Amazing bid by Thiemann to absorb string theory into LQG

[marcus]

this just out
http://arxiv.org/hep-th/0401172
quoting from the abstracts:


The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
Authors: Thomas Thiemann
Comments: 46 p.

"We combine

I. background independent Loop Quantum Gravity (LQG) quantization techniques,

II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and

III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space.

While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincare invariance and 8. without picking up UV divergences.


The existence of this stable solution is, on the one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string.

Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem.

On the other hand, if such solutions are found, then this would prove that neither a critical dimension (D=10,11,26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic.

The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces."

 

[marcus]

From Thiemann's conclusions paragraph

"...Let us conclude by stressing once more that the claim of this paper is certainly not to have found a full solution of string theory. Rather, we wanted to point out two things:

First of all, that canonical and algebraic methods can be fruitfully combined in order to analyze the string.

Secondly, that the specific Fock representation that one always uses in string theory is by far not the end of the story: The invariant representation theory of the quantum string, as we have defined it here, is presumably very rich and we encourage string theorists to study the string from the algebraic perspective and to systematically analyze all its representations.

This might lead to a natural resolution of major current puzzles in string theory,

such as the cosmological constant puzzle [38] (120 orders of magnitude too large),

the tachyon condensation puzzle [39] (unstable bosonic string vacua),

the vacuum degeneracy puzzle [40] (huge moduli space of vacua upon compactification),

the phenomeology puzzle [41] (so far the standard model has not been found among all possible string vacua of the five superstring theories currently defined, even when including D – branes)

and finally the puzzle of proving perturbative finiteness beyond two loops [42].

See the beautiful review [43] for a status report on these issues.

Namely, it might be that there are much simpler representations of the string, especially in lower dimensions and possibly without supersymmetry, which avoid or simplify all or some these problems.


While this would be attractive, the existence of new, phenomenologically sensible representations would demonstrate that D = 10, 11, 26 dimensions, supersymmetry and the matter content of the world are tied to a specific representation of string theory and hence would not be a prediction in this sense...."
---------end of exerpt--------

He seems to be getting rid of the rolled-up dimensions. taking string back down to "phenomenologically sensible" FOUR spacetime dimensions. Ye gods. He says that all those extra dimensions and supersymmetry are not, after all, needed for string theory or necessary predictions of the theory, because he sees a chance to make the theory work without all that extra baggage, in a background independent context using Quantum Gravity tools.

BTW in his references he gives 2004 as the publication date for Rovelli's book. He says "Quantum Gravity, Cambridge University Press 2004." I suspected it would be out this year since Rovelli has sent in the MS and how long can it take?

Lots happening.

 

[jeff]

Originally posted by marcus
this just out
http://arxiv.org/hep-th/0401172


The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
Authors: Thomas Thiemann
Comments: 46 p.

We combine

I. background independent Loop Quantum Gravity (LQG) quantization techniques,

II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and

III. the theory of integrable systems resulting in the invariant Pohlmeyer Charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space.

While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincare invariance and 8. without picking up UV divergences.


The existence of this stable solution is, on the one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string.

Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem.

On the other hand, if such solutions are found, then this would prove that neither a critical dimension (D=10,11,26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic.

The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper we treat the more complicated case of curved target spaces.

Originally posted by marcus
"...Let us conclude by stressing once more that the claim of this paper is certainly not to have found a full solution of string theory. Rather, we wanted to point out two things:

First of all, that canonical and algebraic methods can be fruitfully combined in order to analyze the string.

Secondly, that the specific Fock representation that one always uses in string theory is by far not the end of the story: The invariant representation theory of the quantum string, as we have defined it here, is presumably very rich and we encourage string theorists to study the string from the algebraic perspective and to systematically analyze all its representations.

This might lead to a natural resolution of major current puzzles in string theory,

such as the cosmological constant puzzle [38] (120 orders of magnitude too large),

the tachyon condensation puzzle [39] (unstable bosonic string vacua),

the vacuum degeneracy puzzle [40] (huge moduli space of vacua upon compactification),

the phenomeology puzzle [41] (so far the standard model has not been found among all possible string vacua of the five superstring theories currently defined, even when including D – branes)

and finally the puzzle of proving perturbative finiteness beyond two loops [42].

See the beautiful review [43] for a status report on these issues.

Namely, it might be that there are much simpler representations of the string, especially in lower dimensions and possibly without supersymmetry, which avoid or simplify all or some these problems.


While this would be attractive, the existence of new, phenomenologically sensible representations would demonstrate that D = 10, 11, 26 dimensions, supersymmetry and the matter content of the world are tied to a specific representation of string theory and hence would not be a prediction in this sense...."
---------end of exerpt--------

He seems to be getting rid of the rolled-up dimensions. taking string back down to "phenomenologically sensible" FOUR spacetime dimensions. Ye gods. He says that all those extra dimensions and supersymmetry are not, after all, needed for string theory or necessary predictions of the theory, because he sees a chance to make the theory work without all that extra baggage, in a background independent context using Quantum Gravity tools.

 

[Urs]

Hi everybody -

Marcus wrote:

We combine

Please let us know who "we" is. Thanks! :-)

I. background independent Loop Quantum Gravity (LQG) quantization techniques,

I am not sure if worldsheet background independence is a new feature here. Since one gets the Virasoro constraints from the Nambu-Goto action just as well as from the Polyakov action and since the quantization of the resulting constraints does not introduce any further background, Thiemann's quantization looks just as worldsheet background indepent to me as the usual Fock space quantization.

Even though in his introduction Thiemann says that usually one jumps from the NG action straight to the Polyakov action, this true only for the NSR F-string, mostly. The Green-Schwarz superstring for instance is always formulated in the NG form, as is the D-string, i.e. the D1-brane.

II. the mathematically rigorous framework of Algebraic Quantum Field Theory (AQFT) and

I don't see yet how Thiemann's approach to quantize the single non-interacting worldsheet is more rigorous than the usual CFT approach. 2d CFTs are also rigorously defined. What is not rigorously defined is the (string) perturbation series. But Thiemann's paper does not address this issue.

Moreover, these new representations could solve some of the major puzzles of string theory such as the cosmological constant problem.

Are there any further hints for this claim except that there are open problems in string theory and that any new approach might offer new answers?

While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge new, non -- trivial solution to the representation problem. This solution exists 1. for any target space dimension, 2. for Minkowski signature of the target space, 3. without tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies (zero central charge), 7. while preserving manifest target space Poincare invariance and 8. without picking up UV divergences.

The tachyon seems to disappear - but with it disappear many other features of the ordinary string quantization, such as the rest of the mass spectrum.

I would like to point out that the tachyon is not an inconsistency of the bosonic string, but merely an indication of an instability of its background. For instance the tachyon of the open bosonic string is very well understood as to be due to the instability of the space-filling uncharged D25 brane. The depth of the tachyon potential is precisely the energy density of this unstable brane and while the tachyon rolls down its potential well the brane decays. This way open bosonic string theory decays to closed bosonic string theory and the fact that this process can be understood in terms of single strings, branes, and string field theory shows that it gives a consistent physical picture.

This is analogous to all the tachyons that appear when superstrings stretch between brane-antibrane pairs. They are not a problem in the theory but a physical feature: The brane and antibrane pairs annihilate as the tachyons roll down their potential.

In fact, there is an interesting flavor of string cosmology, where the tachyon serves as the inflaton field and the high initial value of the inflaton is nicely explained by the collision of a brane-antibrane pair. (The initially parabolic potential of the tachyon evolves into a Mexican-Hat type potential as the branes approach.)

My point is that the disappearance of the tachyon is not a advatage per se. Even the closed bosonic string (in the usual quantization) might not be inconistent at all, but might decay into a conistent 2d bosonic string or 10d superstring. See this discussion at the String Coffee Table for more details.


Now let me ask some random questions:

Can you recover the usual Fock representation within the GNS construction framework?

(I guess not, since no anomaly will ever show up in this framework, right?)

If yes, is there anything (apart from the obvious differences) that distinguishes the Fock representation from other representations obtainable by doing the GNS construction ?

What happens when Thiemann's approach is applied to the Polyakov action? A priori this looks like the case more closely related to LQG, since it is the Polyakov action which describes 1+1 dimensional gravity on the worldsheet.

There are several independent ways to arrive at the usual quantization of the F-string. There is the old covariant approach, the light-cone quantization, the BRST quantization, the path-integral quantization. These quantization schemes have superficially many differences, and yet they all give the same result - which disagrees with Thiemann's result. How could that be understood? What is going on here?

This is maybe the most interesting question. H. Nicolai initiated the recent research into LQG quantization of 1+1 dim gravity coupled to scalar matter (at the "Strings meet Loops" symposium in Potsdam last year) by asking if LQG methods can reproduce the very well established results concerning the quantization of the Polyakov action. The idea is that 1+1 dimensions may be an accesible laboratory for understanding how the LQG approach is different from other quantization approaches.

And finally: When Thiemann's approach is generalized to the fermionic string, might it be of any help to know that the constraints of the superstring are deformed exterior (co)derivates on the form bundle over the configuration space of the string and that all massless bosonic backgrounds manifest themselves as deformations of these exterior operators as described in this paper? :-)


Urs

 

[selfAdjoint]

"We" is Thiemann. Marcus is quoting from the abstract as you will see if you check the link.

And although the background free worldsheet is indeed a product of traditional (how easily that word slips off the tongue today!) string theory, the analysis of the world sheet by LQG methods is novel. These are complete and rigorous in 2 dimensions, IIRC.

Finally, as Thiemann repeatedly states, this paper is as yet an incomplete presentation of string theory. No tachyons, yes, but also no gravitons, yet.

Nevertheless, what a thrilling breakthrough! And based on overlooked results from 1982!

 

[marcus]

Originally posted by Urs
Hi everybody -

Marcus wrote:



Please let us know who "we" is. Thanks! :-)

Hi Urs, that whole post is a quote from Thiemann, the abstract(s) of his paper. So you have to ask him who "we" is, but normally it is the author's (or the king's royal) pronoun. I just edited quote marks in the post to make it more obvious that it was from the arxiv link I gave.

You mentioned Nicolai and the October 2003 Berlin symposium
and indeed Thiemann says in his acknowledgments that it was partly at Hermann Nicolai's urging that he pursued this research.

So this can be seen as the program of Nicolai (the string theorist who co-hosted the October "string meet loop" symposium)

 

[marcus]

BTW Urs, I remember you went to that "string meet loop"
symposium and wrote a message to PF in early November when
you had just gotten back from it.

or anyway that is how I remember it---that was you?
unless I'm confusing you with someone else, you said you were still catching up on sleep.

You must know all these people, Hermann Nicolai, Thomas Thiemann
and so on.

 

[marcus]

Originally posted by selfAdjoint

Nevertheless, what a thrilling breakthrough! And based on overlooked results from 1982!

Yes! By the 1982 results you evidently mean
K. Pohlmeyer "A Group Theoretical Approach to the Quantization of the Free Relativistic Closed String"
from Phys. Lett. series B.

You said you were exchanging email with Thiemann last fall
about his continuing work on defining the Hamiltonian IIRC,
I'd be tempted to write congratulations at this point
the paper is apt to be widely cited and to
stimulate a truckload of new research however the details sort out
Nicolai will turn his grad students and postdocs loose on it
and things like that

 

[Urs]

Hi -

re: who is "we", sorry for being dense! :-)

or anyway that is how I remember it---that was you?

Yup, that's me. :-)

stimulate a truckload of new research however the details sort out. Nicolai will turn his grad students and postdocs loose on it and things like that

I am not sure that this will be the case, but let's see.

As I have mentioned before, I don't think that the purpose of the exercise was to find a better quantization of the string. String theory has some problems, but they are usually not considered to be related to the question of how to quantize it. There are many independent ways to quantize the (super)string that all yield the same consistent result - which seems to be differ from the one Thiemann obtains.

Instead, Nicolai's idea originally was, I think, to test LQG by seeing if it can reproduce this familiar quantization of the string, i.e. if it can reproduce well knwon results in quantum gravity in cases where these are obtainable also by other means, which is the case in 1+1 (and also in 2+1) dimensions.

It now looks like this is indeed not the case. This sort of confirms a former result in

A. Starodubtsev, String theory in a vertex operator representation: A
simple model for testing loop quantum gravity, gr-qc/0201089 .

where also the string was quantized by LQG-like methods and a completely non-standard result was found.

 

[marcus]

Originally posted by Urs

...Instead, Nicolai's idea originally was, I think, to test LQG by seeing if it can reproduce this familiar quantization of the string, i.e. if it can reproduce well knwon results in quantum gravity in cases where these are obtainable also by other means, which is the case in 1+1 (and also in 2+1) dimensions.

Whatever you think his original idea was we have a chance to
see what he thinks now by reading his remarks at the October
symposium. As the local organizer-host he laid out the goals of the symposium where he was encouraging just the kind of research direction it seems that Thiemann has taken.
His concluding remarks at the end of the conference, summing up the situation, would offer clues as to what he thinks now.

Maybe I can get some at the AEI website, or at least give a link.

I really can't say what Nicolai's original idea was when he got interested in connecting string with loop. But whatever it was he seems to have learned something, and to have been encouraging Thiemann (and some others) in a much broader program.

Thiemann has some names I want to watch for: Dorothea Bahns, Gerrit Handrich, Catherine Meusburger, Karl-Henning Rehren. Do you happen to have met some of them?

the Pohlmeyer paper I found the most helpful to look at was one of the more recent ones he cited:

http://www.arxiv.org/hep-th/9805057

"The Nambu-Goto Theory of Closed Bosonic Strings Moving in (1+3) Dimensional Minkowski Space: the Quantum Algebra of Observables"

Don't you think it's a bit tacky to have Tachyons? I should feel like a dog with fleas, and be interested in any approach that would get rid of them. However you provide reasons why it may be a good thing to have Tachyons, condensing out of the blue, in one's theory. So it is presumably a matter of taste whether one likes them or not.

 

[Urs]

There are tachyons in very respectable theories. Write down phi^4 field theory with a Mexican Hat potential and do perturbation theory about the local maximum of the potential. You'll find tachyons. They indicate that the system rather wants to sit in the local minima.

Squared mass of a field is nothing but the quadratic term of the field's potential at a local extremum. If it's postive the extremum is a minimim and mass squared is positive, the point is stable. If it's negative the extremum is a maximum, mass squared is negative (tachyonic) and the system is unstable at that point.


Urs

 

[marcus]

I found the Nicolai slide-talk comparing you-know-whats

http://www.aei-potsdam.mpg.de/events/StringmLoops/Nicolai.pdf


It is a seven-page slide-talk comparing Loop Gravity and String.
This was his introductory talk, opening the symposium which you
attended. It doesn't say as much as I had hoped but it does give
a sense of his perspective and provide a side-by-side comparison placing two theories on an equal footing.

 

[selfAdjoint]

Physicists seem to have many attitudes like tachyons are your friend, and ghosts are your friend. But I'll bet you anything that, given a fair chance to do their physics without them, they'd jump at it. On the same side of the street, look at Neumayer's characterization of virtual particles as variables of integration, over on s.p.r.

 

[ranyart]

Marcus, great paper!

This has some interesting Astronomical implications, the background dependence of a specific target space, 'Vacuum Background' instead of abnormal CFT.

Having only absorbed the paper once, I am really amazed!

A simplistic overview can be that the 'string-worlsheet' used in some parts of SST, has been leading theorists 'up-the-garden-path!'

The LQG authors can go from the Milkyway to Andromeda and back, CFT cannot by defination according to Thiemann's paper, because the worldsheet and its dimensional consequences change the target space by its corresponding Time Domains.

This paper will re-define our perceived position within GR, namely, Observation Dependant on Location.

 

[Urs]

Hi Marcus!

Could it be that at the heart of the quantization ambiguity which is the basis for Thiemann's new approach is the quantumly ambiguous choice whether, with classical constraints C_I, one imposes

C_I|psi> = 0

as in the usual OCQ/BRST quantization of the string or

exp(C_I)|psi> = |psi>

as in the group averaging scheme used by Thiemann for his 'LQG-string'?

See
this link for a more detailed discussion.

 

[marcus]

Urs, your link to the string coffee table (a "group blog" about string theory-related matters) could be useful to a several other people here

http://golem.ph.utexas.edu/string/archives/000299.html

I gather the particular post is from you yesterday about Starodubtsev's paper, the symposium, your exchange with Ashtekar, and Thiemann's paper.
It seems that you may have been helpful either in setting Thiemann's research in motion----or at least in getting Ashtekar interested in questions along the same line as those investigated by Thiemann.

Actually it is easier for my computer to read spr posts than
those at "coffee table" because of some format-thing. But did you not post much the same thing on spr, yesterday?
I will try to follow the conversation in spr (unless you tell me I am missing something essential). I hope you don't mind posting these thoughts in both places.

 

[marcus]

Originally posted by ranyart

...This has some interesting Astronomical implications, the background dependence of a specific target space, 'Vacuum Background' instead of abnormal CFT...

ranyart, when I saw Thiemann's abstract I thought of you as one who might find it interesting. In fact you may well have discovered the paper and started reading before that, since you keep on the alert for new quantum gravity papers.

You have nudged me in the direction of looking at astronomical implications---but I don't understand so far, maybe will later.

BTW you mentioned Conformal Field Theory (CFT) and I recently became aware that it was one of the topics of a four-month workshop on Infinite-Dimensional Algebras held two years ago at Berkeley's MSRI.
The Mathematical Sciences Research Institute is an interesting show. It doesn't have a large permanent faculty or research team. Instead, the director and staff choose potentially fruitful topics and pick people from all over the world to come to the Institute for just 4 months or so and be together and give seminars to each other and pursue their collective research interest. Then they go home and another chosen bunch of people is brought in. It piques my curiosity to know what topics they believe have such potential that they would "do" them this way.

Of course the mathematics related to string theory would be hot! So
here is this InfiniteDimensional ("Lie-like" I guess) Algebras Representation Theory workshop. It is sort of enlightening what they say about it as over-view. And CFT is one of the main applications listed.

There has been some related work in LQG (Sahlmann, Thiemann, Lewandowski, Okolow), it seems. Maybe Loop is tapping into the same
store of mathematics as String, at this level. Excuse my vagueness.
Maybe Infinite Dimensional Algebra ("IDA") Reps is the mother pig and our litter of quantum theories of gravity are the piglets. This is still a very preliminary impression---dont know if accurate.

Anyway, for whatever it may be worth I will share this MSRI link
with anyone else curious about the "IDA" Reps (my abbreviation for that polysyllabic mouthful) scene.

http://zeta.msri.org/calendar/programs/ProgramInfo/15/show_program

Conformal Field Theory and Supersymmetry

http://zeta.msri.org/calendar/workshops/WorkshopInfo/141/show_workshop

 

[jeff]

Originally posted by Urs
Hi Marcus!

Could it be that at the heart of the quantization ambiguity which is the basis for Thiemann's new approach is the quantumly ambiguous choice whether, with classical constraints C_I, one imposes

C_I|psi> = 0

as in the usual OCQ/BRST quantization of the string or

exp(C_I)|psi> = |psi>

as in the group averaging scheme used by Thiemann for his 'LQG-string'?

See
this link for a more detailed discussion.

If the constant associated with these operator ordering ambiguities is viewed as the casimir energy associated with fundamental bodies due to their having finite spatial extension, than this method of quantization may be missing an essential piece of physics about strings.

 

[Urs]

Jeff wrote:

f the constant associated with these operator ordering ambiguities is viewed as the casimir energy associated with fundamental bodies due to their having finite spatial extension, than this method of quantization may be missing an essential piece of physics about strings.

Good point. Yes, physics will be quite different. What I would like to understand is if we can understand the difference in general terms, conceptually.

Unless I am missing something it seems that we have here two quantizations of the same classical theory with enormously different behaviour. How can that be? What do these two quantizations mean?

On the other hand, not all of Thiemann's solutions can be quantizations that approach the classical bosonic string in the classical limit. For instance the version of his construction which only admits states that are translation invariant in target space is clearly unphysical and not at all related to the classical string.

BTW, looking at it again I realize that I don't quite get what he is saying in the beginning of section 6.4. Seems like he is saying that massless states are translation invariant. That would be nonsense!

 

[marcus]

Originally posted by selfAdjoint
Physicists seem to have many attitudes like tachyons are your friend, and ghosts are your friend. But I'll bet you anything that, given a fair chance to do their physics without them, they'd jump at it...

that sounds like a safe bet!

Originally posted by selfAdjoint
...On the same side of the street, look at Neumayer's characterization of virtual particles as variables of integration, over on s.p.r.

I missed that post by Neumayer and probably won't have time to track it down. Was he trying to say that virtual particles are mere variables of integration and not to be imagined as existing for very brief intervals of time? Wait, I should not speculate. What was the general drift, if it can be said easily and you don't mind relaying.

 

[Urs]

Marcus,

you need to have a MathML-aware browser like Mozilla and the required fonts installed to read the String Coffee Table. Click on the MathML icon for further information.

 

[marcus]

Originally posted by Urs
Marcus,

you need to have a MathML-aware browser like Mozilla and the required fonts installed to read the String Coffee Table. Click on the MathML icon for further information.

thanks for the tip!
I just checked spr and saw your post expanding (at least slightly I think) on what you said here:

-----quote from Urs post on spr-----
I was trying to figure out what exactly it is in Th. Thiemanns quantization hep-th/0401172 of what he calls the 'LQG-string' that makes it so different from the usual quantization. I now believe that the crucial issue is how to impose the constraints.

Let there be a physical theory with constraints C_I, I in some set. Then requiring

<psi|C_I|psi> = 0

is what, in the case where the C_I are the Virasoro constraints, leads to the usual quantum string.

On the other hand the group averaging method used by Thiemann imposes (see below his eq. (5.4))

<psi|exp(C_I)|psi> = 0 .

This may seem like essentially the same thing, but the crucial issue is apparently that the latter form allows to deal quite differently with operator ordering, which completely changes the quantization. In particular, it seems to allow Thiemann, in this case, to have no operator re-ordering at all, which is the basis for him not finding an anomaly, hence no tachyon and no critical dimension.

--------end quote from the post----

tho not able to understand everything, I did have a look at the equation you mentioned, eq. (5.4) and below
and I did not see

"exp(C_I)|"

exactly (sometimes my eyes don't catch details as well as I would like)
but I did find something like that in eq. (5.2)

where he defines U(t)
as an exponential
exp(i \Sigma t^I \pi(C_I))

(would you care to explain this for bears of little brain?)

and then right below eq. (5.4)
he uses that U(t) in an equation for which what you wrote may
be a suggestive shorthand.

I gather you believe that this is not kosher or anyway not usual.

the more explanation you give probably the more PF posters you can
include and bring along with your reasoning, but it may require some
"John Baez" type pedagogical talents to do this,

regards

 

[Urs]

Hi Marcus,

I think a very crucial information in Thiemann's paper is in the lines between his equation (5.4) and (5.5), where he says "solves the constraints in the sense that...". In ordinary bra-ket notation what he says there is that the constraints are implied by the equation

<psi| U(t) |psi'> = <psi|psi'> .

(In your quote of my spr post the right hand side of this equation is a stupid typo, of ocurse, sorry.)

This is best understood in terms of a very simple example. Consider a 1-dimensional system, like the free particle on the circle. Let us impose on this system the classical constraint that the momentum shall vanish

p = 0 .

Here we have a single constraint

C_1 = p

in Thiemann's notation. The quantum version of this is the operator

^C_1 = ^p = -i d/dx .

Now consider the condition

^p |psi> = 0 .

It simply means that |psi> does not depend on x and hence is a constant. This is the quantum analogue of the classical constraint p=0.

Consider alternatively the condition

e^(^p)|psi> = |psi> .

As you know, e^(^p) is the operator that translates states around, i.e.

e^(^p)|psi(x)> = |psi(x+1)> .

The condition

e^(^p)|psi> = |psi>

therefore means that

|psi(x+1)> = |psi(x)> .

If we now include an arbitrary constant t in the exponent (as in equation (5.2) of Thiemann's paper) then we get

|psi(x+t)> = |psi(x)>

for all t. It follows that psi(x) = psi(0) must be independent of x. Hence in this case the constraint

^p|psi> = 0

is equivalent to

e^(^p)|psi> = |psi> .

This is pretty trivial in this case, because there is no doubt about how all these operators in this example are to be defined.

In more complex cases however the exact representation of classical observables by operators on some Hilbert space may be a subtle issue and the two sorts of constraints need not be equivalent anymore.

 

[selfAdjoint]

Thiemann seems well aware of the subtleties of what he is doing, and as he says, this is a familiar technology to him. Up at the top of this same page 18 he says:
_____________________________________________________________
Since, by assumption, \mathfrak{A} separates the points of \mathcal{M} it is possible to write every C_I as a function of the f \in \mathfrak{A}, however that function is far from unique due to operator ordering ambiguities and in field theory usually involves a limiting procedure (regularization and renormalization). We must make sure that the resulting limiting operators \pi(C_I) are densely defined and closable (i.e. their adjoints are also densely defined) on a suitable domain of \mathcal{H}_{K_{in}}. This step usually severely restricts the abundance of representations. Alternatively in rare cases it is possible to quantize the finite gauge transformations generated by the classical constraints provided they exponentiate to a group. This is actually what we will do in this paper.
_________________________________________________________________

(emphasis added)

So his group averaging quantization method which you critique is actually only possible in this specific case, and the remarkable fact is (or is claimed) that this is sufficient to quantize the closed bosonic string.

 

[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? In this case the formulation of the constraint should be equivalent to the conventional one (though I haven't looked at this carefully). Thus it may be more the choice of representation than the method of quantization that's at the heart of this.

Then his comments about the appearance of a critical dimension in conventional string theory being a consequence of rep would be correct in that those reps capture more physics than his do.

 

[jeff]

Originally posted by selfAdjoint
...his group averaging quantization method...is actually only possible in this specific case, and the remarkable fact is (or is claimed) that this is sufficient to quantize the closed bosonic string.

But is this the same string as in conventional string theory?

 

[selfAdjoint]

Originally posted by jeff
But is this the same string as in conventional string theory?

Well that depends on the Pohlmeyer charges doesn't it? I imagine there will be a lot of digging into just what they represent and how they do it. Thiemann's intro didn't do anything for me. He does start from the Nambu-Goto action, and as far as I recall, there isn't a lot of variation in the theory from there on. It's all pretty cast iron.

 

[Urs]

'selfAdjoint' wrote:

So his group averaging quantization method which you critique is actually only possible in this specific case, and the remarkable fact is (or is claimed) that this is sufficient to quantize the closed bosonic string.

Yes, in more general cases his 'Master Constraint Programme', i.e. the 'Direct Integral Method', can be used instead. The basic idea is essentially the same, as one can see by comparing the Dirac observables in equation (5.5) and (5.8), namely to 'smear' objects over their entire gauge orbit, thereby projecting onto the gauge invariant part.

I would like to concentrate on the group averaging method, though, becasue I'd like to see first the simplest non-trivial LQG quantization scheme compared with the usual approaches. When we understand this to our heart's content we can still worry about fancier setups.

Indeed, I am beginning to wonder how the LQG-like quantization of the boring old relativistic particle would look like, i.e. that of the Nambu-Goto action in only 1+0 dimensions. That's because the crucial issue of the center-of-mass motion of the string seems to be kind of swept under the carpet in Thiemann's paper, unless, of course, I am missing something.

So what is really happening in section 6.4? Judging from equation (6.25) it is pretty obvious that the Hilbert space constructed on the 'vaccum' Omega_omega (sorry, but how again can I insert pretty-printed math here?) is only that of the oscillatory modes, not including the center-of-mass momentum. It is tempting to roughly identify Omega_omega with the usual |0> Fock vacuum state of string oscillators. But even that cannot be quite true, because of the strange relations discussed on the top of page 24, which roughly say that the expectation value of exp(i pi(s)) (where pi is the canonical momentum) vanish in Thiemann's representation.

Anyway, in section 6.4 the space built on Omega_omega is augmented by a momentum index, roughly speaking, and a new (unless I am confused) operator pi_mu(p_nu) is introduced which has eigenvalues p_nu when applied to the new vacumm, wgere p_nu is supposed to be the center-of-mass momentum. Thiemann seems to argue that by choosing this p_nu he can give the string any com-momentum whatsoever.

I don't follow this reasoning. To me it seems that we should identify the already existing 0-mode pi(S^1) of the canonical momentum with this pi_mu(p_nu) above, the way it is done in the usual approach (note that, unfortunately, pi stands for canonical momentum as well as for the representation map). This would seem to be a step in the right direction, because it would couple the energy of the com-motion of the string with that of its oscillators, the way it should be! As far as I can judge from reading section 6.4 there it is proposed that the com-motion, i.e. the mass of the string, is completely unaffected by its internal state. We can choose it essentially to be any value we like! Even if this should be mathematically consistent it is hardly reasonable from a physical point of view.

I have written an email to Thiemann asking him why the com-momentum pi_nu(S^1) should not have eigenvalues p_nu on Omega_omega_p. This would relate the com-momentum p to the oscillator modes for instance in the L_0 virasoro constraint (as usual) and would make the question about the mass spectrum of the LQG string reasonable. Let's see what (and if) he answers. Right now I don't see how any claim about the mass spectrum of the LQG string (tachyon or not) is sensible. (But of course that may well be my fault.)


Jeff wrote:

In this case the formulation of the constraint should be equivalent to the conventional one (though I haven't looked at this carefully).

Hm, but it obviously is not the same, otherwise there would be the usual effects such as normal ordering constants, etc. On the other hand, as I tried to discuss above, I am not really sure that Thiemann properly deals with the com momentum, and I currently think any conclusions about the mass spectrum of the LQG string are premature. (Corrections are very welcome.)

Thus it may be more the choice of representation than the method of quantization that's at the heart of this.

Yes, but isn't that the same thing here? The question is how to represent the classical observables as operators on some Hilbert space. That's quantization.

 

[lumidek]

101 silly errors in a very bad paper by Thomas Thiemann I.

On 27 Jan 2004, Urs Schreiber wrote:

> I was trying to figure out what exactly it is in Th. Thiemanns
> quantization hep-th/0401172 of what he calls the 'LQG-string' that
> makes it so different from the usual quantization. I now believe that
> the crucial issue is how to impose the constraints.

Exactly. If physics is done properly, the (Virasoro) constraints are not
arbitrary constraints that are added by hand. They are really Einstein's
equations, derived as the equations of motion from the action if it is
varied with respect to the metric - in this case the worldsheet metric.
The term R_{ab}-R.g_{ab}/2 vanishes identically in two dimensions, and
T_{ab}=0 is the only term in the equation that imposes the constraint. The
constraints are really Einstein's equations, once again.

Moreover, because the (correct) theory is conformal, the trace
T_{ab}g^{ab} vanishes indentically, too, and therefore the three
components of the symmetric tensor T_{ab} actually reduce to two
components, and those two components impose the so-called Virasoro
constraints (which are easiest to be parameterized in the conformal gauge
where the metric is the standard flat metric rescaled by a
spacetime-dependent factor). For closed strings, there are independent
holomorphic and independent antiholomorphic generators - and they become
left-moving and right-moving observables on the Minkowski worldsheet
after we Wick-rotate.

Thomas Thiemann does not appreciate the logic behind all these things, and
he wants to work directly with the (obsolete) Nambu-Goto action to avoid
conformal field theory that he finds too difficult. Of course, the
Nambu-Goto action has no worldsheet metric, and therefore one is not
allowed to impose any further constraints. They simply don't follow and
can't follow from anything such as the equations of motion.

Thiemann does not give up, and imposes "the two" constraints by hand. It
is obvious from his paper that he thinks that one can add any constraints
he likes. Of course, there are no "the two" constraints. If he has no
worldsheet metric, the stress energy tensor has three components, and
there is no way to reduce them to two. Regardless of the effort one makes,
two tensor constraints in a general covariant nonconformal theory can
never transform properly as a tensor - because a symmetric tensor simply
has three components - and therefore his constraints won't close upon an
algebra. His equations are manifestly general non-covariant, in contrast
with his claims.

Equivalently, because he obtained these constraints by artificially
imposing them, they won't behave as conserved currents. (In a general
covariant theory without the worldsheet metric, we can't even say what
does it mean for a current to be conserved, because the conservation law
nabla_a T^{ab} requires a metric to define the covariant derivative.) If
they don't behave as conserved currents, they don't commute with the
Hamiltonian, and imposing these constraints at t=0 will violate them at
nonzero "t" anyway (the constraint is not conserved).

If one summarizes the situation, these constraints simply contradict the
equations of motion. It is not surprising. We are only allowed to derive
*one* equation of motion for each degree of freedom i.e. each component of
X, and this equation was derived from the action. Any further constraint
is inconsistent with such equations unless we add new degrees of freedom.

I hope that this point is absolutely clear. The equations of motion don't
allow any new arbitrarily added constraints unless it is possible to
derive them from extra terms in the action (that can contain Lagrange
multipliers). The Lagrange multipliers for the Virasoro constraints *are*
the components of worldsheet metric, and omitting one component of g_{ab}
makes his theory explicitly non-covariant (even if Thiemann tries to
obscure the situation by using the letters C,D for the two components of
the metric in eqn. (3.1)).


The conformal symmetry is absolutely paramount in the process of solving
the theory and identifying the Virasoro algebra - isolating the two
generators T_{zz} and T_{zBAR zBAR} per point from the general symmetric
tensor. Conformal/Virasoro transformations are those that fix the
conformal gauge - i.e. the requirement that the metric is given by the
unit matrix up to an overall rescaling. Conformal theories give us T_{z
zBAR} (the trace) equal to zero, and this is necessary to decouple T_{zz}
and T_{z zBAR}. In two dimensions, the conformal transformations -
equivalently the maps preserving the angles - are the holomorphic maps
(with possible poles), and the holomorphic automorphisms of a closed
string's worldsheet are generated by two sets of the Virasoro generators.

This material - why it is necessary to go from the Nambu-Goto action to
the Polyakov action and to conformal field theory in order to solve the
relativistic string and quantize it - is a basic material of chapter 1 or
chapter 2 of all elementary books about string theory and conformal field
theory. I think that a careful student should first try to understand this
basic stuff, before he or she decides to write "bombastic" papers boldly
claiming the discovery of new string theories and invalidity of all the
constraints (such as the critical dimension) that we have ever found.

In fact, I think that a careful student should first try to go through the
whole textbook first, before he publishes a paper on a related topic.
Thomas Thiemann is extremely far from being able to understand the chapter
3 about the BRST quantization, for example.

Thiemann's theory has very little to do with string theory, and very
little to do with real physics, and unlike string theory, it is
inconsistent and misled. String theory is a very robust and unique theory
and there is no way to "deform it" from its stringiness, certainly not in
these naive ways.

> This may seem like essentially the same thing, but the crucial issue is
> apparently that the latter form allows to deal quite differently with
> operator ordering, which completely changes the quantization. In particular,
> it seems to allow Thiemann, in this case, to have no operator re-ordering at
> all, which is the basis for him not finding an anomaly, hence no tachyon and
> no critical dimension.

A problem is that you don't know what you're averaging over because his
"group" is not a real symmetry of the dynamics.

By the way, if you want to define physical spectrum by a
Gupta-Bleuler-like method, you must have a rule for a state itself that
decides whether the state is physical or not. In Gupta-Bleuler old
quantization of the string, "L_0 - a" and "L_m" for m>0 are required
to annihilate the physical states. This implies that the matrix element of
any L_n is zero (or "a" for n=0) because the negative ones annihilate the
bra-vector.

It is important that we could have defined the physical spectrum using a
condition that involves the single state only. If you decided to define
the physical spectrum by saying that all matrix elements of an operator
(or many operators) between the physical states must vanish, you might
obtain many solutions of this self-contained condition. For example, you
could switch the roles of L_7 and L_{-7}. However all consistent solutions
would give you an equivalent Hilbert space to the standard one.

The modern BRST quantization allows us to impose the conditions in a
stronger way. All these subtle things - such as the b,c system carrying
the central charge c=-26 - are extremely important for a correct
treatment of the strings, and they can be derived unambiguously.

> If this is true and Group averaging on the one hand and Gupta-Bleuler
> quantization on the other hand are two inequivalent consistent quantizations
> for the same constrained classical system I would like to understand if they
> are related in any sense.

No, they are not. What is called here the "group averaging" is a naive
classical operation that does not allow one any sort of quantization. You
can simply look that at his statements - such as one below eqn. (5.2) -
that in his treatment, the "anomaly" (central charge) in the commutation
relations (of the Virasoro algebra, for example) vanishes, are never
justified by anything. They are only justified by their simple intuition
that things should be simple. This incorrect result is then spread
everywhere, much like many other incorrect results. It is equally wrong as
simply saying that we have constructed a different representation of
quantum mechanics where the operators "x" and "p" commute with one
another.

The central charge - the c-number that appears on the right hand side of
the Virasoro algebra - is absolutely real and unique determined by the
type of field theory that we study (and the theory must be conformal,
otherwise it is not possible to talk about the Virasoro algebra). It can
be calculated in many ways and any treatment that claims that the Virasoro
generators constructed out of X don't carry any central charge is simply
wrong.

There is absolutely no ambiguity in quantization of the perturbative
string. Knowing the background is equivalent to knowing the full theory,
its spectrum, and its interactions. There is no doubt that Thiemann's
paper - one with the big claims about the "ambiguities" of the
quantization of the string - is plain wrong, and exhibits not one, but a
plenty of elementary misunderstanding by the author about the role of
constraints, symmetries, anomalies, and commutators in physics.

 

[lumidek]

101 silly errors in a very bad paper by Thomas Thiemann II.

Let me summarize a small part of his fundamental errors again. He believes
many very incorrect ideas, for example that

* artificially chosen constraints can be freely imposed on your Hilbert
space, without ruining the theory and contradicting the equations of motion
* two constraints in 2 dimensions can transform as a general symmetric
tensor, and having a tensor with a wrong number of components does not
spoil the general covariance
* he also thinks that the Virasoro generators have nothing to do with the
conformal symmetry and they have the same form in any 2D theory
* in other words, he believes that you can isolate the Virasoro generators
without going to a conformal gauge
* classical Poisson brackets and classical reasoning is enough to
determine the commutators in the corresponding quantum theory
* anomalies in symmetries, carried by various degrees of freedom,
can be ignored or hand-waved away
* there is an ambiguity in defining a representation of the algebra of
creation and annihilation operators
* the calculation of the conformal anomaly does not have to be treated
seriously
* the tools of the so-called axiomatic quantum field theory are useful
in treating two-dimensional <conformal> field theories related to
perturbative string theory
* if a set of formulae looks well enough to him, it must be OK and the
consistent stringy interactions and everything else must follow

Once again, all these things are wrong, much like nearly all of his
conclusions (and completely all "new" conclusions).

Thiemann himself admits that this is the same type of "methods" that they
have also applied to four-dimensional gravity. Well, probably. My research
of the papers on loop quantum gravity confirms it with a high degree of
reliability. Every time one can calculate something that gives them an
interesting but inconvenient result, they claim that in fact we don't need
to calculate it, and it might be ambiguous, and so on. No, this is not
what we can call science. In science, including string theory, we have
pretty well-defined rules how to calculate some class of observables, and
all things calculated according to these rules must be treated seriously.
If a single thing disagrees, the theory must be rejected.

The inevitability of conformal symmetry for a controlled quantization of
the relativistic string - and for isolation (in fact, the definition) of
the Virasoro generators - is real. The theorems of CFT about its being
uniquely determined by certain data are also real. The conformal anomalies
of certain fields are also real. The two-loop divergent diagrams in
ordinary GR are also real. We know how to compute and prove all these
things, and propagating fog and mist can only obscure these
well-established facts from those who don't want to see the truth.

I guess that this paper will demonstrate to most theoretical physicists -
even those who have not been interested in these "alternative" fields -
how bad the situation in the loop quantum gravity community has become.
There are hundreds of people who understand the quantization of a free
string very well, and they can judge whether Thiemann's paper is
reasonable or not and whether funding of this "new kind of science"
should continue.

All the best
Lubos

 

[Urs]

My reply to Lubos can be found at the this entry of the.String Coffee Table.

 

[lumidek]

Singularities in quantum field theory

Dear Urs,

concerning your (more precisely, Thiemann's) comments that you can get rid of all ordering constants by exponentiating something, I hope that you don’t really believe it because this would count as a rudimentary misunderstanding of the singularities (and anomalies) in quantum field theory. The exponentials of something always store the same information as “something”, and if one of them has some ordering constant contribution, you see it in the other as well.

For example, X(z) X(0) have logarithmic OPEs - it behaves as ln(z). This implies that exp(i.K.X(z)) has a power law OPE with exp(-i.K.X(z)). It’s totally nonsensical at quantum level to imagine that exp(-i.K.X(z)) is an inverse operator to exp(i.K.X(z)). Do you understand why? This is a very important point. The singularities are not invented by people who want to make their lives more difficult. They can be derived from the rules of quantum field theory. In quantum field theory, everything (fields) fluctuate, and the total fluctuation summed over all the modes simply implies that the expectation value of X(z)X(z') diverges as z approaches z'. All the operators in QFT are potentially infinite, and one must be very careful in regulating their products. It is not possible to deal with the operators as with ordinary numbers, and most of them can't be inverted.

While for the Virasoro group without the central charge you would be able to write the explicit “exponentiated” elements of the reparameterization group and - because they have a clear geometric interpretation - you could invert them without anomalies, it is simply not true for the Virasoro operators generating the reparameterization of X’s. Because of the term c/z^4 in the OPE of two stress energy tensors, you must know very well that exp(-V) can’t be treated as the inverse of exp(+V). You can only imagine that exp(V) is an honest element of a group if the OPEs of V with itself - and all other “V“‘s that you want to use - only have the 1/z term, corresponding to the commutator. This is the whole point of "anomalies" that quantum mechanics simply prevents us from imagining that the classical, naive symmetries survive in quantum mechanics. A quantum mechanical theory that would respect the naive classical symmetries simply does not exist, if one can calculate the anomalies. The conformal anomaly prevents us from defining the Virasoro "group" where objects can be inverted in a naive way. Recall that

O1(z) O2(0) ~ [O1,O2] (z) / z

the coefficient of 1/z is schematically the commutator of the two operators. If you integrate a stress energy tensor etc., it is also OK to have the 1/z^2 term in the OPEs of the stress energy tensor because it reflects the worldsheet dimension of the stress energy tensor and tells you how should you integrate it to get scalars etc.

But the OPE of the stress energy tensor (of the X^mu CFT) with itself contains an extra 1/z^4 term. This is just a fact that you can calculate in many ways, and this simply means that exp(V) where V is a Virasoro generator, or some integrated combination of the stress energy tensor, does not behave as an honest element of some group, and exp(-V) is not in any naive sense inverse to exp(V) because these two *operators* have singularities.

Note that his naive operation, involving the (wrong) application of the formula e.g.

exp(C.D.C^{-1}) = C exp(D) C^{-1}

which is OK for matrices, is incorrect in our “usual” representation of CFT, because of singularities between C and C itself. Thiemann's sloppy methods would certainly allow you to derive a lot of incorrect "results" in ordinary CFT, too. You can’t imagine that C^{-1} is inverse to C - there are just no meaningful operators on the Hilbert space that would look like C=exp(V) and were inverse to one another. Because C^{-1}.C is not really one, you can’t derive the formula you derived either, unless c=0 where the classical intuition is OK. Note that it even requires you, for C=exp(V), to consider exp(exp(V)…). These are heavily singular operators, and all these confusions simply come from his/their wrong intuition that you can work with the operators in CFT as with ordinary classical numbers. They don’t understand where the normal ordering terms come from, they don’t understand singularities of operators in quantum field theories, they don’t understand the difference between classical and quantum field theory - and perhaps between classical and quantum physics in general.

Even if Thiemann did a better job and counted the quantum contributions properly, the framework of his paper would be terribly far from a construction of a meaningful theory. In a meaningful theory, he would have to consider torus diagrams, for example, and so on. The modular invariance would be definitely broken for his "new representations" if he had no CFT backing him. The modular invariance is derived from the very special functions associated with CFT, Poisson resummation, modular functions, and so on. Without doing the torus diagram, he does not really need to discover the critical dimension (D=26 was first found from unitarity at the one-loop torus level), but if he fails to understand why the critical dimension is necessary, it is very far from having evidence that it does not have to be necessary. All these things are very sensitive and they must be done very exactly, if the theory is supposed to be consistent. His treatment is a naive application of classical reasoning, involving brutal eliminating of the terms that are absolutely essential for consistency of the theory.

It’s just sad. Ignorance about the basics of quantum field theory should not be sold as a "new, revolutionary proposal in physics", and every student in theoretical physics should be able to identify the errors in papers similar to Thiemann's paper.

All the best
Lubos

 

[Urs]

Hi -

Here is my reply to the latest message by Lubos.

 

[marcus]

Originally posted by Urs
My reply to Lubos can be found at the this entry of the.String Coffee Table.

I'm having browser trouble getting the "string coffee table" and
the font is too small on my screen as well, so unless you object I will copy your reply here so I can read it:

-----quote from Urs------
Re: Thiemann’s quantization of the Nambu-Goto action
Hi Luboš,

thanks for your answer!

I see your general point, but would like to look at some of the issues you raised in more detail.

You say that the Nambu-Goto action is ‘obsolete’. But of course the NG action is classically equivalent to the Polyakov action and I think that in the critical number of dimensions the equivalence extends to the quantum theory. Furthermore, the Nambu-Goto action for the string is essentially the Dirac-Born-Infeld action (up to the worldsheet gauge field) of the D-string.

As far as I can see the constraints that Thiemann arrives at in equation (2.4) of his paper follow from standard canonical reasoning. One finds that the canonical momenta π μ of the Nambu-Goto action as well as of the DBI action classically satisfy two identities which can be identified as constraints. At the classical level these constraints are precisely the (classical) Virasoro constraints that one also obtains by varying the worldsheet metric in the Polyakov action. Since the two actions are classically equivalent this is no surprise.

My point is that there should be a priori nothing wrong with looking at the Nambu-Goto action when studying the string. Indeed this is frequently done for instance when F-strings and D-strings are considered at the same time, as for instance in

Y. Igarashi, K. Itoh, K. Kamimura, R. Kuriki, Canonical equivalence between super D-string and type IIB superstring.

In equations (2.3) and (2.4) of this paper the authors in particular give the same two bosonic constraints of the Nambu-Goto action that Thiemann arrives at. Their action also involves superfields and the worldsheet gauge field, but this does not affect the general result that the Virasoro constraints follow from a canonical analysis of the Nambu-Goto action. I have spelled out the derivation (for the bosonic DBI action) in a recent entry. (By setting the worldsheet gauge field and the C fields to zero this derivation directly restricts to that for the ordinary Nambu-Goto action).

My point is that it is maybe not fair to say that Thiemann artificially or freely chooses the constraints - at least not at the classical level. The constraints that he uses are, classically, the Virasoro constraints of the closed bosonic string.

My suspicion is rather that Thiemann devitates from standard reasoning when he defines what he wants to understand under quantizing the Virasoro constraints. Would you agree with this?

Let’s ignore the way on which we arrived at the classical Virasoro constraints (by starting from one of various classically equivalent actions) and concentrate on the question what it means to quantize them.

The standard procedure is to make Gupta-Bleuler quantization and use either creation/annihilation operator normal ordering or CFT techniques to make sense of the quantum representation of the classical Virasoro generators. This leads in the usual way to the anomaly, the shift a in (L_0 - a) and so on.

Thiemann claims (based on a large literature on quantization of constrained systems that is also the basis for loop quantum gravity) that there is an at least superficially different technique that can also be addressed as quantization of the Virasoro constraints. In the simple case at hand this is imposing the constraint the way mentioned right below equation (5.4), which essentially says that <ψ|exp( constraints )|ψ ′>=<ψ|ψ ′>, where the Hilbert space and the representation of the operators is not necessarily the usual Fock representation.

This is not equivalent to and not even implied by saying that <&psi;|constraints |&psi;'>=0. Of course when I write this I am ignoring issues of what we really mean by writing exp (some operator) , i.e. whether this is supposed to be normal ordered or regulated or what. I am trusting that this is taken care of by Thiemann’s rigorous construction of Hilbert spaces and operators on them, but I guess that Luboš disagrees with this. :-)
--------end quote----------

the symbols don't come out but at least we get some idea of Urs reply
i've tried to edit back in some of the symbols
there is another reply further down
all seems pretty interesting

 

[marcus]

Originally posted by Urs
Hi -

Here is my reply to the latest message by Lubos.

So much easier to read in the larger PF font----instead of the coffee table small font on green background!

------quote of Urs next reply-----
Re: Thiemann’s quantization of the Nambu-Goto action
Hi again, Luboš!

Yes, I understand everything that you say here. I know that : exp(-V): is not the inverse to : exp(V): in CFT and I do understand where the 1 /z 4 terms come from. When you go back to my original entry you’ll see that I address precisely this phenomenon by mentioning that things like : exp(k⋅X): have conformal dimension depending on k in CFT, which is another aspect of this phenomenon.

But, yes, I was taking for granted that Thiemann is using a rep of his operators that allows him to ignore all normal ordering issues and work with them as with matrices and hence not as in CFT. He is referring to lot’s of mathematical theorems, using the GNS construction etc. (that I obviously haven’t checked myself and I am trusting that he applies them correctly) and even though he does not say so explicitly I deduced from his paper, in particular from the the third paragraph on p. 20, that he does use exp (C⋅D⋅C - 1)=Cexp(D)C - 1. I do understand that this does not make sense in CFT (or even any other quantum field theory in the usual sense) but I also believe that a large number of mathematically versed people in the LQG camp do think that this can be given good meaning by using all these mathematical constructions that Thiemann alludes to. Unfortunately I am not an expert on this stuff.

I think the key ingredient is the GNS construction, which tells you that a unital *-algebra can be represented faithfully i.e. without normal ordering issues just like matrices on some Hilbert space. That’s the content of the relation in the 9th line from below on p.15: [ a]=[ab]. On the right hand side is the classical multiplication of the algebra, on the left hand side we have operator multiplication. Whenever this is true we do have ( exp( π ω(a) ) ) - 1 =exp(-π ω(a)).

There is some fine print to this construction which I am maybe not fully aware of. In particular things need to be bounded for this to make sense. That’s why Thiemann uses the operators W&Hat; =exp(iY&Hat;) instead of the Y&Hat; themselves, because these would be unbounded.
----------end of quote, sorry symbols not coming out-----

 

[marcus]

recalling the Lubos/Urs discussion

The recent Lubos/Urs discussion was carried on in three places on the web, partly here at PF and the "coffee table" and also at SPR.
To try boil it down and get an idea of the general drift of this interesting exchange, I will exerpt Lubos recent PF posts (look back for detailed assertions) to recall the general tenor. And quote an Urs reply just posted at SPR which seems to sum up his response in the most concise way. First here are exerpts from two Lubos posts:

Originally posted by lumidek
Let me summarize a small part of his fundamental errors again. He believes many very incorrect ideas, for example that...
...Once again, all these things are wrong, much like nearly all of his
conclusions (and completely all "new" conclusions).

...Every time one can calculate something that gives them an
interesting but inconvenient result, they claim that in fact we don't need to calculate it, and it might be ambiguous, and so on. No, this is not what we can call science...

...There are hundreds of people who understand the quantization of a free string very well, and they can judge whether Thiemann's paper is
reasonable or not and whether funding of this "new kind of science"
should continue.

[from the next post]

...It’s just sad. Ignorance about the basics of quantum field theory should not be sold as a "new, revolutionary proposal in physics", and every student in theoretical physics should be able to identify the errors in papers similar to Thiemann's paper.

Here now is Urs SPR post which I just saw a few minutes ago:
----------quote from SPR----


Lubos and I had a little exchange about my original question over at
http://golem.ph.utexas.edu/string/archives/000299.html#c000504 .


It turns out that a crucial point for talking about the methods used in Thiemann's paper (hep-th/0401172) is that the ordinary lore of quantum field theory does not (or is not supposed to) apply in that framework.

For instance, in the ordinary version of quantum field theory the operator

:exp(-V):

is not the inverse of

:exp(V):,


due to the normal ordering, which is indicated by the colons. But in
Thiemann's paper (and, as far as I understand, in similar LQG papers) it is used that there is a representation of QFT operators, obtained by means of the GNS construction, which satisfy

pi(a) pi(b) = pi(ab),

where a and b are classical observables and pi(a) is the operator
representation of the observable a. This would imply that

pi( exp(-V) )

is indeed the inverse of

pi( exp(V) )

and I believe that this is a relation which is used heavily in Thiemann's paper. For instance this seems to be the basis for the claim in the 3rd paragraph of p. 20 that

alpha(W(Y_+-)) = W(alpha(Y_+-)) ,

where Y^mu_+- = p^mu +- X'^mu are essentially what is usually written as

partial X and bar partial X,

i.e. the left- and right-moving bosonic fields on the worldsheet,
W(Y) is the exponentiation of Y
and
alpha is the action of the exponetiated Virasoro constraints.

It is pretty obvious that if this is true then no anomaly does appear, since the elements generated by exponentiating the operator constraints behave exactly as those generated by exponentiating the classical constraints.

Is it hence true that we can alternatively have QFTs that have neither
normal ordering issues, nor anomalies, nor non-trivial OPEs, etc? If not, is there something wrong about Thiemann's assumptions? What is going on here?

-------------end quote--------------

Urs post contains a link to the "coffee table" discussion, which
will be useful to those whose browsers accommodate it gracefully.

 

[Urs]

Hi Marcus -

sorry for the problems caused by the Coffee Table layout that you encountered. I don't know why that occured. For me it is just the other way round: I have a large and nicely readable font when reading the String Coffee Table, with very nice pretty-printed formulas! :-) I also like that I can use the usual LaTeX commands to write math at the Coffee Table - and - well, that when composing messages I am not disturbed by a pack of smileys sitting next to the editor pane staring at me and doing weird things! ;-) But of course I don't want to pull away any discussion from the PF! So if you'll copy the Coffee Table comments any I could just as well post them here myself. :-)

Anyway, Lubos tells me that he will look more closely at the GNS construction and related things after lunch. I am looking forward to hearing his comments.


Urs

 

[marcus]

Originally posted by Urs
Hi Marcus -
... when composing messages I am not disturbed by a pack of smileys sitting next to the editor pane staring at me and doing weird things! ;-)

I know what you mean, especially this one
although this one is, in its own way, equally terrible[zz)]

Actually I appreciated your linking from here to the "coffee table"
and keeping us apprised of the discussion by links. Personally I don't really have to copy everything---I just put my nose closer to the monitor and I can read the "coffee table".

So don't feel you have to copy everything over from there (unless you just want to!)

I hope that some clarification and discussion occurs at SPR since
at least for the time being that is where the widest collection of knowledgeable people seems to be. It is always good to hear from a variety of viewpoints. Maybe your recent post at SPR will get some replies.

[imagine friendly smiley here]

 

[lumidek]

Group averaging & Pohlmeyer charges

Yes, the GNS theorem is alright as a piece of abstract math, and for a unital *-algebra that satisfies certain properties, it returns the Hilbert space on which the algebra acts, and so on. There are many subtleties that would not work in this case, because of the noncompact character of the gauge group, and so on. But there is something more serious going on here. The GNS construction is a formal mathematical theorem, but it is creating less optimism if one sees what Thiemann wants to insert into this theorem, and what can he get.

On page 18, equation (5.5), he's getting the "Dirac observables" by averaging over the whole group. Of course, the Virasoro/reparameterization group is noncompact and the integral does not converge, so he defines the integral as the limit of the average over a finite piece of the (infinite-dimensional) group manifold. The infinite-dimensional integrals are usually ill-defined, but all these things are irrelevant technicalities and it is much easier to see why his construction is not physical.

It is easy to see that the only polynomial operators in the finite oscillators of x,p that can survive - that can average to anything nonzero - are the zero modes of the momentum, i.e. the total momentum of the string. There are just no other operators that would be invariant under the whole Virasoro group. In Gupta-Bleuler quantization, there are *states* that are invariant under one half of the generators (the positive frequency ones), and this fact follows from the special properties of the ground state that is annihilated by all positive frequency oscillators. In modern covariant BRST quantizations, we can find states that are BRST-invariant, which is nearly equivalent to saying that their matter part is invariant under the whole Virasoro group.

But there are no nontrivial "finite energy" Virasoro-invariant operators constructable out of the standard oscillators. So the intersection of the algebra of "Dirac observables", that he inserts into his Hilbert-space-generating theorems, with the regular stringy operators is really the algebra of "D" components of the momentum zero modes. The resulting representation that he can derive implies that particles can have various momenta. What a big deal! It has nothing to do with the internal dynamics of the usual physical string, derived from the oscillator modes.

The operators that Pohlmeyer and Thiemann want to consider are the Wilson lines of a gauge field. The vector potential of this GL(N,C) gauge field is taken to be the derivative of X^mu, contracted with some constant complex N times N matrices T_mu. Of course, a Wilson line is independent of the parameterization. The Wilson line is a path-ordered exponential, and we can Taylor-expand this exponential to get the individual terms, the Pohlmeyer charges (multiplied by a trace of a product of the complex constant matrices T_mu).

The first Pohlmeyer charges - linear in T_mu - are just the string's overall momentum and winding. The second Pohlmeyer charge is still well-defined in the correct spectrum of string theory, but it gives no new information. Look at eqn. (3.29), page 11 of Thiemann's paper. Z is the charge - it is gotten as a sum over cyclical rearrangements of a similar charge called R. For N=2 - the 2nd term in the expansion - it is easy to see that we integrate over all pairs of positions on the string because two points on a circle are always cyclically arranged in the right way - there is one way only - and therefore this charge is simply the first Pohlmeyer charge squared. Again, the overall momentum and possible the overall winding of the string, nothing more.

OK, so now we must go to the first nontrivial case N=3, the third Pohlmeyer charge. Now we only integrate over some intervals. In terms of the usual stringy oscillators, we get some combination of many terms cubic in the oscillators, with coefficients that don't go to zero too quickly as the mode number goes to infinity. Obviously, we would avoid the name "charge" for such an operator because no finite energy state of the string - and now I am considering the Hilbert space of the *real* string theory, not the LQG/AQFT genetically modified string - is an eigenstate of such a "charge".

At any rate, the Pohlmeyer charges form a very small (and sort of non-finite-energy) subalgebra of the polynomials constructed from the real oscillators in string theory, if you fix the usual gauge. You might think that those people would admit that the real string theory is at least one of their solutions. Of course, it is not, because they define wrong commutation relations for these charges. If you look at page 17, paragraph I (Kinematical algebra...), you will see that they simply define the commutators to be equal to the classical Poisson bracket - of the functions on the phase space. That's not what we have in usual string theory: commutators are more than just the Poisson brackets. They are equal in the "semiclassical" physics, but not in the full quantum physics.

This obscure definition of the commutators was inserted in, and not surprisingly, it comes out. Garbage in, garbage out. Everyone knows that no working quantum theory can be based on this obscure redefinition of the commutators. More importantly, it seems to me that the author confuses the *-algebra with a Lie algebra. While he might define a strange Lie algebra where the commutators are replaced by i.hbar times the Poisson brackets, this is not enough for a *-algebra. For a *-algebra, he would have to define the actual products. But of course, it is impossible. Classical physics gives us its simplified picture of commutators - namely the Poisson brackets - but it can't give us any simplified generalization of the multiplication itself. The multiplication in quantum mechanics is approximately equal to the multiplication of the corresponding classical objects, but not quite - for example, it depends what the order of the factors is.

OK, so they define some classical Poisson bracket algebra that just pretends to be a quantum algebra, but nevertheless they claim that there is a representation of it according to the theorems. As I noted, this conclusion is based on the confusion between the Lie algebra and the *-algebra. They can't define a product itself of the Pohlmeyer operators such that AB-BA will be the Poisson bracket. This is just impossible. OK, let's forget for a while that they don't have the required *-algebra because they only defined the commutators, not the products. Imagine that they would fix this bug in some way. What would be the relation of the resulting representation and the (standard) string theory?

By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets. Even though the task is different because they artificially define not-completely-quantum commutators that they decide to pick, we can see that the constraint for the Hilbert space to form a representation of *this* algebra is much weaker than the constraint that the Hilbert space can be obtained from a regular quantization of the oscillators.

All the best
Luboš

 

[lumidek]

Lie algebra and *-algebra

Concerning Thiemann's bombastic claims that you can define quantum theory with a Hilbert space such that the ordering issues (and double Wick contractions) disappear, let me summarize it by saying what is the main technical error that leads his to this ludicrous conclusion.

He defines some reparameterization-invariant operators, based on the Pohlmeyer charges, as operators formally associated with the functions on the phase space. And on page 17, he defines their commutators as the Poisson brackets, which is OK to define a Lie algebra. But it is not enough to define the *-algebra because in the *-algebra, you must actually know the product of each pair of operators. Of course, there is no product AB such that AB-BA would be always equal to the Poisson bracket (exactly), and therefore he has no *-algebra, and he cannot apply the formal theorems from axiomatic or algebraic quantum field theory.

 

[eforgy]

Naive question about Nambu mechanics (don't shoot me)

Hello,

I know very little about string theory besides that it involves strings. Woohoo! :)

Anyway...

Urs' apparent enthusiasm about Thiemann's paper has got me following along with interest. One of Lubos' comments

"By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets."

particularly caught me attention. For some unrelated reason, I recently read the paper

http://www.arxiv.org/abs/hep-th/9501141
Dynamical Symmetries and Nambu Mechanics

My first question: is this the same Nambu that you guys are talking about?!

Lubos' comment reminded me of this paper even before I put 2+2 together. Anyway, Nambu mechanics looks really interesting to me and I had never seen it before. In a way, it is a generalization of Hamiltonian system, where instead of a symplectic structure W, you have an antisymmetric "Nambu tensor" N and a Nambu bracket defined as

{f1,...,fn} = N(df1,...,df2).

compared to the Poisson bracket

{f1,f2} = W(df1,df2).

Let me quote the conclusion from the paper:

"We have demonstrated that several Hamiltonian systems possessing dynamical symmetries can be realized in the Nambu formalism of generalized mechanics. For all but one of these systems, an extra freedom was found in the choice of the generalized Hamiltonians needed for their Nambu construction. Finally, one may speculate that since the harmonic oscillator is a very important example in quantum mechanics, its Nambu formulation may lead to a better understanding of the yet unsolved problem of the quantization of Nambu mechanics."

This made me think about what a "Nambu commutator" might be, which may give some insight into the apparent unsolved problem of quantizing Nambu mechanics. My first guess would be the Schouten-Nijenhuis bracket, but that is a wild (sort of) guess.

If I hazard a speculative dive bomb, I wonder if there could be made any sense out of the diagram

point particles -> Poisson bracket

strings++ -> Nambu bracket.

Kind of like p-form electromagnetism.

I'm wondering if the "unresolved" problem of quantizing Nambu mechanics has actually been completed already, but it is called "string theory."

Is there a chance I am making any sense at all? I doubt it, but I thought I'd bring it up.

Cheers,
Eric

 

[eforgy]

Originally posted by eforgy
If I hazard a speculative dive bomb, I wonder if there could be made any sense out of the diagram

point particles -> Poisson bracket

strings++ -> Nambu bracket.

Kind of like p-form electromagnetism.

I'm wondering if the "unresolved" problem of quantizing Nambu mechanics has actually been completed already, but it is called "string theory."

Is there a chance I am making any sense at all? I doubt it, but I thought I'd bring it up.

Cheers,
Eric

Holy!

I just clicked the "cited by" button of that paper expecting to see no hits. Holy! :) At the top of the list, I see

http://www.arxiv.org/abs/hep-th/0312048
Branes, Strings, and Odd Quantum Nambu Brackets

The first sentence of the abstract says, "The dynamics of topological open branes is controlled by Nambu Brackets."

Sometimes even a blind man throwing darts randomly can hit a bull's eye :)

Sorry for the naive question...

I'll do some reading, but I wonder (aloud) if there is a "Nambu commutator"? I'll go treat myself to a beer if it turns out to be the SN bracket :)

Eric

 

[jeff]

eforgy,

Generally I don't complain about off topic posts, but I'm making an exception in this case.

 

[eforgy]

Originally posted by jeff
eforgy,

Generally I don't complain about off topic posts, but I'm making an exception in this case.

Hi Jeff,

I thought the issue of Poisson brackets and commutators was sort of on topic and was one of the cruxes of Lubos' post.

I can't help but think that Nambu mechanics and its quantization is somehow relevant here. I could be wrong. It wouldn't be the first time.

I apologize for the "beer" comment, but I was somewhat excited to see that my instinct wasn't too far off the mark.

There is certainly a lot of interesting things going on here and I'm looking forward to following along (more quietly).


Eric Forgy, Ph.D.
MIT Lincoln Laboratory

PS: If any of the experts have any words to say about this Nambu perspective, I'd very much appreciate it.

 

[ranyart]

Originally posted by eforgy
Hi Jeff,

I thought the issue of Poisson brackets and commutators was sort of on topic and was one of the cruxes of Lubos' post.

I can't help but think that Nambu mechanics and its quantization is somehow relevant here. I could be wrong. It wouldn't be the first time.

I apologize for the "beer" comment, but I was somewhat excited to see that my instinct wasn't too far off the mark.

There is certainly a lot of interesting things going on here and I'm looking forward to following along (more quietly).


Eric Forgy, Ph.D.
MIT Lincoln Laboratory

PS: If any of the experts have any words to say about this Nambu perspective, I'd very much appreciate it.

Being no expert I have an interest in this thread, Having been busy the last couple of weeks doing some heavy reading, I can point you to this paper, it has relevence, and Marcus's original thread will not be comprimised:http://uk.arxiv.org/PS_cache/gr-qc/pdf/0401/0401114.pdf

Abay Ashtekar has a number of recent papers, with a good insight to the development of Hamiltonian methods, this recent paper will be interesting coupled to Thiemann's paper, when one take the crux of it into consideration, a new 'space-time' perception is emerging and it great!

 

[jeff]

Originally posted by ranyart
Abay Ashtekar has a number of recent papers, with a good insight to the development of Hamiltonian methods

For example?

Originally posted by ranyart
...a new 'space-time' perception is emerging and it great!

Describe this "new 'space-time' perception" and explain why it's "great!"

 

[selfAdjoint]

Originally posted by jeff
Right. Early in thiemann's paper on p3 he says:

"...by definition we only consider representations without Virasoro and
Lorentz anomalies, the central charge is zero by definition."

Just to be clear, you do agree that it's fair to view this as basically ground zero of thiemann's disaster?

"By definition"" here just means within the combination of the three frameworks specified above. Probably he meant "by construction". His English is very good but not perfect, and I noticed a couple of lapses into Germanisms, such as a.a. for almost everywhere.

 

[Urs]

Hi -

let me first answer Eric's question: Nambu brackets are related to the quantization of strings and branes in general because they give you a nice way to write the determinant of the pull-back of the target space metric to the worldvolume, and that's what enters the Nambu-Goto action of the p-brane (and the string), which is , slightly generalized (with an additional gauge field on the wolrdvolume), rather known as the Dirac-Born-Infeld action. Of course this detreminant simply (or rather its square root) simply measures the proper volume of the worldvolume.

Here is why this is the way it is:

On a p-dimensional worldvolume the Nambu bracket

{X^m1,X^m2,...,X^mp}

of the embedding coordinates X^mi(sigma^a) (a = 1,2,..p) that map every point sigma of the worldvolume to a point X(sigma) of the target space (our spacetime) is defined as

{X^m1,X^m2,...,X^mp}
= epsilon^{a1 a2 ... ap} (partial_a1 X^m1)(partial_a2 X^m2)...((partial_a1 X^m1)) .

Using this it is pretty easy to convince yourself that the expression

{X^m1,X^m2,...,X^mp}G_m1n1 Gm2n2 ...Gmpnp {X^n1,X^n2,...,X^np}

is, up to a combinatoric prefactor, equal to det(G_ab), where G_ab is the pull-back of G_mn (the target space metric) to the worldvolume.

You can find the details of this construction reviewed in my diploma thesis (analog of master thesis, roughly)

http://www-stud.uni-essen.de/~sb0264/sqm.html

See pp. 175 for the canonical analysis of the general Nambu action using Nambu brackets and page 179 for definition and discussion of the Nambu bracket themselves.

You might in this context also want to look at the Coffee Table Entry

Canonical analysis of D-string action

where I use the Nambu-bracket for the string while canonically analyzing the DBI action of the bosonic D-string.

2d Nambu brackets are particularly nice, because they can be approximated by commutators of large matrices (just like Poisson brackets can be approximated by operator commutators).

For the string, the 2d Nambu brackets appear in the
Lagrangian (in this context one usually speaks of the Green-Schwarz string in Schild gauge). When you replace these Nambu brackets for the string by commutators of large matrices you get the IKKT or Type IIB matrix model.

For the membrane, the Lagrangian usus 3d Nambu brackets, and people are still trying to get a better handle on these, because that would facilitate the covariant quantization of membranes.

However, the 2d Nambu brackets appear in the Hamiltonian of the membrane, obviously. If you replace these Nambu brackets in the membrane Hamiltonian with matrxi commutators you arrive at the BFSS matrix model of M-theory.

Ok, enough for now. Lubos can surely say more about this.

 

[Urs]

Thiemann's Hilbert space is not seperable

Now let me reply to Lubos' message:

Thiemann indeed does not show that the group averaging in (5.5) is mathematically well defined and that it does indeed project onto the space of Pohlmeyer charges. But I am willing to believe that the proof could be given. In any case, (5.5) does not seem to be essential to his construction. What is essential is that the classical Pohlmeyer charges are classical invariants of the string in the sense that they Poisson-commute with the classical Virasoro constraits. What Thiemann is looking for are quantum versions of these invariants, namely operators that commute with all the operator Virasoro constraints.

Now the big question is: How are these operators defined?

Let me say that I do think that his construction of a Hilbert space and of the quantum operators is well defined and that indeed the commutators that he considers reproduce just the classical Poisson brackets. I have tried to indicate why this can be true and why there are no higher order Wick contractions in Thiemann's framework in this comment.

We should not use CFT reasoning when thinking about Thiemann's quantization. There are no OPEs in his framework, no Wick contraction, no normal ordering, etc. His commutators are essentially the same as Poisson brackets. We have to find out what exactly it is that allows him to do away with all this - and if it is viable. Under the shower this morning I had an insight: I believe that the crucial thing is that Thiemann's Hilbert space is non-seperable, i.e. has no countable basis. This implies that it is much larger than ordinary Hilbert spaces and that operators with small worldsheet distance may sit next to each other without feeling each other's presence in form of singular terms of OPEs.

Let me try to recapitulate Thiemann's construction of the Hilbert space:

He starts with the classical algebra of phase space functions

W(I) = exp(int_I Y),

where I is a Borel subset of the circle, i.e. a union of closed intervals.

We have the well defined product relation

W(I)W(J) = phase factor times W(I + J) .

Now the absolutely crucial and non-standard step is to built a Hilbert space where every single one of the W(I) for I in a set of pairwise disjoint closed subsets of S^1 defines a linearly independent state. That's because states are of the form

W(I) Omega

(where Omega is some sort of "GNS-vacuum state"). And states for disjoint Borel sets are orthogonal

< W(I) | W(J) >_Omega = 0 if I disjoint J .

This follows directly from the algebra of the W and the definition of the scalar product <|> by (6.20).

But since there are non-countable many sets of pairwise disjoint closed subsets of the circle (simply because there are uncountably many points) this means that a basis for Thiemann's Hilbert space also is not countable and hence the space is not seperable. This is a mathematically consistent but physically highly pathological Hilbert space. It's non-seperability explains why there are no OPEs and the like, i.e. why the W(I) are not sensitive to 'neighbouring' W(J): The Hilbert space is by construction so large that W({x}) and W({x+epsilon}) can sit right next to each other without noticing each other. They just commute. This is so by construction. It is not a mathematical inconsistency, I think. But it is apparently physically pathological.

Maybe the use of non-seperable Hilbert spaces is the crux of loop quantum gravity. Because in full LQG, too, the Hilbert space of spin networks is taken to be non-seperable. This is carried over by Bojowald to "loop quantum cosmology" and leads to the very non-standard quantization of the Wheeler-DeWit equation.

It would be tempting to conclude that all these LQG constructions are nothing but a proof by counterexample that nature does not want to be quantized on non-seperable Hilbert spaces. :-)

 

[jeff]

Originally posted by eforgy
Hi Jeff,

I thought the issue of Poisson brackets and commutators was sort of on topic and was one of the cruxes of Lubos' post.

Only in relation to thiemann's paper. It sounded to me like your looking for a general discussion.