Amazing bid by Thiemann to absorb string theory into LQG

[Urs]

Hi selfAdjoint -

you wrote:

It seems to me that Thiemann is saying "Ignore everything in sections 1 through 5, ignore group averaging and all of that. Here in section 6.1 is what I am really doing." And indeed if we look at 6.1, it does seem to be independent of what has gone before.

It kind of looks this way, yes. The most recent discussion at the Coffee Table shows, though, that Thomas might, after all, have made the same mistake that I did in the beginning, namely assuming that there is a quantization of the Virasoro algebra without an anomaly.

What he does is take the Borel intervals on the circle (which he did remark in your discussion are orthogonal if they differ anywhere - as you pointed out to me earlier!). He smears them in a particular special way with functions fk and asserts that the "handed" smeared functions Yk close to a Poisson *-algebra.

Yes, that's totally uncontoversial. It is, after all, nothing but an exotic reformulation of the fact that the usual worldsheet oscillators form a Poisson algebra.

Then he introduces the Weyl elements W = exp(iYk), and invokes the Baker-Campbell-Hausdorff formula to get a value for their product and concludes from this that the W's for right handed and left handed Y's commute.

Here a certain problem is beginning to show, which, at least for me, is a general one in this paper: It is not clear what, at this point, is assumption, definition and derivation.

The problem is that the Ys themselves are not represented as operators on Thomas Thiemann's Hilbert space. So how can we apply BCH to them, if they are not even operators? Of course we know that the Ys could be easily represented on some Hilbert space and we could compute their commutator there and it is the one that Thomas is using in the exponent of the BCH formula. But that's no real help either, because on Hilbert spaces where the Ys are represented (such as the usual Fock Hilbert space) their exponentiations are not unambiguously defined, unless we specify some rule of normal ordering. This gives, in the usual treatment, rise to the peculiar conformal dimension of such exponentiated operators, that you can see for instance in equation (2.4.17) of Polchinski. Therefore, whichever way I try to look at Thomas' equation (6.7) as something derived from previous input it makes me feel uneasy. I can accept (6.7) as a definition of the algebra of the Ws, though. But, just as with the definition of the Us by fiat, this is, while mathematically consistent, not manifestly related to physics-as-we-know-it, I think.

He then deduces from the general intersection geometry of intervals on the circle that "a general element of A (that is, a Weyl element W) can be written as a finite, complex linear combination of elements of the form [...]

Ok, given the algebra of the Ws, somehow, this follows without doubt.

He now defines the gauge group to by two copies of the diffeomorphism group of the circle plus the Poincare group

This is the point that we have been discussing in some detail with Thomas over at the Coffee Table. This way of defining the quantum gauge group means to simply copy the classical gauge group. That's mathematically possible, but not related to any standard quantization procedures. Jacques Distler has today given a further example for why this procedure is usually unphysical.

is there any anomaly visible to you in this work? Is there any reson why the GNS will not work?

No, the anomaly is indeed not there in this approach. But the reason is that by definition Thiemann is using a rep of the classical symmetry group on his Hilbert space. This is not the usual quantization procedure. There is no standard quantum anomaly because there is also no standard quantization.

The GNS theorem will work fine for the algebra of the Ws. The problem is that it is not clear what this algebra has to do with the standard quantization of the system at hand.


I can see that you are trying hard to escape the conclusion that is beginning to force itself upon us. I very much appreciate it. In a way I am delighted that the LQG-string is doing exactly what Nicolai has intended it to do: To show in terms of a simple example what is really going on in LQG. As long as we are dealing with 3+1d nonperturbative quantum gravity nobodoy knows what to expect and hence criticism of new proposals is very difficult. But now we are dealing with a case where we know much better what to expect and it has been possible to spot a very crucial difference of the LQG quantization approach to the standard procdedure:

LQG does not attempt to canonically quantize all the first-class constraints.

Actually, this is hardly a suprprise because, as Jacques has kindly reminded me, the ADM constraints of gravity simply cannot, even in priciple, be canonically quantized. LQG apparently circumvents this by not representing the constraints themselves on some Hilbert space but instead representing the symmetry group generated classically by them (at least for the spatial diffeos).

But this means breaking with a fundamental principle of quantum mechanics and can, at best, be addressed as an alternative quantization procedure. There are many people who are proposing alternatives to standard quantization, for various reasons. I am open-minded and willing to consider all alternatives to standard physics as potentially interesting. But one should be fully aware of what is standard physics and what is a radically new and speculative proposal.

In fact, I am currently thinking about asking Ashtekar, or someone similar, if it is really technically correct to say that LQG is about canonical quantization.

 

[selfAdjoint]

Urs, just on this one point:
The problem is that the Ys themselves are not represented as operators on Thomas Thiemann's Hilbert space. So how can we apply BCH to them, if they are not even operators?

I looked up the BCH theorem on google. There are various definitions, but some of them do not require the elements to be operators on a linear space, just members of a Lie algebra. Well, Thiemann has the Y's as the members of a Poisson algebra, so I'll be he can quote chapter and verse to defend this transition.

Being bone ignorant, I just am not as sensitive to the awful non-standardness of Thiemann's work, but I am sensitive to things that just don't work. My problem right now is that all that section 6 material I quoted does sound to me like a string! Borel intervals, momentumful smearing, yes, I can see it. And I've read enough in LQG literature to recognize what he does with this. When I thought he couldn't rigorously apply GNS or group averaging I was ready to give up on him, but rereading this later material brings me back to the table.

I still have doubts like this: His string is all by itself. As I remember it, Virasoro comes out of string interaction. You have the circle where the other world tube joins this one, and you "projectively" represent that tube as a punctured disc, and develop a Laurent series, and th coefficients of that generate the Virasoro algebra, up to ordering. So can his representation, his quantization, do interaction?

I do know that if you say Foch space he will not agree; he thinks of this work as freeing physics from Foch space arguments.

 

[selfAdjoint]

Originally posted by eforgy
Hi Jeff,

Thanks. I tried to read through the paper. I can't say that I understand it (yet), but I do see that what he means by Gauss' law is not the same as what I mean by Gauss' law. To me (and most geometers I would think), Gauss' law is just an incarnation of the generalized Stokes theorem. The generalized Stokes theorem is valid in general. I'll have to make more effort to understand their meaning of Gauss' law. Thanks. I'm making progress.

Eric

Eric, I think your version of Gauss law is LOCAL. The problem is to extend it over the whole parameter space, to a GLOBAL law. And the twist obstructs that extension.

 

[eforgy]

Originally posted by selfAdjoint
Eric, I think your version of Gauss law is LOCAL. The problem is to extend it over the whole parameter space, to a GLOBAL law. And the twist obstructs that extension.

Hi selfAdjoint,

The problem is not with the locality of my version. I assure you that generalized Stokes theorem is not a local theorem. It is defined globally. I still didn't put my finger on exactly what the issue is, but based on Jeff's comment, I am thinking that it might be related to the existence of a Hodge star. Gauss' law on an n-manifold usually refers to (n-1)-forms A with

int_M dA = int_@M A.

It we want to think of this (n-1)-form as coming from vector field X, we need to convert this vector field to a 1-form alpha via the metric. Then we convert this 1-form alpha to an (n-1) "pseudo" form A = *alpha. I think the paper probably refers to some vector field appearing in the integrand as Gauss' law. I could accept this. So is it that when you quantize, the configuration space has no Hodge star, which means you can't define Gauss' law, which in turn gives you anomalies?

Eric

 

[marcus]

Recap. Does anyone challenge Thiemann's conclusions

Now that we have had a chance to get used to the idea of the "LQG-string" what conclusions, if any, do you think could be incorrect?
The thread is long, with posts apparently containing criticisms that were later dropped. Perhaps it would not be a good time to sum up the main points---so that the busy reader does not have to sift through these many (often contradictory) posts.

The core of the paper is section 6. (pages 19-40).
I gather that a critical reading of pages 19-40 did not produce
any conclusive finding of error. There was plenty of it that some of us, especially string theorists, did not like or found unfamiliar, and Urs said he might ask Abhay Ashtekar about something. (That sounds like a good idea, hopefully he has done this already.)
But after listening to the critiques one was not left with the certainty that anything was actually wrong with Thiemann's math.
(If not some overlooked detail which he could correct and still sustain his conclusions.)

So now the question is which of the conclusions does anyone wish to challenge?

Thiemann concluded for instance that string theory does not, after all, require 11 dimensions, or 26 dimensions. There is no critical dimension, after all, that it must have in order to work, because he models it in LQG in all dimensions including ordinary 4D spacetime.

He also concluded that string theory does not, after all, require supersymmetry. Nor, when modeled in a LQG context, does it have the
undesirable "ghosts" and "tachyons".

So as to recall what is the topic of this thread, I will give the link to Thiemann's paper again, and quote the abstract:

http://arxiv.org/hep-th/0401172

"The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space"

"We combine

I. background independent Loop Quantum Gravity (LQG) quantization techniques,

II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and

III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space.

While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincare invariance and 8. without picking up UV divergences.


The existence of this stable solution is, on the one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string.

Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem.

On the other hand, if such solutions are found, then this would prove that neither a critical dimension (D=10,11,26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic.

The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces."

 

[selfAdjoint]

Marcus, here is where I am.

I do not believe the string people have made their case because Thiemann's technique bypasses everything they know and of course they can only argue on the basis of what they know. So we see Thiemann and Distler going at each other on the Coffe Table site, talking past each other until it's almost "'Tis so" - "'Tis not".

Distler's final shot is that it's mathematically inconsistent to get a string theory without an anomaly - he means that every mathematical technique he or Urs has tried infallibly produces the anomaly. Thiemann's retort is that all those things are just partial views and products of the way they go about quantizing the string. So there. Thiemann points out that there is no rigorous development of all this, so to talk about mathematical consistency is a bit rich.

That said, I am uneasy about Thiemann's theory. The paper, as we have discovered, is hastily slapped together. What we all thought were logical trains of thought, weren't. So for me there's smoke. I can't find any fire. We'll have to wait for bigger guns than we've seen so far. Probably at that Mexico meeting.

 

[marcus]

Originally posted by selfAdjoint
...We'll have to wait for bigger guns than we've seen so far...

I tend to agree. BTW great quote from H the V.
"Once more unto the breach, dear friends, once more!"
also delighted by that reference to the "awful non-standardness".

 

[Haelfix]

Yes but now its not clear to me, why Thiemanns method can't be used in other contexts (Distler picks the Y-M eqns).

 

[eigenguy]

Originally posted by marcus
Now that we have had a chance to get used to the idea of the "LQG-string" what conclusions, if any, do you think could be incorrect?

I'm not sure who you are asking. What specific conclusions do you think could be incorrect? Since you asked the question, I don't think it is unfair to ask you your opinion. Or maybe you are asking some more knowledgeable member. If so, who would this be?

Originally posted by marcus
I gather that a critical reading of pages 19-40 did not produce
any conclusive finding of error. But after listening to the critiques one was not left with the certainty that anything was actually wrong with Thiemann's math.

Would you mind substantiating this a bit for the other members? I just don't think these kinds of broad superficial comments are fair given the difficulty of these issues. I guess what I'm asking is if you would mind explaining your feelings the same way you do with other topics which you know well. Again, I think these are fair questions given your post and the nature of the topic.

Why did you stay out of the technical discussion when it moved into full swing? If you don't feel qualified to comment, I'm not sure why you would feel comfortable posting this, or at least without some clear qualification.

Originally posted by marcus
We'll have to wait for bigger guns than we've seen so far.

Jacques distler, the guy who was arguing with thomas, is one of the worlds most brilliant theorists, even more so than ashtekar et al. So it is quite safe to take distler's point of view seriously.

 

[marcus]

Originally posted by selfAdjoint
Marcus, here is where I am.

I do not believe the string people have made their case because Thiemann's technique bypasses everything they know and of course they can only argue on the basis of what they know. So we see Thiemann and Distler going at each other on the Coffe Table site, talking past each other until it's almost "'Tis so" - "'Tis not".

Distler's final shot is that it's mathematically inconsistent to get a string theory without an anomaly - he means that every mathematical technique he or Urs has tried infallibly produces the anomaly. Thiemann's retort is that all those things are just partial views and products of the way they go about quantizing the string. So there. Thiemann points out that there is no rigorous development of all this, so to talk about mathematical consistency is a bit rich.

That said, I am uneasy about Thiemann's theory. The paper, as we have discovered, is hastily slapped together. What we all thought were logical trains of thought, weren't. So for me there's smoke. I can't find any fire. We'll have to wait for bigger guns than we've seen so far. Probably at that Mexico meeting.

this seems like a fair-minded summation and as I said before I tend to agree with your "waiting for bigger guns" comment.
the context of a conference is a good arena for probing the soundness and implications of new work and some of that probably did go on
at the Mexico meeting---I only have a secondhand report from nonunitary though.

At the May conference in Marseille Thiemann will give the main talk
at the "dynamics and low-energy limit" session. I have posted the program on the surrogate sticky.
So he will be discussing latest developments with LQG Hamiltonian.
I should imagine he will be asked to discuss this paper as well.

But what I personally think would constitute "bigger guns" would be
more in MPI-Potsdam. Hermann Nicolai's institute trains both string and loop theorists, and appears to me to have expert people in both lines of research.

 

[marcus]

In case anyone's interested here's the program for the May
conference where Thiemann will be doing the Hamiltonian and low-energy limit talk:

http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/

http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/programmeprovisoire.html


"Tentative list of morning talks.

Loop Quantum Gravity:
Abhay Ashtekar (quantum geometry)
Thomas Thiemann (dynamics and low energy)
Lee Smolin (overall results)
Ted Jacobson (devil's advocate)

Applications: ...
...
... etc."

 

[selfAdjoint]

Sorry, I just misremembered Mexico for Marseiles. My idea of a good critique of Thiemann's paper would be someone who is a real expert on string quantization issues, and who will couple to Thiemann's argument on its own terms. This is exactly what Distler did not do. If Thiemann's paper is mathematically inconsistent, as Distler claims, then where is the inconsistency? Urs, who is fair-minded found physical reasons he couldn't accept the work, but didn't find any inconsistency.

 

[eigenguy]

Marcus, I now have no doubt that you are an unbelievable phony and the fact that you have that physics of the year award thing even if it is just for fun disgraces this site for people who unike you are in rational and knowledgeable. SelfAdjoint isn't a phoney, but I think he's out of his depth here as well. Someoen should start a new thread to let people know what actually happened with thomas and jacques distler.

 

[marcus]

Originally posted by selfAdjoint
Sorry, I just misremembered Mexico for Marseiles. My idea of a good critique of Thiemann's paper would be someone who is a real expert on string quantization issues, and who will couple to Thiemann's argument on its own terms. This is exactly what Distler did not do. If Thiemann's paper is mathematically inconsistent, as Distler claims, then where is the inconsistency? Urs, who is fair-minded found physical reasons he couldn't accept the work, but didn't find any inconsistency.

Yes! If I can hazard an opinion, the importance of Thiemann's paper is as a straw in the wind. If his method extends, or if other methods can extend his results, then it seems to have major consequences. This is how I read his introduction that I quoted 5 or 6 posts back:

"...The existence of this stable solution is, on the one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string.

...

...if such solutions are found, then this would prove that neither a critical dimension (D=10,11,26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic.

..."

The full quote is in
https://www.physicsforums.com/showthread.php?s=&postid=1431999#post143199
5 or 6 of my posts back, on the preceding page.

 

[eigenguy]

You know marcus, the site guidelines require one responds to any fair question about the claims they make.

In fact, I think anyone would rather prove someone wrong than sticking their head in the ground and hope no one notices. It then stands to reason that if you won't respond, people here will simply believe you can't back up your claims.

 

[marcus]

Originally posted by eigenguy
Marcus, I now have no doubt that you are an unbelievable phony and the fact that you have that physics of the year award thing even if it is just for fun disgraces this site for people who unike you are in rational and knowledgeable. SelfAdjoint isn't a phoney, but I think he's out of his depth here as well...

no comment

 

[eigenguy]

Originally posted by marcus
no comment

Hey, your the one who refuses to answer fair questions about comments you made about physics. This is a physics forum you know. So what's behind all your bluster. What was about the "critiques" that left you with the impression that nothing was resolved? You said it, I didn't. I'm just asking what you are talking about because my reading of it is that thiemann was shown quite clearly that his paper made neither physical nor mathematical sense. Just look at the thread-ending final exchange between him and distler.

 

[selfAdjoint]

You may believe your last three posts are fair comments deserving responses, but they look to me a lot like intemperate ad hominem slurs. Any comment on that?

 

[eigenguy]

Originally posted by selfAdjoint
You may believe your last three posts are fair comments deserving responses, but they look to me a lot like intemperate ad hominem slurs. Any comment on that?

You could say exactly the same thing about the exchanges at the string coffee table, so this is complete baloney. Even if I was rude, my questions are valid and are owed an answer. In a similar position, I would have simply backed up what I said by answering the questions directly and ignored the rudeness, that is, if I had the answers, of course.

I think you know quite well selfadjoint that there is no way anyone could frame the basic question I asked marcus so that he wouldn't find some way to weasle out of it. He did the same thing the one and only other time I was here. At that time I asked why LQG was not taken seriously by physicists. You should review those first exchanges between marcus and me and tell me who was rude. Just search under my name.

Anyway, the fact that marcus would put you in a position were you felt you had to fight his batttles for him should make you wonder about his character, but not his physics because I think you know he pretends to know much more than he actually does, a fact which while monitoring the thread I saw demonstrated quite clearly by the exchanges between marcus and both lubos and urs, after which marcus left the thread and came back only after he thought the "coast was clear". The only reason marcus always turns to you is that he knows he can trust you not to challenge him in a way that would show him up. I think the choice you've made to help marcus keep the wool over everyone's eyes is questionable to say the least. Specifically, after urs made a tremendous effort to explain why the LQG-string is senseless, and it is senseless, marcus makes a completely false pronouncement on what actually happened, summarily dismissing by implication what one of the worlds leading physicists said. Talk about arrogance!

I'm not surprised by the fact that people like jeff, urs and lubos motl don't participate very much around here. You guys are so ignorant that you don't even understand that you don't understand. For example, I notice that when it comes to complicated physics, you aren't able to actually put your finger on the relevant issue in a paper. Instead you just go through the methodology figuring that this way, at least what your saying is probably not wrong.

I think just as in the physics community, we need more people here to be tough and keep the membership honest about the physics and I don't think that anyone can rely on you to do that.

 

[selfAdjoint]

Well, my own lack of technical expertise is no secret, but I am not stupid, either, and I believe I have an insight here that all you experts, Lubos, Distler, Jeff, yourself, and even Urs haven't faced up to. There is something Thiemann has found, distinct from dumb mistakes, that leads him to make his assertions. Perhhaps we should take his hint and blame the Polyakov action. If you didn't make the worldsheet the center of your original analysis, and didn't derive the conformal and Weyl invariances on it, what would your physics be like? If you weren't able to deflect criticism by reference to school excercises, what then?

Generally, your nasty tone, Lubos' fury, and Distler's sarcasm only suggest to me that you are all suffering from mauvais foix - that you fiercely assert this must be, because if it weren't you would all be at sea without a compass.

 

[eigenguy]

Originally posted by selfAdjoint
There is something Thiemann has found, distinct from dumb mistakes, that leads him to make his assertions. Perhhaps we should take his hint and blame the Polyakov action. If you didn't make the worldsheet the center of your original analysis, and didn't derive the conformal and Weyl invariances on it, what would your physics be like?

Okay, so let's talk about this. Firstly, you should take what I say about the physics with a large grain of salt because I know little about LQG. But I have been studying polchinski volume I which covers the string related issues thiemann raises.

In a nutshell, here's what I think are the crucial parts of the exchange at the "string coffee table":

Thiemann claimed to have shown that the existence in the quantized closed bosonic string of a critical dimension, a virasoro anomaly, and a tachyon state which requires supersymmetry to remove, was simply a result of the representation used by string theory guys, the one that follows from the polyakov action. In particular, thiemann claimed his rep to be anomaly free.

Distler pointed out that urs's calculation of the virasoro anomaly depended only on the canonical commutation relations, the point being that these are essentially the same for any quantization of the closed bosonic string.

So thiemann tried to salvage his paper's main results by arguing that whether the virasoro algebra has an anomaly is irrelevant since he was quantizing the group elements directly and not their lie algebra generators. But then distler made the obvious point that one only has to consider group elements near the identity to see that this argument also fails.

What do you think?

 

[selfAdjoint]

eigenguy, these are good questions. Give me till tomorrow and I will try to answer them. I have an idea about the neighborhood of the identity objection but I want to think it over and check it out before I expose it.

Otherwise I saw the thread with Distler on the Coffee Table site as falling into two segments. In the first, Distler convinces Urs that Thiemann is not doing what Urs believes, but is really outside the bounds of proper string theory. In the second, Distler and Thiemann trade high level counterarguments.

I'll do my best to respond to your questions, and I suggest we adopt Jeff's sig line and Keep It About the Physica.

 

[eigenguy]

Originally posted by selfAdjoint
eigenguy, these are good questions. Give me till tomorrow and I will try to answer them. I have an idea about the neighborhood of the identity objection but I want to think it over and check it out before I expose it.

Otherwise I saw the thread with Distler on the Coffee Table site as falling into two segments. In the first, Distler convinces Urs that Thiemann is not doing what Urs believes, but is really outside the bounds of proper string theory. In the second, Distler and Thiemann trade high level counterarguments.

I'll do my best to respond to your questions, and I suggest we adopt Jeff's sig line and Keep It About the Physica.

At the very end of the thread distler has given this link to his summation of the discussion which as it turns out is for the most part pretty much the same as mine.

 

[Urs]

Conclusion

Let me rephrase again what the conclusion of the discussion is - and Thomas Thiemann did agree about this point, just not about it's relevance:

The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.

In his last message Thomas confirmed that he hence thinks that the issue is one that has to be resolved by experiment. Certainly existing experiments strongly confirm standard quantum theory, which is Jacques Disler's point, quoting YM theory.

So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra) it is speculative physics.

In particular, the method used by Thomas is not "canonical quantization" as usually understood. It is not Dirac quantization of first-class constraints.

Often LQG is advertised as a very 'conservative' approach to quantum gravity. I no longer see how this can be claimed. Modifying the basis of quantum theory is hardly a conservative approach. There is so far no hint that the LQG way to impose the constraints is realized in nature.

 

[arivero]

Originally posted by Urs
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.

Fine :-D . Just I take the work, against my own desire, of pointing out a hint of the relationship between area quantization and standard quantum theory, and it seems that the whole congress has concluded on the contrary ! This is a real sincronicity.

 

[marcus]

Originally posted by Urs


In particular, the method used by Thomas is not "canonical quantization" as usually understood. It is not Dirac quantization of first-class constraints.

Often LQG is advertised as a very 'conservative' approach to quantum gravity. I no longer see how this can be claimed. Modifying the basis of quantum theory is hardly a conservative approach. There is so far no hint that the LQG way to impose the constraints is realized in nature.

Urs, you are making a blanket statement about LQG.
Please have a look at Rovelli's book "Quantum Gravity"
(which Thiemann cites in his references) and tell us if you see
anything which you would like to declare non-standard.
It would be extremely interesting if you would point out a section of
the book where the quantum theory is not kosher according to you.

 

[marcus]

Originally posted by Urs
...

The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.

...

So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra) it is speculative physics.

Urs, I appreciate the fact that you have just taken part in a lively discussion at what I take to be Jacques Distler's message board. I'm glad to hear from you what you believe can be concluded from that discussion!

Please tell me at what point in the "LQG-String" paper does TT use a procedure that is not related to standard quantum theory. I assume this has nothing specifically to do with String (which is not so-far "standard quantum theory") but is a LQG procedure which you find non-standard. I would very much like to know what this is and have the paper printed out here. So if you tell me a page number and quote some lines, I will be closer to understanding what this non-standardness is, or at least be able to ask for clarification.

I also have Thiemann's "Lectures on Loop Quantum Gravity" which I gather Springer Verlag published last year---a kind of textbook on LQG. It is available, as you know, online (gr-qc/0210094) and is less than 100 pages long. It would be great if you could find the non-standard procedure in "Lectures" and explain it in that context. That way the issues would be kept separate from string theory, making it easier to judge what is speculative and what is not speculative.

Thanks in advance

 

[eigenguy]

Originally posted by Urs
So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra)

Then what's the significance of distler's remark that thiemann's approach of quantizing $$Diff(S^1)$$ directly can't avoid the virasoro anomaly issue?

Originally posted by Urs
Often LQG is advertised as a very 'conservative' approach to quantum gravity. Modifying the basis of quantum theory is hardly a conservative approach.

So the LQG-string framework is in fact that of standard LQG?

 

[eigenguy]

Originally posted by marcus
String (which is not so-far "standard quantum theory")

Yes it is standard quantum theory, but applied to strings. Keep in mind that in the low energy limit ST reduces to ordinary QFT.

 

[selfAdjoint]

Not

Urs, let's discuss this once more
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.

Every step that Thiemann takes is based on some previous result, mostly from classical mathematical physics. His use of the GNS construction is exactly as in Haag's book Local Quantum Physics, his quantization is per the Giulini-Marolf paper. Maybe this isn't the way string, or particle - physicists go about things, but it's a valid way within mathematical physics.

I do have a question, in that the symmetry group in all those prior theorems^* is assumed to be locally compact, which pretty much much means finitely generated, and Diff(S¹) isn't. I think that in earlier LQG papers we saw GNS extended to infinitely generated groups (Marcus, help me out here!), but if not, then his work is invalid. But then that would make his work mathematically wrong, not physically meaningless.

^* I'm thinking here especially of Corollary 4.1 where a "G-invariant state" is introduced out of the blue. The parallel discussion in Haag has attention paid to the nature of G, which is assumed to be locally compact.

 

[marcus]

Originally posted by selfAdjoint
... I think that in earlier LQG papers we saw GNS extended to infinitely generated groups (Marcus, help me out here!),...

selfAdjoint,
for starters I will put out some arxiv numbers of papers which
we looked at or discussed at PF some months back. then I will
have a look-see if any of these fill the bill

Okolow and Lewandowski
"Diffeomorphism covariant representations of the holonomy-flux *-algebra"
http://arxiv.org/gr-qc/0302059

this was Jerzy Lewandowski's reaction to the work of Hanno Sahlmann, then at AEI-Potsdam with Thiemann. Then there were some papers of Sahlmann and of Thiemann/Sahlmann. Here are a couple, which would have references to others.

Hanno Sahlmann
"Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity"
http://arxiv.org/gr-qc/0207111

Sahlmann and Thiemann
"Irreducibility of the Ashtekar-Isham-Lewandowski Representation"
http://arxiv.org/gr-qc/0303074

Sometime while we were discussing these and related papers I recall
getting out my old copy of Naimark's book "Normed Rings" and
studying up on the Gelfand-Naimark construction. Or I guess one calls it the "GNS" for Gelfand-Naimark-Segal.

I think the role of GNS in Loop Gravity goes back to much earlier work---by Ashtekar, Lewandowski and others. I am responding too quickly perhaps, not sure if this is to the point.

But I will have a look at some of these papers and see if I can reply better.

 

[selfAdjoint]

GNS and the Symmetry Group

-The only restriction on G is that it be locally compact, and I now think that DIFF(S¹) is, because the circle is compact. Take a neighborhood of the identity - diffeomorphisms that don't move any point as much as some small $$\epsilon$$, then inside that we can lift pointwise convergence to diffeomorphism convergence.

-GNS seems to have been introduced into LQG in a 1992 paper by Ashtekar and Isham, hep-th/9202053.

Notice also that Thiemann never claims to have a general representation theory of G; he says that will have to be a topic of further research, and offers instead just about the simplest example you could think of, one that takes the value 1 on all his momentumized networks and is zero only on the empty network (6.20).

I am now digging into the details of his implementation of the Pohlmayer charges, and the development of the algebraic representations, sections 6.5 and 6.6. I should have done this in the first place.

 

[marcus]

You beat me by 2 years
you found a 1992 paper and I just came back with a 1994 paper
by Ashtekar, Lewandowski, Don Marolf, Jose Mourao, and Thomas Thiemann
It is called
"Coherent State Transforms for Spaces of Connections"
http://arxiv.org/gr-qc/9412014
page 9, for instance, uses the Gelfand-Naimark construction

 

[marcus]

This will give some of the flavor of the 1994 paper that Thiemann co-authored with Ashtekar, Lewandowski, Marolf and Mourao.
I cannot easily reproduce the symbols from their gothic and script fonts. I will leave off the overbar on A/G and write mu for the greek mu and so on. this is just a exerpt to give a feel for how the Gelfand-Naimark theory was used at around that time.

-----quote from page 9 of gr-qc/9412014------
"The classical configuration space is then the space A/G of orbits in A generated by the action of the group G of smooth vertical automorphisms of P. In quantum mechanics, the domain space of quantum states coincides with the classical configuration space. In quantum field theories, on the other hand, the domain spaces are typically larger; indeed the classical configuration spaces generally form a set of zero measure. In gauge theories, therefore, one is led to the problem of finding suitable extensions of A/G. The problem is somewhat involved because A/G is a rather complicated, non-linear space.

One avenue [6] towards the resolution of this problem is offered by the Gel'fand-Naimark theory of commutative C*-algebras. Since traces of holonomies of connections around closed loops are gauge invariant, one can use them to construct a certain Abelian C*-algebra with identity, called the holonomy algebra. Elements of this algebra separate points of A/G, whence, A/G is densely embedded in the spectrum of the algebra. The spectrum is therefore denoted by [A/G bar, can't make the symbol]. This extension of A/G can be taken to be the domain space of quantum states.

Indeed, in every cyclic representation of the holonomy algebra, states can be identified as elements of L2(A/G; mu) for some regular Borel measure mu on A/G. One can characterize the space A/G purely algebraically [6, 7] as the space of all homomorphisms from a certain group (formed out of piecewise analytic, based loops in Sigma) to the structure group G. Another {and, for the present paper more convenient) characterization can be given using certain projective limit techniques [10, 14]: A/G with the Gel'fand topology is homeomorphic to the projective limit, with Tychonov topology, of an appropriate projective family of finite dimensional compact spaces.

This result simplifies the analysis of the structure of A/G considerably. Furthermore, it provides an extension of A/G also in the case when the structure group G is non-compact.

Projective techniques were first used in [10, 14] for measure theoretic purposes and then extended in [13] to introduce
"differential geometry" on A/G

The first example of a non-trivial measure on A/G was constructed in
[7] using the Haar measure on the structure group G. This is a natural
measure in that it does not require any additional input; it is also faithful and invariant under the induced action of the diffeomorphism group of Sigma.

Baez [8] then proved that every measure on A/G is given by a suitably consistent family of measures on the projective family..."

 

[eigenguy]

Originally posted by selfAdjoint
Every step that Thiemann takes is based on some previous result

Originally posted by urs The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.

but it's a valid way within mathematical physics.

I like this post because it touches on my own feeling that much of the friction between the LQG and ST camps is due to their belief that the other's opinion about what constitutes genuine physical research is wrong.

My own opinion is that neither the mathematical consistency of, nor the presence within a theory of analogs or generalizations of ideas whose physical validity has been proven or otherwise generally accepted, is sufficient cause to view it as physicsally viable or valid: theorists should be guided by plausibility rather than mere logical possibility.

What would cause you to abandon your interest in LQG?