Amazing bid by Thiemann to absorb string theory into LQG

[marcus]

Originally posted by Urs


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

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

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

 

[marcus]

Originally posted by Urs
...

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

...

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

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

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

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

Thanks in advance

 

[eigenguy]

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

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

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

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

 

[eigenguy]

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

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

 

[selfAdjoint]

Not

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

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

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

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

 

[marcus]

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

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

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

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

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

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

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

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

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

 

[selfAdjoint]

GNS and the Symmetry Group

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

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

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

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

 

[marcus]

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

 

[marcus]

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

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

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

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

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

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

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

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

 

[eigenguy]

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

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

but it's a valid way within mathematical physics.

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

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

What would cause you to abandon your interest in LQG?

 

[Urs]

Marcus,

have you followed the disucssion over at the Coffee Table? Thomas Thiemann himself confirmed that in LQG the spatial diffeo constraints are imposed in the same way that he imposes the Virasoro constraints in his 'LQG-string' paper. This is precisely the step which is non-standard, as Distler has made quite clear, because it does follow neither from path intergal nor from canonical Dirac quantization but instead conjures up a new principle which says that it is fine to find any rep of the classical symmetry group on the quantum theory's Hilbert space and demand that physical states be invariant under this group.


selfAdjoint,

you write

his quantization is per the Giulini-Marolf paper.

No, it is not. Giulini-Marolf require a rep of the quantum first class constraints which is anomaly free. Thiemann has no rep at all of the first class constraints and cannot even in principle get one that is anomaly free. Instead of Giulini-Marolf what he does is group averaging with a group of operators that does not follow from standard quantization in any way.

This is not controversial, I think, because Thiemann himself confirmed repeatedly at the Coffee Table that this is what he is doing. What is controversial is only whether this 'new' method could have something to do with physics.

Thomas Thiemann says at the Coffee Table that he thinks that only experiment can tell whether his form of quantization is correct or the standard one. I can accept this, but we then have to be quite clear on what this means: This means that Thomas Thiemann is proposing a modification of the quantum principle (at the Planck scale). This means that LQG is not canonical quantization, but a new kind of quantization.

I am the last one to embrace this conclusion, but it is what Thomas Thiemann is saying.

 

[marcus]

Interest is a personal matter

Originally posted by eigenguy

What would cause you to abandon your interest in LQG?

Eating too much of it, like chocolate (if I may be allowed to reply
Interest or non-interest in a developing line of research is a matter of personal taste.
I do not ask you Eigenguy and/or Jeff to justify NOT being interested.
Perhaps you are interested in String---well, I do not ask you to explain this (although I am not interested in String myself)

this is diverting a physics discussion to argument about personality issues

"keep it about the physics"

 

[eigenguy]

I have a question for urs.

What is your feeling about the view that any attempt to quantize GR directly is naive because the assumption that the einstein-hilbert action isn't just the leading term of a more general effective theory is naive?

 

[marcus]

Originally posted by Urs
Marcus,

have you followed the disucssion over at the Coffee Table? Thomas Thiemann himself confirmed that in LQG the spatial diffeo constraints are imposed in the same way that he imposes ...

Urs, so nice to hear from you! I am glad you are concerned with an issue that is purely about Loop Gravity, in isolation from String.

That is, you fear something might be wrong in the development of LQG proper, not just in this particular analysis of a string within a LQG model by Thiemann.

The thing to do, I feel sure, is to learn what is exactly that we are talking about.
Regardless of what you understood Thiemann to have said in some discussion, we should find in his "Lectures on LQG" where what you are worried about happens. Or in some other textbook.

It is the old idea of actually looking in the horses mouth to count the teeth.

Early in the thread I gave you a reference to a page in Rovelli textbook where the spatio diffeomorphisms are imposed.
Two network states are made equivalent if they differ by a diffeo.
Thus the states become "equivalence classes" and equivalence classes of network states are knot states. It is a common algebraic procedure to factor something down to equivalence classes. This is all familiar to you! Anyway, I referred to that part of Rovelli very early on in the thread. Unless I misunderstand your question, you can see how it is done there (I think around pages 170-173) and see if you like it or not!

I would be delighted to know if you do not like how Rovelli takes care of invariance under spatial diffeo! This would be a choice topic of discussion.

Also it seems to me very clean and easy to understand. He does it quickly without much notation and trouble--then you can say this is kosher or not-kosher, traditional or not-traditional, according to how you think.

Since Rovelli is one of the main Loop Gravity textbooks that would
be reasonable basis for general statements about how things are done.
If you think it is bad----or if I misunderstand your question--I would very much like to know.

BTW you asked if I followed TT and JD on the other board, no because I don't want to change browsers and its very hard to read with the Microsoft browser (no symbols, fine print, as we discussed). But this issue is much broader----how diffeomorphism invariance (a basic feature of General Relativity) is handled in LQG---in particular how LQG handles spatial diffeos. We should be clear about whether or not it's kosher quantum theory by your standards.

 

[eigenguy]

Originally posted by marcus
I do not ask you Eigenguy and/or Jeff

Just for the record, I don't necessarily agree with everything jeff thinks and I'm perfectly capable of forming my own ideas without anyone else's help.

 

[selfAdjoint]

I believe I have some surpising information about this issue:

No, it is not. Giulini-Marolf require a rep of the quantum first class constraints which is anomaly free. Thiemann has no rep at all of the first class constraints and cannot even in principle get one that is anomaly free. Instead of Giulini-Marolf what he does is group averaging with a group of operators that does not follow from standard quantization in any way.

It is this. Thiemann does not use Giulini-Marolf or group averaging in his actual construction in this paper! He does use GNS intensively. But nearly everything he does in his specific example (which is the only quantization he does, as opposed to talks about) is careful manipulation of Hilbert space issues. For example he does not actually exponentiate the Pohlmayer charges; he regulates them and develops a specific expression in terms of the regulator that show the Pohlmayer charges as functions of the W's.

Maybe his further reaserches on the representation theory of his algebra will involve these issues, but the construction of a string quantization which he actually exhibits in this paper does not.

 

[Urs]

Marcus,

the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.

 

[Urs]

Hi selfAdjoint -

yes, he does a couple of things on this Hilbert space which are probably all fine and dandy. But what he does not do is impose the constraints in the usual way. This is particular implies that the Pohlmeyer charges do not commute with the usual constraints. They are merely invariant under the classical symmetry group action that Thomas Thiemann is using.

I have mentioned a way around this problem: Use the classical DDF invariants instead of the Pohlmeyer charges. Then quantize correctly, find the anomaly in the longitudinaly DDF invariants, include the logarithmic counter term to cancel these and - voila - one is left with the standard string! :-)

 

[Urs]

Einstein-Hilbert as leading order term

Hi eigenguy,

in the discussion with Jacques Distlet I was reminded of a simple fact which I apparently did not sufficiently appreciate before: There is no canonical quantization in principle of the ADM constraints of the EH action. LQG only avoids/ignores this no-go-fact by quantizing only the Hamiltonian constraint and imposing the classical diffeo constraints by hand. So if I were to believe that gravity has to have a canonical quantization, then I would hope that EH is only a leading order term, because otherwise I'd have to give up immediately.

To me this insight is a completely new perspective on the old discussion about what is conservative about LQG and about strings.

But, personally, I don't know if I hope that gravity can be quantized canonically. I feel much more comfortable with quantizing really small things than really big ones! :-) I find it much more trustworthy to apply quantization to a tiny string than to the entire universe. We are more likely to get the former right, I'd say.

On the other hand, string theory of course has the promise of giving us tools to quantize the entire universe in one stroke by means of Matrix Theory.

 

[marcus]

Originally posted by Urs
Marcus,

the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.

Great! Thanks for looking it up in a standard LQG source Urs. I have that article printed out in a pile of papers
by my desk and I will look it up and see what you mean.

 

[Urs]

Marcus -

the analogous problem in Rovelli's book http://www.cpt.univ-mrs.fr/~rovelli/book.pdf is indeed on pp. 170. Consider a diffeomorphism \phi that leaves orientation and ordering of links of some graph \Gamma invariant. Then according to the first in-line equation in section 6.4 Rovelli sets
<br /> U_\phi|\Gamma\rangle<br /> =<br /> |\phi\Gamma\rangle<br /> \,.<br />
This is the precise analogue of equation (6.25) in Thiemann's paper. And this is the problem, because this relation only holds because the U_\phi are constructed in a way to satisfy precisely this relations. That's certainly possible, the operators U_\phi undoubtly exist. What is problematic is that nothing in the world so far tells us that we should demand quantum states to be invariant under the classical gauge group induced by these U_\phi, which is however the content of equation (6.43) in Rovelli's book.

The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation \langle \psi|\pi(C)|\psi\ranfgle = 0.

In the last paragraph on page 34 of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf Rovelli seems to claim that the latter is possible. This is in contradiction to what Thomas Thiemann said at the Coffee Table.

 

[eigenguy]

Originally posted by Urs
There is no canonical quantization in principle of the ADM constraints of the EH action.

Whoa! Is this widely known?

Originally posted by Urs
LQG only avoids/ignores this no-go-fact by quantizing only the Hamiltonian constraint...

...which seems to be virtually impossible to solve, while the constraints that have been solved are imposed...

Originally posted by Urs
... by hand.

There is a message in here somewhere.

 

[selfAdjoint]

Equation 6.25

Urs,

Thiemann's equation 6.25,
U_{\omega}(g)\pi_{\omega}(b)\Omega_{\omega} = \pi_{\omega}(\alpha_g(b))\Omega_{\omega}

Comes directly from his corollary 4.1 (which I commented on above):

U_{\omega}\pi_{\omega} := \pi_{\omega}(\alpha_g (a))\Omega

Which is, he claims, given to him by GNS, and modulo my doubts about his handling of the group, this is true according to Haag. He may have pulled part of Corollary 4.1 out of the blue but that is not true of this unitary relationship.

If his GNS is kosher, then this U can be assumed to exist as part of the construction. In that case to reject it as not proper quantum mechanics is to reject GNS and the whole enterprise of algebraic quantum field theory too.

 

[marcus]

Now Urs has said what he thinks is wrong with LQG and what, in his view, invalidates the paper under discussion. And he refers me to what are, for me, standard texts of LQG (rovelli 1998 livingreviews and rovelli 2004 "Quantum Gravity" book)

I am very content with this. I don't have to try to say whether Urs is wrong or right or whether Rovelli is right or wrong. The important thing is Urs has said what he thinks is wrong and I can study it and give it the appropriate consideration. This is a big benefit and improvement!

So some thanks are due to both of you selfAdjoint and Urs for steering the rowboat of this conversation thru the rough waters
of unfriendly argument and finally into some calm understanding!
I am impressed with the patience shown by both of you! It is even
surprising me that we didnt tip over and all sink at some point.

 

[eigenguy]

Originally posted by selfAdjoint
Urs,

Thiemann's equation 6.25,
U_{\omega}(g)\pi_{\omega}(b)\Omega_{\omega} = \pi_{\omega}(\alpha_g(b))\Omega_{\omega}

Comes directly from his corollary 4.1 (which I commented on above):

U_{\omega}\pi_{\omega} := \pi_{\omega}(\alpha_g (a))\Omega

Which is, he claims, given to him by GNS, and modulo my doubts about his handling of the group, this is true according to Haag. He may have pulled part of Corollary 4.1 out of the blue but that is not true of this unitary relationship.

If his GNS is kosher, then this U can be assumed to exist as part of the construction. In that case to reject it as not proper quantum mechanics is to reject GNS and the whole enterprise of algebraic quantum field theory too.

Maybe GNS allows more than the usual quantum theories, with some being more viable physically than others. In thiemann's implementation of it, the quantum states are assumed, in urs's words, "invariant under the classical gauge group induced by these U_{\omega}.

 

[Urs]

selfAdjoint,

yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.

Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.

The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!

 

[marcus]

Originally posted by Urs
Marcus,

the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.

Incredibly enough this (33) was the equation I was trying to tell you about early in the thread, and now we have come round to it again.
To be a little finicky about language it does not say to "fillet" or take out the invariant states.
It says to take the quotient Hilbert space by a certain eqivalence relation.

The vectors in the new vector space are sets of vectors from the old space.

the vectors in the new space (of physical states or H_Diff)
are equivalence classes of old vectors, under the operation of the group Diff(M).

We have all met this in algebra countless times, including for some even the first time they met the complex numbers---which some books define as a quotient of a polynomial ring.

In the new book "Quantum Gravity" Rovelli uses an extended Diff group and gets a reduced quotient that happens to be separable. That was why I was talking about separable earlier in thread. But this does not matter. I don't want to talk about that again!

I am just glad we have finally met at this (equivalence class quotient) algebraic definition of the state space.

I will think more about your objection to it.

 

[marcus]

when you take diffeo equivalence classes of networks you get abstract knots

so Urs, on the page 34 you pointed me to, I see

"The second reason [that diffeo invariance is good for the theory] is that H_Diff turns out to have a natural basis labeled by knots.
...an equivalence class of spin networks under diffeomorphism...
..is characterized by its "abstract" graph (defined only by the adjacency relations between links and nodes), by the coloring, and by its knotting and linking properties, as in knot theory.
Thus, the physical quantum gravity states of the gravitational field turn out to be essentially classified by knot theory"

think how heart-warming this could sound to a mathematician.


So the spatial diffeo invariance "has" to be handled this way because of the nice topological and algebraic outcome that the states of the grav. field are a hilbertspace of knots.
and quantum superpositions of knots.

But don't give up on a real red-blooded constraint too!
There is still a diffeomorphism constraint coming later.
We only dealt with the spatial diffeo invariance. there is still more. so a constraint will be imposed later-----gauss, diffeo, hamiltonian. Three of them.

I am not saying you should agree or be happy. I am just kind of sketching the outlines of how I see your objections.
You point me to (33) which defines
H_Diff
and you don't like it. I will try to digest and understand this.
Again thanks, and in advance for any more explanation of what you find nonstandard!

 

[ranyart]

Originally posted by marcus
Now Urs has said what he thinks is wrong with LQG and what, in his view, invalidates the paper under discussion. And he refers me to what are, for me, standard texts of LQG (rovelli 1998 livingreviews and rovelli 2004 "Quantum Gravity" book)

I am very content with this. I don't have to try to say whether Urs is wrong or right or whether Rovelli is right or wrong. The important thing is Urs has said what he thinks is wrong and I can study it and give it the appropriate consideration. This is a big benefit and improvement!

So some thanks are due to both of you selfAdjoint and Urs for steering the rowboat of this conversation thru the rough waters
of unfriendly argument and finally into some calm understanding!
I am impressed with the patience shown by both of you! It is even
surprising me that we didnt tip over and all sink at some point.

Marcus a paper by Marolf and Rovelli from sometime ago may have a baring on this thread:http://uk.arxiv.org/abs/gr-qc?0203056

Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights

 

[selfAdjoint]

Originally posted by Urs
selfAdjoint,

yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.

Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.

The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!

Urs, I'm going to quit this discussion because we are talking past each other. Thiemann has two things, after the dust settles: he has a very persuasive model of the string, laid out in his section 6.2, and he has the classic results of "local quantum physics" as Haag puts it. His achievement is to apply the latter to the former. Now you say this is not what you are told to do by standard quantum theory. So much the worse for standard quantum theory. Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.

So I can't convince you and I'm afraid you can't convince me.

 

[marcus]

Originally posted by ranyart
Marcus a paper by Marolf and Rovelli from sometime ago may have a bearing on this thread:http://uk.arxiv.org/abs/gr-qc?0203056

Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights

you know ranyart though I don't have the right to judge I have to say I think Rovelli's thoughts about quantum theory are among the most perceptive and sophisticated--especially in connection with relativity. he thinks about situations and measurments in an extremely concrete fashion.

I keep seeing Marolf's name, maybe he is another one who really thinks instead of just operating at a symbolic level.

Rovelli has a section, pages 62-68 in the book, where he talks about
"Physical coordinates and GPS observables"
it uses the Global Positioning Satellite system to illustrate something about general relativity. I haven't grasped it. have you looked at it?

Anyway thanks for the link.

what it means to me relative to this thread is the article you give is further evidence that Rovelli does not just quantize by rote, or by accepted procedures. He is one of the more philosophically astute people in knowing what is going on when he quantizes something. (IMHO of course

 

[arivero]

invariance of diathige and trope

Originally posted by Urs

The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation \langle \psi|\pi(C)|\psi\ranfgle = 0.

Let me to notice the historical remark in Rovelli Living Review:

The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin [182, 163, 183, 184] to ``change basis in the Hilbert space of the theory''
...The immediate results of the loop representation were two: The diffeomorphism constraint was completely solved by knot states (loop functionals that depend only on the knotting of the loops), making earlier suggestions by Smolin on the role of knot theory in quantum gravity [195] concrete; and (suitable [184, 196] extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.

It seems there are so sure of his technique that the review articles already forget to relate it to the constrains.

On other hand, Thiemann Hamiltonian constrain is a later development, dated 1996.

 

[Urs]

Yes, it's kind of strange. The quantum constraints are not even mentioned anymore when it comes to 'solving' diffeo-invariance in LQG reviews. I believe this is a trap. At least people should be well aware that at this point standard canonical quantization is abandoned. Luckiliy, this has become clear now in the simpler example of quantization of the Nambu-Goto action by Thomas Thiemann.

 

[eigenguy]

I know that the following is implicit in what urs said, but it's worth pointing out that on the issue of whether gravity should be quantized, dirac said that it would be hard to see how a theory that treats gravity classically and other interactions quantumly could be consistent. For the same reason, it seems reasonable that gravity should be quantized in the same way as other interactions as well, making LQG seem even less plausible.

 

[Urs]

I have now contacted A. Ashtekar and H. Nicolai. Let's see if they have something to say about the LQG-string.

 

[marcus]

Originally posted by Urs
I have now contacted A. Ashtekar and H. Nicolai. Let's see if they have something to say about the LQG-string.

Bravo Urs! This is a great thread, we are getting our money's worth, so to speak. must again express thanks to you for your carefulness, open-mindedness, patience etc.

whatever they may say, it is only to the good that they answer---but I do hope they respond in timely manner!

 

[Urs]

Hi Marcus,

yes, but they might answer at the Coffee Table! :-) So get a copy of Mozilla. It's free, it's easy, it does not not interfere with anything and Mozilla is more politically correct than MSIE, anyway. ;-)

BTW, anyone who is interested in following the discussion at the Coffee Table but wants to be informed automatically about new comments should download an "RSS News Aggregator" such as Sharp Reader. This allows you to read the Coffee Table just like any usenet newsgroup, plus some extras. Just download, install, and then drag-and-drop the boxes that sit under the headline "Syndicate" at the Coffee Table entry page into the SharpReader window.

 

[eigenguy]

Hi urs,

You need to edit the link to sharp reader.

 

[lethe]

Originally posted by Urs

BTW, anyone who is interested in following the discussion at the Coffee Table but wants to be informed automatically about new comments should download an "RSS News Aggregator" such as http://http://www.sharpreader.net/ . This allows you to read the Coffee Table just like any usenet newsgroup, plus some extras.

Urs-

is there a RSS news reader that supports MathML? i didn't try sharpreader, since i don't run windows. does it support MathML?

 

[Urs]

No, unfortunately I couldn't make SharpReader display MathML. I use the reader to stay in touch with new comments and switch to Mozilla when I need to read equations. That's not the way it should be, of course.

There should be "RSS News Aggregators" for all kinds of operating systems. I'll ask Jacques Distler. He himself is using MacOS.

 

[eigenguy]

Originally posted by selfAdjoint
Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.

From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.

 

[selfAdjoint]

Originally posted by eigenguy
From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.

And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.

 

[eigenguy]

Originally posted by selfAdjoint
And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.

I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.

 

[lethe]

Originally posted by eigenguy
I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.

um, actually, it does. the claymath problem is specifically about QFT.

 

[eigenguy]

Maybe LQG wasn't given a fair shake

Something just occurred to me. Suppose it turns out that LQG is wrong for the reasons that urs discovered. Wouldn't it stand to reason that if other physicists had given LQG a serious look, they would have seen this a long time ago? I believe that feynman said the physicists job is to prove themselves wrong as quickly as possible (Of course, from this point of view, the LQG camp would deserve most of the blame).

 

[eigenguy]

Originally posted by lethe
um, actually, it does. the claymath problem is specifically about QFT.

You will find the following description http://www.claymath.org/millennium/Yang-Mills_Theory/ :

Yang-Mills and Mass Gap

The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory [note the word "field" is not used here or anywhere else in this paragraph] is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

Clearly, no assumption has been, nor should be, made about what the solution will look like.

 

[lethe]

Originally posted by eigenguy

Clearly, no assumption has been, nor should be, made about what the solution will look like.

ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!

 

[eigenguy]

Originally posted by lethe
ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!

Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

 

[lethe]

Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

OK, fine, string theory allows nonabelian gauge theories. but Yang-Mills theory is not string theory, it is a quantum field theory. The positive mass gap conjecture is not about string theory or some other as-yet-undetermined theory, it is about Yang-Mills theory.

 

[selfAdjoint]

Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

Eigen, I am afraid you've got your foot in by your tonsils. The words Yang-Mills, followed by the word theory, refer to a class of Quantum Field Theories. If you want to refer to Yang-Mills symmetry, you say Yang-Mills symmetry. See for example

Peskin & schoeder section 15.2, the field theory associated with a non-commuting local symmetry is termed a non-Abelian gauge theory.

Ryder, section 3.5 The Yang-Mills field.

Both P & S and Ryder have in their indices, Yang-Mills theory, see non-Abelian gauge theory.

Yang-Mills theory was quantized by Veltzmann & 'tHooft, becoming thereby a Quantum Field Theory. It is this theory that is usually referred to as Y-M theory.