Ashtekar's Shadow states 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.

 

[selfAdjoint]

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

I believe we have gone round and round and come out where we started, but with some knowledge.

1. The quantization used by LQG is a valid quantization (Rehrens).
2. It does not agree with the method used in the rest of quantum mechanics (Distler, Urs).
3. The shadow states paper shows a way that LQG can be tied to low energy quantum physics.
4. Urs thinks this shadow states method is "arbitrary" and I (for what I'm worth) don't. I think it was a new initiative, where the authors were seeking a method, and they found one. Their method should be judged on its results, not on the metagame, or else I'm going to bring up the metagame attacks ("just philosophy", "sweeping infinities under the rug") that have been leveled at string physics and quantum field theory.
5. In the upshot, I don't think the LQG physicists have to modify their quantization in the high energy/small dimension range, but they have a long way to go to come up with a theory that is both mathematically consistent and "necessary" in the sense that each feature follows uniquely from prior features (except for alternate ways to the same goal.) Superstring theory does have this property.

And Urs still hasn't responded on the Meusberger & Rehrens paper.

 

[jeff]

Originally posted by selfAdjoint
The quantization used by LQG is a valid quantization

This remark requires some defense since...

Originally posted by selfAdjoint
It does not agree with the method used in the rest of quantum mechanics

I think this is currently our common ground of agreement.

Originally posted by selfAdjoint
The shadow states paper shows a way that LQG can be tied to low energy quantum physics

It shows that fock space reps in quantum theories we know are correct can be reformulated in LQG-like terms as polymer reps, nothing more.

Originally posted by selfAdjoint
Their method [shadow states] should be judged on its results, not on the metagame...

Would you mind explaining what in the shadow states paper indicates that it can be applied to LQG?

Originally posted by selfAdjoint
...I don't think the LQG physicists have to modify their quantization in the high energy/small dimension range...

Moving from the implausible to the down right silly, are you saying that despite all of this, LQG should still be taken seriously because gravity might be quantized differently at different energies? You should think hard about the implications of this.

 

[Urs]

Hi everybody -

I realize that it takes a lot of time to follow the discussion here on PF and filter out the noise produced by some people. Please, if anyone wants to talk with me further about the LQG string, please drop me a note over at the Coffee Table. For instance, just write a comment to this thread.

Thanks!

All the best,
Urs

 

[Urs]

selfAdjoint -

as soon as I find the time I write something about Meusburger&Rehren, math-ph/0202041. Sorry for the delay, there are too many things to do! :-)

 

[selfAdjoint]

Sorry for pushing you Urs. I'm just interested in it.