Ashtekar's Shadow states paper

[marcus]

Ashtekar's "Shadow states" paper

There is this paper by Ashtekar, Fairhurst, and Willis called
"Quantum gravity, shadow states, and quantum mechanics"
http://arxiv.org/gr-qc/0207106

Citebase lists it as having been cited 5 times, one of which was in a paper by one of the authors. The other 4 citations were:

1.Alfaro, J ; Morales-Tecotl, H A ; Urrutia, L F
"Quantum gravity and spin 1/2 particles effective dynamics"
Abstract:"Quantum gravity phenomenology opens up the possibility of probing Planck scale physics..."

2.Lusanna, Luca; Pauri, Massimo
"General Covariance and the Objectivity of Space-Time Point-Events: The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity"
Abstract: "...The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the ..."

3.Pullin, Jorge
"Canonical quantization of general relativity: the last 18 years in a nutshell"
Abstract: "This is a summary of the lectures presented at the Xth Brazilian school on cosmology and gravitation. The style of the text is that of a lightly written descriptive summary of ideas with almost no formulas, with pointers to the literature. We hope this style can encourage new people to take a look ..."

4.Corichi, Alejandro; Cortez, Jeronimo; Quevedo, Hernando
"Note on Canonical Quantization and Unitary Equivalence in Field Theory"

The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of algebraic quantization. In particular, when the phase space of the system is linear and finite dimensional..."
--------------

I could not see from this how the paper fitted into the current mainstream development of LQG. There seemed to be no research line issuing from it, as if it represented a gambit that proved to be a blind alley. Since I wanted to find if and how the paper relates to the main body of LQG, I looked for references in Rovelli's "Quantum Gravity".

Rovelli's citations are not in Citebase so there was some expectation of finding the paper. In fact Rovelli's book has 385 citations including many by Ashtekar (more than I felt like counting) but no reference is made to this "Shadow state" paper.

What is wrong? Is this particular paper unrelated to LQG proper? So I looked at another large bibliography, and found plenty of Ashtekar but no mention of this paper, so I lost hope. Maybe someone else will turn up a citation indicating how this "Shadow state" paper connects to the main Loop Gravity picture.

 

[marcus]

Oh, the fourth citer, the Corichi, Cortez, Quevedo paper seems to have pointed out something odd about the Ashtekar "Shadow state" paper. It may explain why it has not been cited much. I will get the full abstract.

Corichi, Alejandro; Cortez, Jeronimo; Quevedo, Hernando
"Note on Canonical Quantization and Unitary Equivalence in Field Theory"

Abstract: "The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of algebraic quantization. In particular, when the phase space of the system is linear and finite dimensional, the `vertical polarization' provides an unambiguous quantization. For infinite dimensional field theory systems, where the Stone-von Neumann theorem fails to be valid, even the simplest representation, the Schroedinger functional picture has some non-trivial subtleties. In this letter we consider the quantization of a real free scalar field --where the Fock quantization is well understood-- on an arbitrary background and show that the representation coming from the most natural application of the algebraic quantization approach is not, in general, unitary equivalent to the corresponding Schroedinger-Fock quantization. We comment on the possible implications of this result for field quantization."

In case anyone wishes to look at Corichi/Cortez/Quevedo here is the link
http://arXiv.org/abs/gr-qc/0212023

As you can see not much has been heard about the "Shadow states" paper since 2002.

 

[marcus]

Recently I was surprised to learn that the "Shadow states" paper, although seemingly somewhat marginal in its connection to the rest of contemporary LQG research, has been intensely studied here at Physics Forums!

Indeed there is a tone of excitment and discovery as in this post:

"...I have mentioned this paper for quite a while now and many times, here, on s.p.r. and at the Coffee Table, have written summaries and critical discussions of this paper in these three groups, have compared its techniques to those used by Thomas Thiemann,...

Especially I suggest you have a look at the last dozen or so entries of the 'Amazing bid' thread where the content of this paper was part of an 'exercise' which was jointly analyzed by several participants..."

"This paper" in the words of the post refers to this very same "Shadow states" paper of Ashtekar/Fairhurst/Willis! My bolding to show key points. The evident excitement and intense study is all to the good, even if it may be directed at an apparently somewhat off-center paper already a year and a half old(the first preprint posting of was July 2002).

 

[nonunitary]

Marcus,

The "shadow states" paper as you call it is certainly very relevant for LQG. The thing is that there are for instance two representation in a field theory, the Fock representation and the "polymer" representation, as Ashtekar has called it lately. LQG is an example of the "polymer" representation (the paper by Thiemann in Strings in another). The main difference is that in this last representation there is an operator that is not deefined (for instance the position operator). One is led to consider the "exponenciated version" of the operator that IS well defined (i.e. the holonomies). The paper by Ashtekar and colaborators do this quantization for quantum mechanics and ask the question: can this non-equivalent quantization approximate the ordinary Schrodinger QN in some regime? The answer is yes!
It is relevant for LQG because this is precisely the strategy to approximate low-energy physics (where a fock type representation is expected to be valid, for gravitons). Some of these ideas were reported in a paper by Ashtekar and Lewandowski in 2001 (CQG, letters).

 

[marcus]

Originally posted by nonunitary


The "shadow states" paper as you call it is certainly very relevant for LQG...

If what you say is right then the only follow-up paper that correctly gauges the importance of this one by Ashtekar et al is the one by
Corichi/Cortez/Quevedo and I will try to get some understanding of it. As I recall you have been right on target (almost prophetically) a couple of times in the past. So I expect you are this time too!
but it goes against appearances

Shadow states paper strikes me as (if not a blind alley) at least a preliminary and tentative venture into new ground. It doesn't look like typical LQG to me (whatever that is).

 

[nonunitary]

I am sorry, but I don't see the relation between the "small" paper of Ashtekar and the chapter by Rovelli. They are not even talking about the same thing! The paper by ashtekar is an attempt to show that such "weird" representations can indeed approximate the physics of the standard ones. If this were not true even for QM, the the whole program could be, justifiedly, seen as suspect.

On the other hand, I think that not all the LQG people have the same opinion about the relevance of the paper. The natural followup is not the paper by Corichi, Cortez, Quevedo you mention (they are concerned mainly with Fock type representations), but the work in LQC.
In a paper by Ashtekar, Bojowald and Lewandowski about the math of LQC they recognize that the main fuature of the quantization is that it is precisely of polymer type. Hussain and Winkler also recognize this, even when thay don't use the same words.

 

[marcus]

In accordance with your advice want to avoid underestimating Ashtekar's paper and have deleted my earlier post that down-played its potential importance.

 

[marcus]

Originally posted by nonunitary
...
On the other hand, I think that not all the LQG people have the same opinion about the relevance of the paper. The natural followup is not the paper by Corichi, Cortez, Quevedo you mention (they are concerned mainly with Fock type representations), but the work in LQC.
In a paper by Ashtekar, Bojowald and Lewandowski about the math of LQC they recognize that the main fuature of the quantization is that it is precisely of polymer type. Hussain and Winkler also recognize this, even when thay don't use the same words.

that Bohr compactification of the real line again! All three papers have it. Hussain/Winkler, ABL, and AFW. I have not yet managed to feel comfortable with R_Bohr.

 

[marcus]

Urs Schreiber's criticism of Ashtekar's "Shadow states" paper

Urs, so far the criticisms I have heard of Ashtekar et al "Quantum gravity, shadow states, and quantum mechanics" are more like unsubstantiated rumors and hints, from my standpoint or that of some visitor to PF.

You say you have made extensive criticism but you have refused to give me links. I have gotten no explanation with explicit references to lines in this comparatively little-known paper.

The thread you refer to is over a 300 posts long. It probably has references to Ashtekar and the title of his paper, but this is like searching for a needle in a haystack! There is no keyword search engine here at PF to find the relevant posts in that long thread, which you say are there.

The closest thing to an explicit pointer to the criticisms of Ashtekar et al you claim you have made is your remark that I should look at "the last dozen or so posts" in the TT thread. Well I did! I examined the last TWO dozen posts and there was no mention of Ashtekar or the paper by title! How is a random visitor to PF to find the discussion that these allegations of yours refer to? I cannot. Could you have discussed the paper somewhere without giving the title and authors?

Here is a pointer to a post currently 23 from the end of the Thiemann thread. There is nothing that says Ashtekar between there and the end.

https://www.physicsforums.com/showthread.php?s=&postid=149676#post149676

You see how difficult it would be to find things, especially for the infrequent guest, if ordinary scholarly openness and transparency is not observed.

I have absolute confidence that you have made, plenty of criticism of Ashtekar's paper which would be useful if it were openly posted so knowledgeable people including visitors could find it.

I was also impressed by your sincerely agrieved tone when you said, about the Ashtekar paper,

"...I have mentioned this paper for quite a while now and many times, here, on s.p.r. and at the Coffee Table, have written summaries and critical discussions of this paper in these three groups,..."

But as you know I don't follow "Coffee Table", and I looked at SPR and found no thread about Ashtekar's paper. If you have some criticisms of it, isn't the obvious thing to do to start a thread? And you provided no clear pointer to any discussion here at PF.

If you would like to construct pointers to specific posts at PF, here is the format. This is to a post that is a dozen or so from the end of the TT thread.

https://www.physicsforums.com/showthread.php?s=&postid=150286#post150286

Perhaps you can use this format to point me to a PF post of yours that gives the authors and title of this paper and outlines your main objections.

 

[jeff]

Originally posted by marcus
Urs, so far the criticisms I have heard of Ashtekar et al "Quantum gravity, shadow states, and quantum mechanics" are more like unsubstantiated rumors...

All those who've wasted time reading this pointless thread need to know is that ashtekar's shadow states paper makes no difference with respect to the basic issue - confirmed by the LQG camp, including ashtekar himself - which is that LQG quantization is unfortunately fundamentally different than the standard method. This is a problem because the equivalence principle requires gravity couple to everything including itself in exactly the same way so the idea that a self-consistent quantum theory of all interactions in which gravity is quantized one way and all else another - this other way having been revealed by experiment over and over again to be correct - is implausible in the extreme.

 

[marcus]

...the basic issue - confirmed by the LQG camp, including ashtekar himself - which is that LQG quantization is unfortunately fundamentally different than the standard ...

It would be nice to have a link to a paper by Ashtekar that says this.
Or a link to a post by him on some message board.
I would be interested in reading what he says in his own words,
in context, not just someone's interpretation of it.

 

[ranyart]

Originally posted by marcus
It would be nice to have a link to a paper by Ashtekar that says this.
Or a link to a post by him on some message board.
I would be interested in reading what he says in his own words,
in context, not just someone's interpretation of it.

Marcus, the Ashtekar paper that started the ball rolling with 'area operators' goes back to this paper:A. Ashtekar and J. Lewandowski, Quantum theory of gravity I:Area operators, Class. Quantum Grav. 14, A55 (1997), [gr-qc/9602046].

I believe?

The cite base for this paper reads somewhat more interesting?:http://citebase.eprints.org/cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9602046

The monthly citations shows the evidence of a '6.2 Earthquake',if one was to compare it to a Richter scale!


Bettered only by this paper:http://citebase.eprints.org/cgi-bin/citations?id=oai%3AarXiv%2Eorg%3Agr%2Dqc%2F9504018

Which registers about '6.5' on the equivilent 'interesting scale'!

 

[jeff]

Originally posted by marcus
It would be nice to have a link to a paper by Ashtekar that says this.
Or a link to a post by him on some message board.
I would be interested in reading what he says in his own words,
in context, not just someone's interpretation of it.

Just email him and ask. His homepage is here.

 

[marcus]

...the basic issue - confirmed by the LQG camp, including ashtekar himself - which is that LQG quantization is unfortunately fundamentally different than the standard ...

It was claimed here that Ashtekar has said "LQG quantization is unfortunately fundamentally different than the standard" or words to that effect.
No source was offered. I'd like an online source for what he actually said. I requested substantiation and was told to write him---I guess to ask him if the rumor was true!
If no more justification is offered, like a communication from Ashtekar, it seems more reasonable to assume its just an
unsubstantiated fabrication or distortion.

As far as I know there is no universally accepted proceedure quantization in physics. See page 2 of Quevedo Tafoya
"Towards the deformation quantization of linearized gravity"
http://arxiv.org/gr-qc/0401088
Quantization is not yet thoroughly understood
(See for example Stefan Waldmann
http://arxiv.org/hep-th/0303080 )
and a variety of techniques are used in a more or less ad hoc
fashion.
It is misleading to suggest that the picture is monolithic with a large split between LQG and other theories.
I judge there is at least as much diversity in the rest of physics as there is in LQG (where there is no one single "LQG-like" style that is the same in every detail)

 

[jeff]

Originally posted by marcus
It was claimed here that Ashtekar has said "LQG quantization is unfortunately fundamentally different than the standard" or words to that effect.

No source was offered.

This behaviour merely confirms once again that you have no interest in what ashtekar or anyone else who threatens to burst your bubble thinks. Do you really think anyone here would hold it against you if you admitted that maybe LQG isn't what you thought it was? Well, LQG isn't what anyone thought it was! Physicists have to give up on pet ideas all the time. Feynman said that the job of a physicist is to prove themselves wrong as quickly as possible. That's the nature of doing hard science. Grow up.

 

[selfAdjoint]

The shadow states paper does state that the "polymer" representation is different from the usual one, and in particular lacks a momentum operator. They motivate this as to be expected in a theory with discrete spacetime such as LQG. The point of the paper, as nonunitary has already posted, is to see if in the low energy limit, the polymer representation can reproduce the results of the usual Schroedinger representation. And as nonunitary said, the paper concludes that they can. In other words, this difference, which has been represented here as fatal to LQG is both natural and adequate within the actual situation of LQG.

The case of Thiemann's quantization of the string is different, so it seems, in that he has not worked through expamples to show that in a common energy regime, his predictions are asymptotic to the usual ones.

 

[jeff]

Originally posted by selfAdjoint
The point of the paper...is to see if in the low energy limit, the polymer representation can reproduce the results of the usual Schroedinger representation...the paper concludes that they can.

Yes, but the question is can this result be extended to full LQG?

Originally posted by selfAdjoint
...this difference...is both natural and adequate within the actual situation of LQG.

I'm sorry selfAdjoint, but it really isn't. As ashtekar points out in the paper, the constuction is in terms of only a simple quantum mechanical toy model meant to help people understand some of the characteristic features of LQG and in particular, understand how LQG might make contact with ordinary quantum theory. The thing is that the issue we've been discussing is centred around the spatial diffeomorphism group of which there is no analog in the very simple shadow states construction.

Originally posted by selfAdjoint
The case of Thiemann's quantization of the string is different, so it seems, in that he has not worked through expamples to show that in a common energy regime, his predictions are asymptotic to the usual ones.

Again selfAdjoint, I'm sorry, but as I just pointed out, there's nothing in ashtekar's construction that indicates it can be extended to the LQG-string or full LQG. The onus is really on the LQG people to show that it can, not on urs that it can't.

 

[marcus]

Originally posted by selfAdjoint
The point of the paper, as nonunitary has already posted, is to see if in the low energy limit, the polymer representation can reproduce the results of the usual Schroedinger representation. And as nonunitary said, the paper concludes that they can.

Yeah here is an exerpt from their conclusions section, page 30:

"The fact that the Schroedinger semi-classical states can be recovered
in the polymer framework is thus non-trivial and suggests how standard low energy physics could emerge from the polymer framework.

Thus, our analysis provides useful guidelines for more realistic theories, pointing out potential pitfalls where care is needed and suggesting technical strategies."

Nonunitary (with you helping, sA) has succeeded in getting me interested in this paper.

So I want first to understand what it is trying to do--which seems to be just this: recover Schr. states in low energy limit.
which they conclude they can.

 

[marcus]

I see that on page 3, the authors warn against trying to learn real LQG from this paper, because the toy model differs at certain points fromt the full theory. This seems to be a mistake that it is possible to make---drawing conclusions about LQG from a toy model intended to illustrate something but differing in essential respects. So it is a good warning to give the reader.

--------quote from Ashtekar et al page 3------

For readers who are not familiar with quantum geometry, this example can also serve as an introduction to the mathematical techniques used in that framework.

However, as is typically the case with toy models, one has to exercise some caution. First, motivations behind various construction often become obscure from the restrictive perspective of the toy model, whence the framework can seem cumbersome if one's only goal is to describe a non-relativistic particle.

Secondly, even within mathematical constructions, occasionally external elements have to be brought in to mimic the situation in quantum geometry.

Finally, because the toy model fails to capture several essential features of general relativity, there are some key differences between the treatment of the Hamiltonian and other constraints in the full theory and that of the Hamiltonian operator in the toy model.

With these caveats in mind, the toy model can be useful in understanding the essential differences between our background independent approach and the Fock-space approach used in Minkowskian,
perturbative quantum field theory.
-----end quote------

 

[marcus]

On page 4 they describe the organization of the paper, which helps me understand what they are driving at and how they plan to get there. Section V seems to be the ultimate goal. Here is an exerpt from the organization paragraph on page 4

-------exerpt from Ashtekar et al--------
...In section V we discuss dynamics in the polymer particle
representation. To define the kinetic energy term in the Hamiltonian, on can mimic the procedure used to define the Hamiltonian constraint operator in quantum general relativity.

However, in the toy model, this requires the introduction of a new structure by hand, namely a fundamental length scale, which can be regarded as descending from an underlying quantum geometry. The resulting dynamics is indistinguishable from the standard Schroedinger
mechanics in the domain of applicability of the non-relativistic approximation. Deviations arise only at energies which are sufficiently high to probe the quantum geometry scale. In particular, shadows of the Schroedinger energy eigenstates are excellent approximations to the `more fundamental' polymer eigenstates.
--------end quote------

 

[selfAdjoint]

Originally posted by jeff
Yes, but the question is can this result be extended to full LQG?



I'm sorry selfAdjoint, but it really isn't. As ashtekar points out in the paper, the constuction is in terms of only a simple quantum mechanical toy model meant to help people understand some of the characteristic features of LQG and in particular, understand how LQG might make contact with ordinary quantum theory. The thing is that the issue we've been discussing is centred around the spatial diffeomorphism group of which there is no analog in the very simple shadow states construction.



Again selfAdjoint, I'm sorry, but as I just pointed out, there's nothing in ashtekar's construction that indicates it can be extended to the LQG-string or full LQG. The onus is really on the LQG people to show that it can, not on urs that it can't.

Oh, indeed. Yes they are doing a toy model, just to show if the derivation of compatible results is POSSIBLE, and that is what they have shown to be true. No, this paper does not by itself demostrate that LQG has the desired property. But like any such demonstration, it does encourage one to suppose that it might be, and that it is a problem that is worth working on.

You don't have to be sorry to point that out, it doesn't disapoint me, but I should point out in turn that the fact that the paper is about a toy model does not constitute a flaw in LQG. All the people who work in LQG are constantly stating that it is a work in progress.

The challenges LQG poses seem to me to be of a different kind than the challenges of string physics. The LQG people are still trying to do what string physicists did back in the 80's, set up the basic physical relationships upon which everything else has grown. The exciting part of that is that PERHAPS, they have broken through into a fresh area of representation and quantization, an area that was never required in string work, but which has been forced on the LQG community by the nature of their physics. And IF that pans out it will be a wonderful enlargement of the world of physics.

 

[Urs]

nonunitary wrote:

The paper by Ashtekar and colaborators do this quantization for quantum mechanics and ask the question: can this non-equivalent quantization approximate the ordinary Schrodinger QN in some regime? The answer is yes!

The answer is yes if you take care that the non-equivalent quantization preserves some characteristics of the ordinary quantization. The crucial step in the paper, as I have said before, is that right above and leading to equation (IV.5), where the BCH formula in the usual quantization is used as a template to model (IV.5). This way the crucial information about the Heisenberg algebra (which is not represented on the polymer Hilbert space) is carried over by hand. The \alpha^2 term plays precisely the same role as the anomaly does for the string.

The problem is that you could replace \exp(-\alpha^2/2) in (IV.5) with any other scalar term and still have an LQG-like quantization. But with different factors you loose even the faint resemblance of ordinary quantization. Note that using a factor of unity corresponds to having an operator rep of the classical algebra. This is the analogue of what Thomas Thiemann does for the LQG string.

 

[jeff]

Originally posted by Urs
The answer is yes if you take care that the non-equivalent quantization preserves some characteristics of the ordinary quantization...

In this paper it's the fock and polymer representations of states of a non-relativistic particle that are being compared. (Also, the lqg string was constructed on a fixed spacetime background). But in full lqg, we're dealing with states of geometry itself. Can the imposition of the structure required to make contact with fock reps of graviton states be inserted by hand analogous to the shadow states paper without ruining background independence?

 

[Urs]

I think the answer to your question is no. Not because background independence would be ruined, but because it is not possible in principle.

The point is that in the LQG-quantization of the nonrelativistic point particle the standard quantization does exist, hence we can compare with its algebra and model the LQG construction accordingly. But for gravity the standard quantization does not even exist. So there is nothing to compare with. Maybe there is even a choice of the analogue of \alpha^2 in gravity which yields a sensible result! That would be exciting. But the problem is that current approaches to LQG just use the classical algebra, which is analogous to setting \alpha = 0. That this cannot give anything like ordinary quantum mechanics is illustrated by Thomas Thiemann for a 1+1 dimensional theory. In 3+1 d things will get worse.

It is really hard to make definite statements about LQG-like quantizations because of this ambiguity. Note that Thomas Thiemann could have chosen operators U(\phi) on his Hilber space which do represent the Virasoro group modified by the anomaly. That would have allowed doing the analogue of gr-qc/0207106 for the string. He just did non choose to do so.

 

[Urs]

One should underline the phrase on p.14 of gr-qc/0207106:

'[...] ideas motivated by results in the Schroedinger representation, we are now led to seek the analog [...]'.

For an approach which is supposed to solve the outstanding problem of theoretical physics, namely finding quantum gravity, this is surprisingly vague.

For 3+1 gravity there are no 'results in the Schroedinger representation' by which we could motivate construction in LQG. If there were, we would not need any further attempts at quantum gravity!

 

[selfAdjoint]

Originally posted by Urs
One should underline the phrase on p.14 of gr-qc/0207106:

'[...] ideas motivated by results in the Schroedinger representation, we are now led to seek the analog [...]'.

For an approach which is supposed to solve the outstanding problem of theoretical physics, namely finding quantum gravity, this is surprisingly vague.

For 3+1 gravity there are no 'results in the Schroedinger representation' by which we could motivate construction in LQG. If there were, we would not need any further attempts at quantum gravity!

I don't see this argument. They are hot on the trail of deriving low order results from their theory (see Thiemann's Phoenix program). This toy model is built to test out ideas. They find a good route by considering the parallel quantization.

You can't have it both ways, that this is only a toy model which doesn't deserve to have a fuss made over it, and that its every rhetorical shortcoming is to be used to beat the big LQG theory with.

 

[jeff]

Originally posted by selfAdjoint
They are hot on the trail of deriving low order results from their theory (see Thiemann's Phoenix program).

Hot on the trail?

Originally posted by selfAdjoint
You can't have it both ways, that this is only a toy model which doesn't deserve to have a fuss made over it, and that its every rhetorical shortcoming is to be used to beat the big LQG theory with.

Oh come on! There's nothing in urs's post you quoted that detracts from his basic point which is that he doesn't think this method can work for lqg.

 

[selfAdjoint]

No Jeff, it was your post earlier that had that insinuation. Sorry if I conflated the two of you but it does seem you are doing a double team effort against LQG.

Understand I think it's OK to:

• Point out flaws in Thiemann's LQG String paper
• Point out that the shadow states paper is only a toy model
• Attack any specific error in any specific LQG paper

But not just to mix and match.

 

[jeff]

I'm sorry if it looks like I'm ganging up with urs against you. Really, urs doesn't need my help and I haven't offered it since he understands the in's and out's of lqg better than I do. I also wouldn't describe anything I say as insinuation. I said that I know that the methods of the shadow states paper can't be extended to lqg. But what I didn't feel like figuring out for myself was if there was some other method to modify lqg quantization so as to make contact with the fock space representation of graviton states, wouldn't it ruin background independence anyway? So I asked urs and he said that he doesn't think it's possible even in principle to modify lqg this way. My question may have been a bit cavalier, but his answer wasn't entirely unequivocal.

Let's try to recognize and concentrate on the central physical issue of the consequences for the lqg program of the peculiar nature of it's quantization, and not so much on the precise wording of posts, and distinguish between comments of direct physical relevance and those that are more or less beside the point (however annoying they may seem).

 

[Urs]

selfAdjoint -

I am not sure if you are right that the 'shadow-states' paper is criticized for being about a toy model. I, for one, consider it highly desirable to apply LQG-like formalism to toy models like the 1d nonrel particle or the NG string, to see what happens.

And in this respect I found the 'shadow states' paper very helpful, because I could easily spot the step which contained the crucial assumption.

And of course everybody wants to learn what the toy model teaches us about the LQG quantization of 3+1 gravity. That's why the paper was written in the first place.

And I learned two important things from the 'shadow states' paper:

1) LQG-like quantization is not canonical but can, at best, approximate canonical quantization is a certain limit. (That's not controversial. It is the very point of this paper.)

2) To make this approximation possible in the first place the algebra of the canonical/Schroedinger quantization has to be closely mimicked by the LQG formalism. If instead the classical algebra is mimicked (as in Thiemann's 'LQG-string' or for the spatial diffeo constraints of 3+1d gravity) there is no reason to expect any relation to standard quantization. This is strongly confirmed by Thiemann's paper.

 

[selfAdjoint]

Urs, I thoroughly agree with these points, and I have a question. Do you think the BHC modeling in the shadow states papers could be adapted to Thiemann's LQG string development? Is this what he's missing?

 

[Urs]

It could be done, but I am pretty sure that there would be no convergence to the standard results by taking any limit.

That's because there is a crucial difference between Thiemann's approach to the string and Ashtekar,Fairhurst&Willis approach to the 1d nonrel particle: AF&W take care to copy the ordinary quantum corrections (the alpha^2 term) to their algebra. But in Thomas Thiemann's approach the analogous quantum corrections are not taken into account. The 'LQG-string' is like setting alpha^2 = 0 in the AF&W paper.

Note that in the LQG-like quantization of the string one could do the following: Instead of constructing operators U which represent the classical diff algebra
U(\phi)U(\psi) = U(\phi\circ\psi) one could choose operators U which incorporate the quantum correction to this algebra that is induced by the anomaly as found in the ordinary quantization (what AF&W call the 'Schroedinger quantization'). Doing this would make the 'LQG-string' compliant with the methods in the AF&W paper and presumably the 'shadow states' technique would then be applicable to the string and approximation to the standard quantization would be found.

 

[Urs]

selfAdjoint -

after rereading your message I realized that I did not pay due attention the first time. You were asking if the approach by Thomas Thiemann could be modified in such a way that the analog of the Baker-Campbell-Hausdorff step in the AF&W paper is incorporated.

Yes, it can. I said so in my previous answer, but didn't realize that this was what you were asking, too.

Yes. The AF&W paper demonstrates a quantization of the 1d nonrel particle where all essential algebraic relations are copied from the standard quantization and the only difference is that exp(i p) is represented non-weakly continuously.

Similarly, I think one could find operators which represent the Virasoro group including the usual anomaly term but such that these operators are non-weakly continuous. This would be the AF&W analog of the 'LQG-string'. This would allow to construct shadow states and all that for the 'LQG-string'.

Yes, this should be possible. I just don't know what this would be good for.

 

[selfAdjoint]

Urs, I don't know either. But Thiemann's string paper seems to be a side show now. If somebody can show it works in some sense of the word it may become interesting but right now it doesn't seem to have any use (except that it caused you to investigate some stuff and gt started on an idea you might not otherwise have come upon :=) ).

The AF&W shadow states idea looks a lot more promising. May I ask if you really think this looks arbitrary? It looked to me as if they said, LQG development is no good if we can't at least asymptotically reproduce ordinary quantum results, so how can we do that, out of our existing way of quantizing? And modeling on the BHC theorem is certainly a reasonable way to go. No?

 

[Urs]

Hi selfAdjoint -

hm, yes, basically I agree, but, you see, the problem is this 'modeling on the BCH theorem'.

They abandon the Heisenberg algebra, use a Weyl-like algebra instead and then fix the ambiguity in how exactly to define this algebra by looking at the results that one would have obtained with the Heisenberg algebra in the first place.

What they end up with is essentially an unfamiliar representation of the usual relations. In particular this means that all the usual quantum corrections to the classical algebra are there - because they have simply been copied from the usual formalism.

But this does not work for gravity. There is no standard Schroedinger rep-like quantization of gravity on which to model a Weyl-like algebra. If there were, we wouldn't need LQG anyway! Instead of modeling their algebra on a Schroedinger-like quantization, people in LQG model the quantum constraint algebra on the classical algebra of constraints.

But obviously this modeling on the classical algebra looses contact with everything in the AF&W paper. No shadow states in the AF&W paper could be built if the algebra were not modeled on the BCH theorem but on the classical algebra.

I am currently really at a loss with the point of the entire LQG approach. There is a symposium of the DPG in Ulm.Germany in two weeks where I should get the chance to talk to some LQGists. Maybe that will help.

 

[selfAdjoint]

They abandon the Heisenberg algebra, use a Weyl-like algebra instead and then fix the ambiguity in how exactly to define this algebra by looking at the results that one would have obtained with the Heisenberg algebra in the first place.

Their approach doesn't give them a Heisenberg algebra, but it does give them something they have some reason to call a Weyl algebra. They want to have this algebra to behave as much as possible like a real Weyl algebra and they find they do have the freedom to do that, in this one case.

I take your point about the fact that there is a Schroedinger representation to copy in this model that wouldn't be available in full LQG, but is it really "copying" they are doing here? Wouldn't a better term be modelling? The difference being that they might hope they could still enforce this BHC-like behavior, at least in some limit, on the full LQG Weyl algebra? Quite apart from the Schroedinger context?

 

[Urs]

Hi selfAdjoint -

I am glad that we now apparently agree on what the issues are and are now having a constructive discussion of these issues. Thanks!


Ok, so you are asking if I think there is hope that one can use the (large) freedom that the LQG approach offers to construct
something interesting.

First of all let me emphasize that I think this is indeed the right question to ask, since the LQG-like quantization is a generalization of standard quantization. To fully appreciate this, note that, if we understand under LQG-like quantization the prescription 'Quantize the group, not the generators', the standard quantization is just the special case where we use precisely that Weyl algebra which comes from the Heisenberg algebra, or, more generally, use that quantization of the group that comes from the quantization of the generators.

So in this sense we might say that LQG-like QM is a more general concept than ordinary QM. It is not at all clear that the full generality of this approach makes physical sense, but technically it is a generalization.

Hence, and I assume that this is roughly (maybe not explicitly) what happened historically, one might ask if some choice of quantized group algebra could be used in cases where the quantized algebra of generators is not available, as is the case in 3+1d gravity.

Yes, in principle I think this is a possibility. I don't see any compelling argument why it should be true, but since we know so little about non-perturbative QG it would probably be premature to exclude this possibility altogether. Maybe, indeed, some quantization of the classical group of spacetime diffeomorphisms gives the correct description of QG. Maybe.

Here by quantization of the group I of course mean a set of operators U(phi) that satisfy the classical group algebra up to quantum corrections, something like U(phi)U(psi) = U(phi o psi)V, where V is a quantum correction. For instance for the quantization of the Virasoro algebra V would be a phase factor that comes from the anomaly in the Virasoro algebra.

But of course there is a big problem: So far in the LQG-like quantization of gravity people have simply set V=1 identically (for the spatial diffeomorphisms or for all group elements in Thiemann's LQG-string). I am convinced that this cannot possibly be the right choice because it amounts to eliminating all quantum effects whatsoever. Thiemann's string shows how very different this choice is from the usual V=phasefactor choice.

I believe that LQG-like quantization of gravity could make sense for some V=complicated correction. Maybe.

But how should we find this V? And is anyone looking for it?

One thing I could imagine one could do is the following: Choose a certain V and then check if the corresponding physical states contain smooth space and a nice semiclassical limit of gravity and graviton excitations as fluctuations about this limit. If anybody could do this he would immediately be very famous! :-)

Of course people are currently trying to show this for V=1. But, having seen the LQG-string which also uses V=1 and is way off standard physics, I have severe doubts that the current approach to LQG does have a sensible semiclassical limit.

And the fact that papers which are supposed to go into this direction, like AF&W do not set V=1 in their derivations, but use a V that comes from the BCH formula of the usual quantization, doesn't make me confident that the thing is going in the right direction.

Ok, so it seems that I am proposing a new LQG program:

Find a V=something quantum correction to the diffeomorphism group such that a sensible semiclassical limit is obtained.

:-)

 

[Haelfix]

It isn't so different than the quagmire String theory is in really. Far too general a theory, with no good way to pick the right special case that matches reality. The difference is LQG is at an earlier stage of development. (Although the fact that they have had some theoretical successes in describing BH's, is curious)

What was it Smolin said, there's a nasty little theorem about randomly picking constraints until we figure out the right one. Something like, we might be at this for ~ the age of the universe.

Contrast this with General relativity, Einstein's eqns were pretty much the *second* simplest thing you could think of.

 

[selfAdjoint]

Urs, I have been looking through some of the more recent general sources on LQG, Thiemann's habilitation thesis and Rovelli's textbook, to see how your approach,

Here by quantization of the group I of course mean a set of operators U(phi) that satisfy the classical group algebra up to quantum corrections, something like U(phi)U(psi) = U(phi o psi)V, where V is a quantum correction.

would fit in. It seems to me that one could move back from the actual quantization to the point where the classical kinematic states are being defined. There is already an inmcompleteness here, with several candidates being offered. At this point some new candidate could be introduced that would upon the definiton of the Poisson algebra yield the V phase just as the Schroedinger quantization does in the AF&W paper. But what that new candidate might be so far escapes me.

 

[jeff]

Hi urs,

I'm very curious to learn how much LQG interests you at this point and who in the LQG camp have you heard from about all of this?

 

[Urs]

Jeff wrote:

I'm very curious to learn how much LQG interests you at this point

I used to find the general idea of LQG interesting. I thought that just canonically quantizing gravity cannot be that wrong and that maybe one learns something useful by seeing how it does or does not work.

But since I have studied the 'LQG-string' paper by Thomas Thiemann and had lots of discussion about it I realized a couple of things about LQG which were not clear to me before.

Most importantly, I learned that it is not true that LQG is just a standard textbook attempt at quantizing gravity, but involves a notion of quantization which is alien to physics as we know it.

This greatly reduces my willingness to, for instance, find the recent development by Bojowald and others in 'loop quantum cosmology' interesting.

I have to say I am glad that my research is not related to LQG.

With string theory we certainly know it is about physics, even if experimental tests are difficult. With LQG we don't even seem to know that it is physics rather than some arbitrary construction.

and who in the LQG camp have you heard from about all of this?

Well, I have heard talks by A. Ashtekar, T. Thiemann, J. Lewandowski, M. Bojowald, L. Freidel, have had private discussion with A. Ashtekat, T. Thiemann, H. Sahlmann, L. Freidel, had a couple of usenet discussions with J. Baez and I have read a bit here and there in the LQG literature. I am absolutely no LQG expert, though I feel that in the last couple of weeks my understanding of the details of the approach has improved.

 

[selfAdjoint]

Urs, one other question. At one point you said that you were reading the Meusebergr & Rehrens paper. Could you share what you thought about it? It's not easy for some of us to get to it.

 

[jeff]

Originally posted by Urs
...LQG...involves a notion of quantization which is alien to physics as we know it...

...we don't even...know that it is physics...

With string theory we certainly know it is about physics...

...I am glad that my research is not related to LQG

This is pretty much how most theorists have felt about LQG from it's inception. From this perspective, the title of this forum, Strings, Branes, & LQG, seems a bit silly. The topics here are quantum cosmology, quantum gravity and theories of everything, strings remaining our only bonafide example of the last two. Genuinely interesting non-stringy research in relation to quantum gravity continues, but not so much with theories of everything. So I'm thinking maybe PF members would be better served if the name of this forum was changed to something like Quantum Gravity, Quantum Cosmology, and String/M-Theory.

 

[marcus]

----quote----

...Most importantly, I learned that it is not true that LQG is just a standard textbook attempt at quantizing gravity, but involves a notion of quantization which is alien to physics as we know it.

This greatly reduces my willingness to, for instance, find the recent development by Bojowald and others in 'loop quantum cosmology' interesting.

I have to say I am glad that my research is not related to LQG.

With string theory we certainly know it is about physics, even if experimental tests are difficult. With LQG we don't even seem to know that it is physics rather than some arbitrary construction...

----end quote---

Bojowald's best-known result is the removal of the BB singularity. This has been reproduced outside the LQC (loop quantum cosmology) context by Husain and Winkler
"On singularity resolution in quantum gravity"
http://arxiv.org/gr-qc/0312094
They use ADM (metric) variables instead of the Ashtekar variables used in LQG. And they get similar results.


Another result in LQC is agreement with solutions of the Wheeler-DeWitt equation away from the singularity, see page 24 of
Ashtekar Bojowald Lewandowski "Mathematical Structure of Loop Quantum Cosmology"
http://arxiv.org/gr-qc/0304074

Here is what Ashtekar et al say:

"We established two results to show that this expectation is indeed correct. First, there is a precise sense in which the difference equation of loop quantum cosmology reduces to the Wheeler-DeWitt differential equation in the continuum limit. Second, in the regime far removed from the Planck scale, solutions to the Wheeler-De
Witt equation solve the difference equation to an excellent accuracy. Thus the quantum constraint of loop quantum cosmology modifies the Wheeler-DeWitt equation in a subtle manner: the modification is significant only in the Planck regime and yet manages to be 'just right' to provide a natural resolution of the big-bang singularity."

 

[marcus]

----quote----

...Most importantly, I learned that it is not true that LQG is just a standard textbook attempt at quantizing gravity, but involves a notion of quantization which is alien to physics as we know it.

This greatly reduces my willingness to, for instance, find the recent development by Bojowald and others in 'loop quantum cosmology' interesting.

I have to say I am glad that my research is not related to LQG.

With string theory we certainly know it is about physics, even if experimental tests are difficult. With LQG we don't even seem to know that it is physics rather than some arbitrary construction...

----end quote---

I would say that the signs are that LQG is not an arbitrary construction and that it agrees in important ways with physics as we know it.

It is signficicant that Husain Winkler can quantize the ADM (metric) variable model of cosmology and confirm Bojowald's result and that LQC matches Wheeler-DeWitt a few hundred Planck times away from the singularity. This fits right in with physics as we know it.

I question the word "alien" applied to LQG as a whole. It seems to be based on your reading of two papers:
Thiemann's Loop-String
Ashtekar Fairhurst Willis "Shadow states" paper
These are highly atypical and the latter is explicit in saying not
to take it as representative of the main theory. It seems risky to draw conclusions about LQG as a whole from only two papers and especially from such exceptional ones.

I have been reading a new LQG paper and cannot find any evidence of
the "alien" approach to quantizing which you have so often talked of.
the paper is by Karim Noui and Alejandro Perez

"Three dimensional loop quantum gravity: coupling to point particles"
http://arxiv.org/gr-qc/0402111

I would be delighted if someone would point out where, in the "Loop Quantization" section----I assume pages 15-22----Noui and Perez do something alien in the matter of quantization.

 

[marcus]

Originally posted by selfAdjoint
Urs, one other question. At one point you said that you were reading the Meusebergr & Rehrens paper. Could you share what you thought about it? It's not easy for some of us to get to it.

this whole discussion is so scattered, I have been reading pieces of it over at Coffee Table and I came across reference to Urs reading a 1988 paper by Pohlmeyer and Rehrens. (However so far I missed any reference to Meuseberger.)

over at CoffeeTable at one point Urs wrote:

"I have now managed to get hold of a copy of

K. Pohlmeyer and K.-H. Rehren, The Invariant Charges of the Nambu-Goto Theory: Their Geometric Origin and Their Completeness, 1988"

have to compliment you on your ability to follow the discussion
(follow it at all, not to mention doing so with courtesy)

Did Urs ever reply to the question about Meuseberger and Rehrens?
I don't understand the thread structure at coffeetable, seems cross-linked and tangled, so I'm not sure I've found where all the parts are.

 

[marcus]

Originally posted by Urs
Hi selfAdjoint -
...Ok, so you are asking if I think there is hope that one can use the (large) freedom that the LQG approach offers to construct
something interesting.

First of all let me emphasize that I think this is indeed the right question to ask, since the LQG-like quantization is a generalization of standard quantization. To fully appreciate this, note that, if we understand under LQG-like quantization the prescription 'Quantize the group, not the generators', the standard quantization is just the special case where we use precisely that Weyl algebra which comes from the Heisenberg algebra, or, more generally, use that quantization of the group that comes from the quantization of the generators.

So in this sense we might say that LQG-like QM is a more general concept than ordinary QM. It is not at all clear that the full generality of this approach makes physical sense, but technically it is a generalization...

this quote is from two Urs-posts back in this thread.
there is an interesting train of thought, but it got interrupted
(perhaps by more basic issues of identity/group loyalty)

pity Urs seems to have abandoned this train of thought

"quantize the group not the generators" sounds like
quantize the group not the Lie algebra
almost like something I could understand and use
as a criterion to check other LQG papers.

without some clarification like that it is impossible to tell
if Urs term "LQG-like quantization" applies just to the two papers he studied (which are quite atypical ones) or more generally to a large number of LQG papers.

so far I could not find "LQG-like quantization" in the LQG paper I am currently reading by Noui and Perez----may be missing something obvious, paper's difficult a mon avis.

 

[jeff]

Originally posted by marcus
...without some clarification like that it is impossible to tell if Urs term "LQG-like quantization" applies...to the two papers he studied (which are quite atypical ones) or more generally to a large number of LQG papers.

IT APPLIES TO ALL LQG PAPERS, AS URS HAS PATIENTLY BUT REPEATEDLY EXPLAINED TO YOU IN A VARIETY OF DIFFERENT WAYS MARCUS!

I believe that selfAdjoint for example has acknowledged this (though I can't say he's given up on LQG).

 

[marcus]

---quote---

IT APPLIES TO ALL LQG PAPERS, AS URS HAS PATIENTLY BUT REPEATEDLY EXPLAINED...
---end quote---

Urs has not read all LQG papers and therefore cannot know.
I am skeptical and wish to check his statements myself.
So far two papers have been read in detail.
I would like a simple criterion which I can apply to
whatever LQG paper I happen to be reading that will tell me
whether this paper shares or does not share the feature.

I am also interested by Urs general comment, which I quoted, and some things TT said over in CoffeeSquabble which I may fetch later. Whole business is fascinating.

 

[jeff]

Originally posted by marcus
Whole business is fascinating.

Following urs around has been a lot of fun for a bunch of us which, despite our disagreements, is the most important thing anyway.

Originally posted by marcus
Urs has not read all LQG papers and therefore cannot know.

I guess you'd have to ask urs why he doesn't believe there's a more conventional approach to LQG-quantization being floated in papers he hasn't heard about. I think though he's spoken with a number of LQG people, so given the subject of their conversations, I'd imagine they would've alerted him to any such research.

Originally posted by marcus
I am skeptical and wish to check his statements myself. I would like a simple criterion which I can apply to whatever LQG paper I happen to be reading that will tell me whether this paper shares or does not share the feature.

It's perfectly understandable that you'd like to be able to recognize counterexamples when you happen across any. However, I do think urs has already supplied you with the tools to do so. You just need to put the elbow grease on and study his remarks. But you'd think that if there was more than one approach to quantization in LQG, it would've been mentioned in reviews of the subject.