# Ashtekar's Shadow states paper

1. Feb 27, 2004

### 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 [Broken]

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.

Last edited by a moderator: May 1, 2017
2. Feb 27, 2004

### 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.

Last edited: Feb 27, 2004
3. Feb 27, 2004

### 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).

Last edited: Feb 27, 2004
4. Feb 27, 2004

### 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).

5. Feb 27, 2004

### marcus

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 doesnt look like typical LQG to me (whatever that is).

6. Feb 27, 2004

### 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.

Last edited: Feb 27, 2004
7. Feb 27, 2004

### 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.

Last edited: Feb 28, 2004
8. Feb 27, 2004

### marcus

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.

Last edited: Feb 28, 2004
9. Feb 28, 2004

### 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 dont 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.

Last edited: Feb 28, 2004
10. Feb 28, 2004

### jeff

Re: Urs Schneider's criticism of Ashtekar's "Shadow states" paper

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.

11. Feb 28, 2004

### 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.

Last edited: Feb 28, 2004
12. Feb 28, 2004

### ranyart

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 [Broken]

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 [Broken]

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

Last edited by a moderator: May 1, 2017
13. Feb 28, 2004

### jeff

Just email him and ask. His homepage is here.

14. Feb 28, 2004

### 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. 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 [Broken]
Quantization is not yet thoroughly understood
(See for example Stefan Waldmann
http://arxiv.org/hep-th/0303080 [Broken])
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)

Last edited by a moderator: May 1, 2017
15. Feb 28, 2004

### jeff

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.

16. Feb 28, 2004

### selfAdjoint

Staff Emeritus
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.

17. Feb 28, 2004

### 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.

18. Feb 28, 2004

### marcus

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.

Last edited: Feb 28, 2004
19. Feb 28, 2004

### 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------

20. Feb 28, 2004

### 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------

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