes to defend yourself).
There are BTZ black holes in 3d. If CS is the correct way to quantize gravity than there should be extremely many states in the topological CS theory. What are they ? 🙂
Of course, there are no such states (since CS is a topological theory), which is the main point in Witten’s paper (and the strings talk). This means that CS has nothing to do with the full quantum theory of pure gravity.
I would like to recall ori that is possible to written the BTZ black hole as a solution of CS theory. It was done many years ago by Malrof and other people. Look at ArXives.
Thus, the quantum CS theory also contains BH, as is part of they are part of its classical spectrum.
Ori,
I certainly don’t claim that Chern-Simons is the “correct way to quantize gravity”. For one thing, as Witten states, non-perturbatively the situation of 3d gravity remains unclear. The problem is not necessarily even well defined: maybe there is more than one consistent non-perturbative theory that has the right classical limit. Witten has made a very interesting proposal about a distinguished boundary CFT that might deserve to be called the “right” boundary CFT for non-perturbative pure gravity. I don’t think he claims at all to know what if any 3d variables and Lagrangian treated non-perturbatively with conventional path integral techniques would lead to this theory.
This problem corresponds to something that has always bothered me in the simpler compact Chern-Simons-Witten theory. The standard thing people say there is that, even non-perturbatively, the theory is defined by “integrating [tex] e^{ikCS[A]} [/tex] over the space of connections A”. This is kind of nonsense. Try actually doing this integral, e.g. by putting it on a lattice, and you start to see what the problems are. While the CS action is fine for generating a perturbation theory, non-perturbatively it doesn’t completely define the theory. What Witten does in that case is use the CS action to motivate what the boundary theory is (WZW), and then uses that to actually do computations. Even in that case, the CS action isn’t the full story about the non-perturbative theory, but it definitely is true that you want to be using gauge theory variables.
Anyway, there’s a very interesting story going on here, and gauge theory variables are definitely part of it.
Thanks dbm,
I was wondering about Distler’s claim that “We would not, for instance, ever see the BTZ blackhole from the gauge theory”, which didn’t seem to make sense since the gauge theory gives the same classical equations and the BTZ black hole is a classical solution.
Peter,
your links to lubos’ blog get dumped to a unwelcoming page, you should know this by now, or is it an attitude to ignore the fact ? 😉
This time though I will second the auto-embargo of Lubos. I intended to read his rant, but now I changed my mind… He is probably happier that way.
Cheers,
T.
Tommaso,
I’m not ignoring it, just figure it adds to the entertainment value…
What I think Jacques meant (although I haven’t asked him) is the microstates of the BTZ black hole — it’s hard to imagine where they are in a topological theory.
The Einstein-Hilbert action functional can be rewritten as a Chern-Simns action functional in 3d. So if the former has an extremum somewhere (like a black hole solution), the latter must have, too. After all, they are equal. No? What am I missing?
One more thing:
consider pure gravity in any dimension d. Consider a d-dimensional topological space, possibly wtih boundary. Suppose you could make sense of the EH path integral over all metrics on that space, possibly with prescribed boundary data. Shouldn’t the answer depend solely on the topology of the chosen space? Shouldn’t it in fact give you a functor from d-dimensional topological cobordisms to some flavor of vector spaces? Wouldn’t that be exactly as for Chern-Simons theory?
The only real distinction which I can see between the EH formulation and the corresponding gauge theoretic (Palatini) formulation of gravity is that the former vaguely suggests not to include degenerate metrics, while the latter vaguely suggests to include them.
No? What am I missing?
Microstates. You’re thinking about the classical solution which is a macrostate.
Microstates. You’re thinking about the classical solution which is a macrostate.
Hm.
Suppose I did the CS path integral over connections with the constraint that the vielbein be invertible.
What would be the difference between using the EH action functional in the path integral from using the CS action functional? Both are equal. Both are defined on the same domain. What’s the difference?
(Sorry, there is a blockquote missing. The comment preview did correctly show the nested quotation which I did include.)
Isn’t that question aking to, suppose I did the path integral for the EH action in 4D? Do you know how to quantize CS theory with that constraint?
(Witten talks about this in his original paper, BTW.)
Right, of course I don’t.
The only thing I really know here (beyond knowledge of folk lore and plausibility arguments that all get so confusing after a while) is this:
for every modular tensor category, I can construct
A) a 3-dimensional TQFT
B) a 2-dimensional rational CFT
C) such that they are holographically related.
If that tensor category happens to be the representation category of loops in G, then
A) is G-Chern-Simons theory
B) is the WZW model on G
C) is the fact that states of Chern-Simons are conformal blocks (precorrelators) of WZW.
It is strongly assumed that this can be extended beyond rational CFTs. But only for rational CFTs this has been done in detail.
In any case, there should be other modular tensor categories, or variations thereof, and there should be other realizations of A,B,C. I would like to see if what Witten is talking about could be (classes of) other realizations of this. It ought to be, since he is in effect arguing that A should be defined in terms of B.
The videos seem to be working for me (USA, west coast). Somewhere else, someone mentioned some download bandwidth issues.
I need to contact the program chair (name & email) to get permission to do some tests with their videos. Can’t find it on the Strings ’07 website, just the webmaster emails. Help? (I went to the IFT sponsor site, but the Miembros section has a generic list of staff.)
Hi, “dont be misleading!”
You seem to completely misunderstand the point. It is correct that the classical BTZ solution was written long time ago, as any other classical solution (eg gravitons near the boundary). However, the BTZ entropy can not be accounted for by the quantum CS theory in any reasonable way (and here Witten says “although there were some attempts in the past…”).
Thus, the CS theory does not appreciate the thermodynamics of black holes and can not explain their entropy. Hence, CS can not describe in the quantum level anything like gravity (this is a correct invariant physical conclusion which means that the gauge theory variables are meaningless after quantization and can at best serve for some unmotivated guesses).
A similar thing happens in 4d as well, where the gauge theory variables of LQG dont make sense after quantization.
Urs,
>”The only real distinction which I can see between the EH >formulation and the corresponding gauge theoretic (Palatini) >formulation of gravity is that the former vaguely suggests not to >include degenerate metrics, while the latter vaguely suggests to >include them.”
You are correct, and this difference is very very crucial !! if not this difference the theories were equivalent. In all known examples in string theory the non perturbative formualtion has never included degenerate metrics. Also here, this is the fundamental reason for the failure of CS variables in the quantum level. Similarly, in 4d, the gauge theory variables (and LQG) break down for the same reason.
In 4D, LQG has enough degrees of freedom to account for black hole entropy, therefore I do not buy Ori’s A similar thing happens in 4d as well, where the gauge theory variables of LQG dont make sense after quantization..
It may very well be that LQG does not make sense for other reasons.
hi uri,
I think that despite phylosphical considerations, facts are facts.
And the first fact is that you don need to consider the CS formulation of GR to admit configurations wih a degenerated vielbein, is enough to consider the first order formalism in which the vielbein and the spin connection are regarded independent fields, this theory, automatically, admits a solution where all the fields are zero. Furthermore this first order formulation of GR can be done in any dimension.
The second fact is that it has been proved, long ago, that the first order formalism, is off shell equivalent to the usual metric formulation, that is: the set of configurations where the vielbein is not inverible is of measure zero.
I would say that all this bla bla, of witten and the degrees of freedom of GR, and that in string theory the invertibility of the metric is a serious thing, contradict not only his paper on the soulbility of GR, but also his papers on topological quantum field theory, and topological gravity of the 80’s.
I really dont like his arguments and I even think that his last paper deserve further analysis, for instance is no possible that the guy doesnt obtain the right value for the entropy for BH of mass=2 and says, ok, for larger k a better agreement will be obtained.
Come on!
I want to live in the dimension where a story can earn you a Fields medal, and shouting loud enough releases a graviton.
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dear dbm,
The entropy for k=1 is not expected to match since you compare a semi-classical entropy given by the BH formula to an exact calculation of microstates.
In this regime, semi classical analysis makes no sense since the black hole is very small and gravity is strongly curved. Therefore, the fact the results are similar is a good surprise.
The fact that for larger k there is a better and better agreement is a strong indication that what Witten did is correct.
Of course, this is correct regardless of you liking or disliking this.
How do black holes form in 2+1 classical gravity? (i.e., with no gravitational radiation)?
Dear ori,
So, please explain me, why in every realistic situation in which we can actually compute the entropy beginning with the microscopical description it COINCIDE with our thermodynamical result.
Thanks!
Dear ori,
>So, please explain me, why in every realistic >situation in which we can actually compute the >entropy beginning with the microscopical >description it COINCIDE with our >thermodynamical result.
>Thanks!
I assume you refer to systems which are not gravitational. The explanation is trivial : The result coincides with the thermodynamic result in the limit where the system is large. For example, even in the system of free spins which contains N spins the result will not coincide with the thermodynamic result for finite N, there are logarithmic corrections (Stirling’s formula) etc.
Note that to count states correctly one also needs to have a weakly coupled description of the degrees of freedom, and for the word thermodynamics to make sense there needs to be a classical description of the system.
In the case k=1 the black hole is very small, so it is not a macroscopic system and also the system has no good classical description. Thus, you basically extrapolate formulas illegaly and you should not be surprised that you get a somewhat different result.
I don’t have any idea what you meant by your comment, anyone who has ever studied statistical physics know that there is never agreement with thermodynamics beyond the strict limit of infinite number of dof, sorry.
Ori,
You are right, in the theormodynamic limit.
However, I still insist in my point of the offshell equivalence of CS theory, and 2+1 GR, I mean the facts that I previously mentioned.
The first fact is that you don need to consider the CS formulation of GR to admit configurations wih a degenerated vielbein, is enough to consider the first order formalism in which the vielbein and the spin connection are regarded independent fields, this theory, automatically, admits a solution where all the fields are zero. Furthermore this first order formulation of GR can be done in any dimension.
The second fact is that it has been proved, long ago, that the first order formalism, is off shell equivalent to the usual metric formulation, that is: the set of configurations where the vielbein is not inverible is of measure zero.
I am happy you agree (and you should withdraw your “come on” comment).
The mistake in your other statement is the sentence
“the set of configurations where the vielbein is not inverible is of measure zero.”
Nobody knows how to formulate the non perturbative gravity path integral (it is not necessarily the naive one with Haar measures)
and hence, you can not deduce the measure of degenerate metrics correctly, sorry.
In fact the work of Witten CLEARLY shows that the effect is not a measure zero one, and leads to substantial non perturbative differences in the two theories (which naively differ by measure zero).
btw, one should expect this set of non invertible metrics to be very important also in higher dimensions.
But, consider the paper gr-qc/0303113. My argument is as follows, if the first order, palatini, formalism admit naturally a degenerated viebein, and these people has shown that it coincide, off shell, with the second order metric formalism. ..
There is some level of tension between that paper and your last setence!
btw, thanks for the discussion, is fun.