Amazing bid by Thiemann to absorb string theory into LQG

[selfAdjoint]

Thanks so much! I have printed it off and I am going to start studying it NOW! What a lot of work went into it.

I am really leery of downloading Mozilla, to fill up my hard drive with unknown software that has unknown interactions with my operating system. Does anyone else have Mozilla with Windows NT? Any war stories?

 

[Urs]

Mozilla and NT

Hi selfAdjoint -

I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.

 

[marcus]

Originally posted by Urs
Hi selfAdjoint -

I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.

Urs, I see you are online at the moment
has anyone found anyone equation in TT's paper that
you can put your finger on and say is wrong?

could you point me to some "equation number such and such on page
soandso" and say what is wrong with it

or is the controversy(if there is still some discussion) only about the verbal interpretations
and what conclusions to draw?

 

[Urs]

problems with the LQG string

Hi Marcus -

1) Jacques Distler say that this step is not allowed. That would invalidate the second and third paragraph on p.20.

2) If the constraints themselves are not represented on H_kin then (5.3) to (5.5) would not make sense. I don't know if they are, see here.

 

[jeff]

Originally posted by marcus
could you point me to some equation...and say what is wrong with it...or is the controversy only about the verbal interpretations and what conclusions to draw?

It was clear right away that the LQG-string is unphysical, and whether it's mathematically sound isn't a matter of interpretation.

 

[marcus]

Originally posted by Urs
Hi Marcus -

1) Jacques Distler say that this step http://golem.ph.utexas.edu/string/archives/000299.html#c000530
is not allowed. That would invalidate the second and third paragraph on p.20.

Urs, thanks for the reply! As I indicated I am looking for some
specific step in the argument or some equation, that is believed mathematically unsound.

I have a better idea now (Jacques says second and third paragraph
on page 20 are invalid)
but I still do not know what false step is supposed to make them invalid because
your link is to coffeetable and covers a lot of ground.
If not too much trouble could someone please point to which particular step
on some page in the TT paper
that Jacques says is wrong.

In the meanwhile I will have a look at those paragraphs on page 20.

 

[marcus]

Are there actually any math problems with the LQG string

does anyone else know what "step" Jacques thought
could be mathematically unsound?

I mean what page and what line on that page
where he says some definite thing that is invalid.

It isn't easy to read the coffeetable and
I don't see any specific reference there to
some equation number or some sentence that Urs
link points to.

what is the wrong "step" that makes second and third
paragraphs of page 20 invalid?

 

[marcus]

Are there actually any math problems with TT's paper

Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.

Here is one of those two paragraphs. What is wrong with it, can anybody say?

"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."

 

[selfAdjoint]

For the benefit of others, here are those two paragraphs.
1. Second paragraph on p. 20_________________________
Next we construct from \mathfrak{P} bounded functions on \mathcal{M} which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on \mathfrak{P} be promoted to commutators divided by i\hbar and the adjoint respectively. Denote the resulting *-algebra by \mathfrak{A}.
____________________________
Notes on 1. \mathfrak{P} is a classical Poisson subalgebra constructed from the Pohlmayer charges. \mathcal{M} is the target manifold (background space).

2. Third paragraph on page 20._____
The automorphism groups \alpha^{\pm}_{\varphi}, \varphi \in Diff(S^1) generated by the Virasoro constraints s well as the Poincare automorphism group \alpha_{a,L} extend naturally from \mathfrak{P} to \mathfrak{A} simply by \alpha (W(Y_{\pm})) = W (\alpha,(Y_{\pm})). A general representation of \mathfrak{A} should now be such that the automorphism groups \alpha are represented by inner automorphisms, that is, by conjugation by unitary operators representing the corresponding group elements. Physically the representation property amounts to an anomaly-free implementation of both the local gauge group and the global symmetry group while unitarity implies that expectation values of gauge invariant or Poincare invariant observables does not depend on the gauge or frame of the measuring state. Finally, the representation should be irreducible or at least cyclic.
_______________________________

 

[selfAdjoint]

Originally posted by marcus
Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.

Here is one of those two paragraphs. What is wrong with it, can anybody say?

"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."

Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.

 

[marcus]

Originally posted by selfAdjoint
Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.

selfAdjoint many thanks for this report. I am glad that you have
Mozilla running and can read coffeetable.

I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.

It is certainly possible that TT's paper is flawless mathematically, I should say, and that no one will be able to point to any specific line in it where TT makes a false move.

But as you know it is not uncommon either for math papers to need corrections when they are first circulated in draft and it would be helpful to TT if anyone can find some error or unclear point, which he could be told about so he could have a chance to fix it.

The overall conclusions certainly are interesting, are they not?

 

[selfAdjoint]

The conclusions are strong in my opinion. BTW I have been trying to fit Urs' DDF operators into Thiemann's scheme, so far without success (I just don't heve this stuff sufficiently at the tip of my mind).

Here is Distler's first comment on THE LQG String:
Why don’t I just close my eyes, click my heels and wish away all anomalies?

What are the rules here?

It is well known that it is impossible to preserve all of the relations of the classical Poisson-bracket algebra as operator relations in the quantum theory.

What principle allows Thiemann to decide which relations will be carried over into the quantum theory?

Where does he discuss which relations fail to carry over?

To which the answer is, see the last five years of LQG theory, especially by the Ashtekar school.

And here is his second comment.
No, I don’t believe there’s any quantization scheme that takes the full Poisson-bracket algebra of the classical theory and carries it over — unaltered — into the operator algebra of the quantum theory.

Depending on the quantization scheme, you may be able to carry over some subalgebra (the prototypical example, being the CCRs).

And his third comment (getting down to some detail).
OK, so you (he) claim(s) that there is a quantization in which the commutation relations of X '(ó) , Ð (ó) , T + + (ó) and T - - (ó) are carried over from the classical Poisson-bracket algebra, unaltered (i.e., the commutators of the T ’s do not pick up a central term)?

Certainly, that’s not true if the T ’s lie in the universal enveloping algebra generated by X '(ó) , Ð (ó) — as is conventionally the case.

Without being sure, I suspect he's still working from inside string theory here.

And finally, the nuv of his argument.
You mean aside from the fact that none of the symbols are well-defined?

Look, this is elementary stuff.

We can expand everything in Fourier modes. If T + + is in the universal enveloping algebra of the Fourier modes of X ' and Ð , then its Fourier modes (conventionally called L n ) are some expressions quadratic in those modes.

Since the Fourier modes of X ' and Ð (the “oscillators”) don’t commute, you need to specify an ordering. I don’t care what ordering you choose, but I insist that you choose one.

Now compute the commutator of two L n ’s. Again, you will obtain something which is at most quadratic in oscillators (there will, in general, also be a piece 0 th -order in oscillators). And it must be re-ordered to agree with your original definition of the L n s.

Carrying out this computation, you obtain the central term in the Virasoro algebra, and I believe that it is a theorem that the result is independent of what ordering you chose for the L n s.

Note that I never mentioned what Hilbert space I hope to represent these operators on. So I don’t see where its separability (or lack thereof) enters into the considerations.

It's pretty clear here that the X' and \Pi he is talking about come out of the echt string context. "Oscillators"!

 

[jeff]

Originally posted by selfAdjoint
The conclusions are strong in my opinion.

Respectfully, what conclusions?

 

[selfAdjoint]

Jeff, I meant the conclusions of the Thiemann paper, The LQG String.

Notice that I hold that criticism of it based on the techniques and constraints of string physics are by the point, or at least that they have to be explicitly shown to bear on what Thiemann is doing. Almost all of his paper, including the Hilbert space and operator algebra parts, is common to the developments of LQG over the last few years, and much of it is common to the work of mathematical physicists over the past several decades - the GNS construction, for example is truly classic.

I believe the weakest point of the paper is the Pohlmeyer charges, which Thiemann seems to have used as he found them, but I also think that Urs has provided the beginning of a fix for that in his generalized (un-string-ized) DDF charges. The problem with these as far as I understand is that he has provided a classical pre-quantum development of them, and Thiemann is set upon introducing his charges post-quantization. That difficulty is only temporary, I am sure.

BTW, thank you for the link to the Mozilla page. As I said above, I have installed it (browser only) and it seems to be working fine.

 

[lethe]

Originally posted by marcus

I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.

mozilla will not threaten your existing Internet Explorer installation. it is perfectly safe. the mozilla suite includes a web browser, email client, and html editor, all rolled into one. if you only need a web browser, you can get just the web browser component alone as an application called http://www.mozilla.org/products/firebird/ .

marcus, i think you should give firebird a try. not only is it the only browser around that will let you read MathML, but pop up advertisements will become a thing of the past, and tabbed browsing is very useful.

furthermore, firebird doesn t "install" in your computer at all. you just download the application, unzip it, and double click. don t like it? got tired of it? just delete it, you don t even have to uninstall. it is completely safe and completely free, and there is a chance that you will like it so well, that you will never know why you stuck with IE.

 

[marcus]

Originally posted by lethe
...there is a chance that you will like it so well, that you will never know why you stuck with IE.

thanks Lethe, that is a persuasive recommendation
my resistance to trying it is weakening, must admit

also message to Meteor:
have been reading parts of the paper by Elias V.
(U. Barcelona) and the Iranian physicist Setare.
Get the impression that QNMs of anything besides
Schw. and ReissnerNordstrom holes are not well
understood at all, which
gives more potential for surprises.

 

[meteor]

have been reading parts of the paper by Elias V.
(U. Barcelona) and the Iranian physicist Setare.
Get the impression that QNMs of anything besides
Schw. and RessnerNordstrom holes are not well
understood at all, which
gives more potential for surprises.

Seems that the speciality of Mr. Vagenas are 2D black holes. Just see his list of publications.
I will call him tomorrow and will ask him about QNM. And will invite also him to join the forum!

 

[Urs]

math problem in LQG-string

Hi -

let me be more precise: Jacques Distler's point indeed pertains to Thiemann's paper. That's not a question of arguing 'from within string theory' or not and it has nothing to do with being perturbative or not. Thiemann claims in these paragraphs on p. 20 that he can construct a quantization of the classical Poisson algebra of the string which completely preserves the Virasoro algebra without adding a central extension. Distler says that that's not possible in general. This should not be a matter of approaches or opninions.

The point is that Thiemann in passing claims that he can find a quantization of the Poisson algebra such that
\alpha(W(Y_\pm)) = W(\alpha(Y_\pm))
holds true on the quantum level, where \alpha is supposed to be an inner automorphism of the algebra. (second sentence of third paragraph on p.20).

(Hm, something is wrong with the parsing engine. I'll try to continue in another post.)

 

[Urs]

Explicitly this means that for K some element of the algebra we set
\alpha(K) :=<br /> \exp(i t^I V_I)<br /> K<br /> \exp(-i t^I V_I)
where V_I are the quantized Virasoro constraints and
t^I are some constants that parameterize
\alpha. This obviously has the advertized property of inducing group transformations only if the V_I generate a group as they do classically. Distler says that this is not possible - no matter what, because the anomaly will appear in any imaginable quantization.

So this discussion at the Coffee Table is directly relevant to Thiemann's paper. There I have provided two versions (the functional and the mode-basis one) of how I think Thiemann is quantizing the Virasoro constraints. Distler's point is that that's too naive, since it ignores a subtle issue. I haven't yet had time to fully solve the excercise that he told me to do, but I understand that his point is the following:

Consider what I wrote inthis comment. Why does Distler say that's not well-defined? Because it is naively multiplying distributions, which is not a well defined operation. Since the Y(\sigma) are technically operator-valued distributions already their product Y(\sigma)Y(\sigma) is not well defined. This becomes more obvious when one ignores this and caclulates the commutator
<br /> [Y(\sigma)Y(\sigma),Y(\sigma^\prime)Y(\sigma^\prime)]<br /> =<br /> 2 \delta^\prime(\sigma,\sigma^\prime)<br /> \left(<br /> Y(\sigma)Y(\sigma^\prime)<br /> +<br /> Y(\sigma^\prime)Y(\sigma)<br /> \right)<br />
<br /> =<br /> 4 \delta^\prime(\sigma,\sigma^\prime)<br /> Y(\sigma)Y(\sigma^\prime)<br /> -2<br /> \delta^\prime(\sigma,\sigma^\prime)<br /> \delta^\prime(\sigma,\sigma^\prime)<br />
Here the first term is the one that gives the usual Virasoro algebra, while the second term comes from re-ordering and should be related to the anomaly. It is however not well defined when written this way. Depending on how you decide to deal with this term it might look like 0 when integrated over or like infinity.

So this does not make sense. The above commutator has be be regulated by introdicing appropriate smearing functions. After computing the regularized commutator these smearing functions can be taken to be delta-functions again. This is the functional version of what Distler proposed to do, namely to introduce a cutoff in the summation over modes in the Virasoro generators. And it should produce an honest and well defined term which is an anomaly.

So the claim is that Thiemann is using the naive quantization where you don't see the anomaly, even though it is really there.

 

[ranyart]

Originally posted by meteor
Seems that the speciality of Mr. Vagenas are 2D black holes. Just see his list of publications.
I will call him tomorrow and will ask him about QNM. And will invite also him to join the forum!

Yes! I have a few papers on my computer which I cannot axcess(PDF-adobe fails).

It is very interesting that dimensionally BHs are not only holding Galaxies intact(Spacetime), but there is increasing evidence that they are Inter-Dimensional crossroads.

QNMs are where a certain dimensional energies meets another ..different dimensional Energy.

 

[marcus]

Originally posted by Urs
...Why does Distler say that's not well-defined? Because it is naively multiplying distributions, which is not a well defined operation. Since the Y(\sigma) are technically operator-valued distributions already their product Y(\sigma)Y(\sigma) is not well defined. This becomes more obvious when one ignores this and caclulates the commutator
<br /> [Y(\sigma)Y(\sigma),Y(\sigma^\prime)Y(\sigma^\prime)]<br /> =<br /> 2 \delta^\prime(\sigma,\sigma^\prime)<br /> \left(<br /> Y(\sigma)Y(\sigma^\prime)<br /> +<br /> Y(\sigma^\prime)Y(\sigma)<br /> \right)<br />
<br /> =<br /> 4 \delta^\prime(\sigma,\sigma^\prime)<br /> Y(\sigma)Y(\sigma^\prime)<br /> -2<br /> \delta^\prime(\sigma,\sigma^\prime)<br /> \delta^\prime(\sigma,\sigma^\prime)<br />
Here the first term is the one that gives the usual Virasoro algebra, while the second term comes from re-ordering and should be related to the anomaly. It is however not well defined when written this way. Depending on how you decide to deal with this term it might look like 0 when integrated over or like infinity.
...

Urs, thanks for taking the trouble to
move things over here to PF and write it out
in this level of detail. I much appreciate
and suspect that others will too!

Also this point about not being able to directly
multiply distributions---this is an understandable
specific criticism. At first sight I cannot agree
or disagree but it is the kind of error that
people (even good mathematicians) can easily make
now and then in a long paper. The distributions look
like ordinary (true) functions and one does not watch
and then one treats them as if they were!

I hope more specifics like this will come to PF
where I will get a chance to see. (still do not have
Mozilla)

 

[marcus]

Originally posted by Urs
...Thiemann claims in these paragraphs on p. 20 that he can construct a quantization of the classical Poisson algebra of the string which completely preserves the Virasoro algebra without adding a central extension. Distler says that that's not possible in general. This should not be a matter of approaches or opninions.

The point is that Thiemann in passing claims that he can find a quantization of the Poisson algebra such that
\alpha(W(Y_\pm)) = W(\alpha(Y_\pm))
holds true on the quantum level,...

I see where he says that (in passing) at the beginning of the third paragraph on page 20.


And then later in the same paragraph he says:

"Physically this representation property amounts to an anomaly-free implementation of both the local group and the global symmetry group..."

So (while I cannot yet form an independent judgement that takes everything into account) I can at least see where Jacques point connects to the paper. This is a real help!

Offhand I would say it sounds like the sort of thing TT would like to be told about, and I hope this happens.

 

[selfAdjoint]

Urs, I can't believe Thiemann made a mistake about multiplying distributions. He's an authority in this field and has a book coming out on it. Marcus, in addition to that book, he cites Rovelli's forthcoming book Quantum Gravity to to support his construction. I'm going to dig into that, but as you're now the expert on Rovelli maybe you can look too to see how he does this algebra transformation.

 

[Urs]

regulating products of distributions

Hi -

I have now checked the calculation that Jacques Disller told me to do. Indeed, when one regulates the non-normal-ordered Virasoro generators that Thomas Thiemann is using, computes their commutators and then removes the regulator afterwards, one picks up an anomaly. Jacque's point is that without doing this regularization one is dealing with ill-defined quantities. I think that's uncontroversial, I have discussed it in my previous message.

Thiemann is in contact with us. I am going to ask him about this issue.

 

[marcus]

Originally posted by Urs


Thiemann is in contact with us. I am going to ask him about this issue.

that sounds like a constructive thing to do!
(i'm not trying to guess if there is an error or not, either Distler or Thiemann could be mistaken I suppose, but asking about it is a helpful initiative)
I am glad to know you are in contact with Thiemann.

 

[Urs]

Thiemann says that he knows about the issue raised by Distler! But he also says that he thinks that it does not affect anything he did.

He says that he does not represent the constraints on the Hilbert space but just imposes the condition
<br /> U(T)W(Y) \Omega_\omega<br /> =<br /> W(\alpha(Y))\Omega_\omega<br />
by hand.

I thought all along that this should follow from his quantization of the constraints, but apparently that's the definition of the action of the U-operators.

So the issue here is not one of mathematical correctness, after all.

But is it physically viable to define the anomaly away like this?

Thomas says he will join the Coffee Table discussion. Let's see what happens.

 

[marcus]

Originally posted by selfAdjoint
Marcus, in addition to that book, he cites Rovelli's forthcoming book Quantum Gravity to to support his construction...

it would save me some fumbling if you point me at a particular page of TT where he uses something from Rovelli's book.

concerning this fracas about the anomaly,
I can't second guess but I do know
that math papers when they first come out are checked over and
nitpicked by friends and colleagues, and even good people
can make errors and really benefit from other people
scrutinizing their work. I hope TT is getting a lot of this
feedback and that Distler is only one of many looking for nits.

 

[marcus]

Originally posted by Urs
Thiemann says that he knows about the issue raised by Distler! But he also says that he thinks that it does not affect anything he did.

...

Thomas says he will join the Coffee Table discussion. Let's see what happens.

great news! glad to hear it.

 

[Urs]

This lastes comment by Distler looks important to me.

 

[jeff]

Originally posted by Urs
Thiemann...just imposes the condition
<br /> U(T)W(Y) \Omega_\omega<br /> =<br /> W(\alpha(Y))\Omega_\omega<br />
by hand.

I thought all along that this should follow from his quantization of the constraints, but apparently that's the definition of the action of the U-operators.

So the issue here is not one of mathematical correctness, after all.

Did I not try to warn you about this (albeit in retrospect using somewhat sloppy phrasing) much earlier in the thread?:

Originally posted by jeff
Urs,

Did you notice thiemann's declaration on p3 that the reps considered in the paper have been taken by definition to be anomaly-free right out of the box? Thus it may be more the choice of representation than the method of quantization that's at the heart of this.

Originally posted by Urs
But is it physically viable to define the anomaly away like this?

This is yet another example of the contrived approach to theory construction so characteristic of the whole LQG program.