View Full Version : Amazing bid by Thiemann to absorb string theory into LQG
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."
[o)]
"...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.
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.
[o)]
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.
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 (http://golem.ph.utexas.edu/string/archives/000265.html#c000490) 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 (http://xxx.uni-augsburg.de/abs/hep-th/0401175)? :-)
Best,
Urs
selfAdjoint
Jan26-04, 09:26 AM
"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!
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 acknowledgements 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)
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.
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
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.
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.
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.
Best,
Urs
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 doesnt 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
Jan26-04, 02:20 PM
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
Jan27-04, 12:34 AM
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 percieved position within GR, namely, Observation Dependant on Location.
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 (http://golem.ph.utexas.edu/string/archives/000299.html) for a more detailed discussion.
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 dont mind posting these thoughts in both places.
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 dont 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 doesnt 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
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 (http://golem.ph.utexas.edu/string/archives/000299.html) 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.
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!
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 wont 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 dont mind relaying.
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 (http://golem.ph.utexas.edu/~distler/blog/mathml.html) for further information.
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 (http://golem.ph.utexas.edu/~distler/blog/mathml.html) 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 dont 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
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
Jan27-04, 04:10 PM
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.
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.
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
Jan27-04, 05:54 PM
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.
'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
Jan28-04, 08:54 AM
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
Jan28-04, 08:56 AM
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
My reply to Lubos can be found at the this entry (http://golem.ph.utexas.edu/string/archives/000299.html) of the.String Coffee Table (http://golem.ph.utexas.edu/string/index.shtml).
lumidek
Jan28-04, 11:02 AM
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
Hi -
Here (http://golem.ph.utexas.edu/string/archives/000299.html#c000506) is my reply to the latest message by Lubos.
Originally posted by Urs
My reply to Lubos can be found at the this entry (http://golem.ph.utexas.edu/string/archives/000299.html) of the.String Coffee Table (http://golem.ph.utexas.edu/string/index.shtml).
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 <ψ|constraints |ψ'>=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 dont 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
Originally posted by Urs
Hi -
Here (http://golem.ph.utexas.edu/string/archives/000299.html#c000506) 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][b]=[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^ =exp(iY^) instead of the Y^ themselves, because these would be unbounded.
----------end of quote, sorry symbols not coming out-----
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 accomodate it gracefully.
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.
Best,
Urs
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 [o)]
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 dont 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
Jan28-04, 05:07 PM
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! [6)] 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
Jan28-04, 05:22 PM
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.
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
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
eforgy,
Generally I don't complain about off topic posts, but I'm making an exception in this case.
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 relevent 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).
Best regards,
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
Jan28-04, 10:01 PM
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 relevent 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).
Best regards,
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!
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
Jan28-04, 10:46 PM
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.
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)
Supersymmetric Homogeneous Quantum Cosmology (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 (http://golem.ph.utexas.edu/string/archives/000288.html)
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.
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 (http://golem.ph.utexas.edu/string/archives/000299.html#c000507).
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. :-)
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.
Originally posted by Urs
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. :-)
Hi Urs,
To hear you say this does not bode well for LQG. You are always pretty impartial :)
I'm pushing the limits of my knowledge, but what you seem to be saying is related to a concern I've always had about LQG (not that I know much about it). The inner product of two spin network states is zero unless they are precisely the same state. It never made too much sense to me that if you slightly perturb a spin network state that you get something completely orthogonal. There should be a better notion of "closeness". This is somewhat related to some of the old work of Whitney, where he defines a Hilbert space of surfaces that have a "nice" notion of closeness. It boils down to integration. If a surface S is "close" to a surface S', even if they are nonintersecting, you wil have
int_S W ~ int_S' W
for any smooth W. Jenny Harrison has improved upon Whitney's work and defines another Hilbert space of "close" surfaces.
If I understand you, Thiemann's paper was born from some interaction at the "Loops Meet Strings" conference in order to test LQG with well-known results in string theory. I feel a growing concensus that that effort has failed. I also feel a focusing in on the culprit, i.e. the non-separable Hilbert space.
It is probably obvious, but I suggest that the sickness manifesting itself here is due to a bad choice of inner product. With the correct choice of inner product, they may get a nice separable Hilbert space. If I am anywhere near the mark, this will lead them to (finally) start considering things like the discrete Hodge star (which I have been trying to get them to do for ages). I think that our joint work might come to play here:
Discrete Differential Geometry on n-Diamond Complexes
Eric Forgy and Urs Schreiber
http://www-stud.uni-essen.de/~sb0264/p4a.pdf
In other words, I think that LQG as formulated now is sick because of their choice of inner product (leading to non separable Hilbert space). However, I don't think that (if true) would be the death knell for LQG. They would just need to consider alternative Hilbert spaces. Our work provides one such option for them.
Best regards,
Eric
Hi Eric -
this sound intriguing, but you need to give me more details on what exactly you have in mind. So far in our notes on discrete diff calc we have considered inner products for states (differential forms) on a fixed discrete space. In LQG one has an inner prodcut on the space of discrete spaces, so to say.
It is an interesting quetsion how much freedom one has in LQG to change the inner product. Since full LQG is too complex for me I'd like to concentrate on the toy example that Thiemann provides. He is also discussing 'networks' on the string. How would your proposal apply there?
Indeed, there is (maybe, says Thiemann) a huge amount of freedom in choosing the scalar product in the LQG framework, since it depends on that functional omega. But apparently some natural requirements strongly restrict the possible omega again.
selfAdjoint
Jan29-04, 10:22 AM
Urs, I believe you are correct about the inseparability of Thiemann's initial Hilbert space H_{kin}. What he does in this paper is exactly what happens in many LQG papers in the Ashtekar tradition. The great big initial Hilbert space followed by a dense subsetting operation of some kind. This all seems to be accepted in the mathematical physics community. At least LQG papers have been attacked on many points, as you know, but not on that one.
selfAdjoint
Jan29-04, 10:37 AM
Eric and Urs,
See the papers by Giulini and Marolf, gr-qc/9902045 and gr-qc/9812024, referenced by Thiemann. These develop the group averaging procedure in a general constrained physics situation. Although they speak of its utility for LQG there is nothing specific about LQG in their development. They call the initial Hilbert space H_{aux}.
The real question is whether all these behaviors discovered by mathematical physicists beyond the bounds of what theoretical physicists accept are credible or viable. Thiemann's paper is one laboratory for that question to be studied.
Originally posted by Urs
It is an interesting quetsion how much freedom one has in LQG to change the inner product. Since full LQG is too complex for me I'd like to concentrate on the toy example that Thiemann provides. He is also discussing 'networks' on the string. How would your proposal apply there?
Indeed, there is (maybe, says Thiemann) a huge amount of freedom in choosing the scalar product in the LQG framework, since it depends on that functional omega. But apparently some natural requirements strongly restrict the possible omega again.
First, I want to make it clear that I don't really know what I am talking about :)
From my understanding via conversations with Professor Baez on spr, there are two "schools" on LQG: the northern school and the southern school. The difference between the two is essentially in how they deal with the inner product. The northern school (which Thiemann is a part of) has apparently made the most progress, but their inner product is uglier (in my opinion). The southern school has an inner product that "feels" better, but they have struggled to get results.
I know what it feels like to stress over an inner product and it is really the center of all glory or all failure. The fact that we came up with a nice inner product on a discrete space was thrilling for me since I (in addition to several well known researchers much better than me) had fumbled around unsuccessfully for so long.
The only way we got things to work out so nicely was to abandon simplices as the elementary building blocks in favor of diamonds. Without doing this, we would never have come up with a good inner product.
A question that has been on my mind is whether or not we can do LQG on a diamond complex. I understand that the inner product we have is for fields "in" the space and in LQG it is an inner product of "spaces", but with some fiddling we could relate the two. For example, would it be possible to consider two "spaces" to be some subset of some bigger space. Then they would be "in" the bigger space. I know I am rambling. Sorry.
Anyway, to bring it back to Thiemann's paper, I think (based on your interpretations, which I have some level of faith in) that Thiemann's paper is demonstrating some consequences of starting with an ugly inner product (the northern school).
I think, but could be wrong, that the insight we gained could help correct the situation.
Best regards,
Eric
Hi selfAdjoint!
I don't question that group averaging and all these techniques are mathematically sound.
My question is this: Assuming (which I do) that everything in Thiemann's paper is technically correct (no mathematical errors), then where exactly is the point at which he parts company with the usual lore?
I think this is not an issue of string theory, or of loop quantum gravity at this point. That's because we can think of the worldsheet theory of the string as just one particular example of a quantum field theory in a particularly nice (low) number of dimensions with a simple (scalar) field content.
When this 1+1 dim quantum field theory is quantized by the usual methods, essentially the same techniques are employed that are also used to quantize, say QED. We know that this way of quantizing QED is physically correct, because it agrees with experiment. That's why one would assume that the same techniques are physically correct when quantizing this hypothetical 1+1 dimensional field theory.
Now Thiemann et al say they have an alternative way to quantize field theories. It turns out that this alternative way does not reproduce the usual results which were obtained by quantization methods that originate in something which has experimental foundation.
Note that I am speaking of experiment here in a very unspecific manner. I am not talking about measuring particle masses or similar things, but just about the general empirical fact that the world is governed by quantum theory. Apparently there is some fine-print though, because quantum theories of the same classical thing can apparently be vastly different.
That's why I am trying to spot at which point the crucial assumption is made by Thiemann which distinguishes his approach from others. It might be the assumption that it is OK to deal with nonseparable Hilbert spaces.
Originally posted by eforgy
From my understanding via conversations with Professor Baez on spr, there are two "schools" on LQG: the northern school and the southern school.
I just found the reference to my discussion with Professor Baez
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&threadm=3fa8470f.0106232045.77d132a7%40posting.goo gle.com&rnum=1&prev=/groups%3Fq%3D%2522northern%2Bschool%2522%2Bgroup:s ci.physics.research%26hl%3Den%26lr%3D%26ie%3DUTF-8%26group%3Dsci.physics.research%26selm%3D3fa8470f .0106232045.77d132a7%2540posting.google.com%26rnum %3D1
lumidek
Jan29-04, 12:34 PM
Is this categorization correlated with the difference between the spinfoam path integral approach vs. the canonical operator approach? Or do you have a better description? Thanks.
Originally posted by eforgy
Hi Urs,
quote:
---------
Originally posted by Urs
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. :-)
---------
To hear you say this does not bode well for LQG. You are always pretty impartial :)
...discrete Hodge star (which I have been trying to get them to do for ages). I think that our joint work might come to play here:
Discrete Differential Geometry on n-Diamond Complexes
Eric Forgy and Urs Schreiber
http://www-stud.uni-essen.de/~sb0264/p4a.pdf
In other words, I think that LQG as formulated now is sick because of their choice of inner product (leading to non separable Hilbert space). However, I don't think that (if true) would be the death knell for LQG. They would just need to consider alternative Hilbert spaces. Our work provides one such option for them.
Best regards,
Eric [/B]
congratulations on the diamond complexes you two, will have a look.
puzzled about something: lots of separable Hilbert spaces in Rovelli's book. have a look. download it from his website.
I think it has to do with choosing the diffeomorphism gauge group to include maps that can have a finite number of points where they aren't smooth. Maybe I am not remembering correctly so I'll see if I can find a page reference
Yeah. Section 6.4.2 Page 173
"The Hilbert Space K-Diff is Separable"
selfAdjoint
Jan29-04, 01:14 PM
I remember this categorization. It was overtaken by events when the Ashtekar school took up canonical quantization, I believe after the group averaging and other methods were developed to make it possible in the background free context. There's a 1999 paper by Ashtekar, Thiemann and others on generalizing the Osterwalder-Schrader theorem to background free systems that may mark the beginning of the new direction.
Originally posted by selfAdjoint
I remember this categorization. It was overtaken by events...
You may be mistaken about the "Northern/Southern" distinction going away. I looked at Eric's link to his conversation with Baez and Baez defined southern as Gambini and Pullin
Gambini and Pullin's work continues as a distinct offshoot from the rest. I am uncertain about just why, but it is quite different. They seem to consider a different set of problems---having resolved to their satisfaction some problems that bother the others.
Baez classified Rovelli, Smolin, Ashtekar, Thiemann as "northern". I would rather just call them "majority".
IIRC there were several papers in 2002-2003 from the Gambini/Pullin folks. I guess it is nicer to call them "southern" because of living in Argentina and Louisiana, than to call them "splinter group". Baez always tactful.
Sorry not to be more informative about what makes Gambini and Pullin's work different, maybe looking at their recent papers would
make it clear to anyone curious about it.
selfAdjoint
Jan29-04, 02:11 PM
Originally posted by marcus
congratulations on the diamond complexes you two, will have a look.
puzzled about something: lots of separable Hilbert spaces in Rovelli's book. have a look. download it from his website.
I think it has to do with choosing the diffeomorphism gauge group to include maps that can have a finite number of points where they aren't smooth. Maybe I am not remembering correctly so I'll see if I can find a page reference
Yeah. Section 6.4.2 Page 173
"The Hilbert Space K-Diff is Separable"
But look above on page 164
He says his kinematical Hilbert space \mathcal K (parallel to Thiemann's \mathcal H_{kin}) is inseparable, but don't worry, that's just guage.
The point is that you build a huge Hilbert space of "raw states" and then fillet out of it your separable space of "physical states".
Urs, Eric, what am I missing? why do you say Loop hilbert space is non-separable?
Rovelli's book section 6.4.2 page 173
"The 'excessive size' of the kinematical hilbert space K reflected in its nonseparability turns out to be just a gauge artifact."
To make the theory diffeo-invariant you have to mod out the diffeomorphism group so K is not the final version
In rovelli's notation the diffeo group is Diff
and Diff* is the same thing but including maps that have a finite number of points where they arent smooth.
Modding out by Diff*
replaces spin network states by their equivalence classes (knots).
an equivalence class of networks is a knot
the space of knot states
K_{Diff*}
For Rovelli the real kinematical hilbert space is this separable one. When he is done saying this he says
"This concludes the construction of the kinematical quantum state space of LQG...Now it is time to define the operators."
?
Originally posted by lumidek
Is this categorization correlated with the difference between the spinfoam path integral approach vs. the canonical operator approach? Or do you have a better description? Thanks.
Hi Lubos,
The best categorization is probably found in that URL I gave. I probably couldn't do any better so I'll just quote it.
In early work on loop quantum gravity, folks assumed this derivative existed for the kinematical states of interest. But then Ashtekar and Lewandowski constructed a very nice Hilbert space of kinematical states, clearly "right" in many ways, but with the unfortunate property that the derivative does NOT exist.
Now loop quantum gravity is split into two broad schools, which one could call the "northern" and "southern" schools. The northern school uses the Ashtekar-Lewandowski Hilbert space of kinematical states, and gives up on using loop derivatives. The southern school attempts to make sense of loop derivatives, and works with a space of kinematical states with no clear Hilbert space structure. The main exponents o the southern school are Rodolfo Gambini, Jorge Pullin and their collaborators, mostly from South America. The northern school includes Abhay Ashtekar, Carlo Rovelli and Lee Smolin.
Originally posted by marcus
why do you say Loop hilbert space is non-separable?
Because Urs says so? :)
I don't really know if the Hilbert space is separable or not. So far the only thing I've said that was based on my own knowledge is that I don't like the inner product of the northern approach (it's easy to throw mud). I haven't even attempted to read Thiemann's paper yet and I'm sure it would go way over my head if I did try.
I became infatuated with the loop derivative (as you can see if you follow that link I posted), so I've never been a fan of the northern approach. I'm hoping that the southern approach can find a good inner product and I'm halfway hoping that Urs' and my paper may help in that direction, but the chances are admittedly slim.
If Thiemann's paper ends up casting doubt on their approach to LQG, I'd point the finger at the inner product. It's not clear to me that this is directly related to the issue of "seperable"ness though.
Best regards,
Eric
selfAdjoint
Jan29-04, 04:27 PM
Thiemann seems to regard his \mathcal H_{\mathcal Kin} as separable. See his discussion leading into 5.7: "We now use the well-known fact that \mathcal H_{\mathcal Kin}, if separable, can be represented as a direct integral of Hilbert spaces...".
Let us see why this might be so. Everything here concerns the circle S^1 and we have the following theorem (Oresme-Tcehbyschev:
Let k be a coordinate on the circle that is incommensurable with 2\pi, then the integer multiples of k are dense in the circle.
Thus Urs' nondenumberable covering of S^1 by closed intervals has a countable subcovering by intervals beginning and ending on different multiples of k, leading to a separable space of functional states.
Haelfix
Jan29-04, 04:39 PM
I don't care so much about the extended hilbert space, there is quite a bit of literature in the 50s where analysts play around with that idea (and moving to Banach spaces etc etc). Sometimes they even managed to make working (but equivalent) theorems.
What I don't get in Thiemanns paper, is precisely why he expects the Poisson bracket to carry over into quantum mechanics operators. It strikes me that he would have to do incredible violence to the geometry to pull that off.
selfAdjoint
Jan29-04, 04:45 PM
That's just normal canonical quantization. Define the operator algebra and turn the Poisson brackets, mutatis mutandis, into commutators.
Haelfix
Jan29-04, 09:36 PM
The poisson algebra is classical, to properly quantize it you typically only end up with some subalgebra at best, something new in general.
The rules for quantizing depends on the scheme you use, in canonical quantization you *replace* the Poisson bracket with commutator brackets (curly brackets to square brackets rule). Presto Classical mechanics into Operator mechanics.
Here its not so clear, this almost looks like a *semi* classical derivation of the bosonic sector. Its not so surprising no anomalies show up.
Unless i'm missing something deep in the mathematics of LQG that allows this sort of quantization.
ranyart
Jan29-04, 10:39 PM
Originally posted by jeff
Describe this "new 'space-time' perception" and explain why it's "great!"
For JEFF:
Exhibit 'A':http://uk.arxiv.org/PS_cache/hep-th/pdf/0311/0311011.pdf
These may have relavance to others:
This paper is for those scanning the posts who would like a summary of interest:http://uk.arxiv.org/ftp/physics/papers/0401/0401128.pdf
Recent papers by Moffat:http://uk.arxiv.org/abs/gr-qc/0401117
and one by Gambini-Porto-Pullin:http://uk.arxiv.org/abs/gr-qc/0401117
Martinetti and Rovelli:http://uk.arxiv.org/PS_cache/gr-qc/pdf/0212/0212074.pdf really great!
I have just received email by Thiemann where he confirms that his kinematical Hilbert space in hep-th/0401172 (the 'LQG'-string) is indeed non-separable.
selfAdjoint wrote regarding this question:
Thus Urs' nondenumberable covering of by closed intervals has a countable subcovering by intervals beginning and ending on different multiples of k, leading to a separable space of functional states.
It may have a countable subcovering, but it remains true that there are more states in the Hilbert space than associated with this countable sub-covering. Just imagine: Every subset of S^1 which is the union of a finite number of closed intervals defines a state in H_kin which is orthogonal to any state associated with any other such subset! This are clearly uncountably many mutually orthogonal states.
Thiemann seems to regard his as separable. See his discussion leading into 5.7: "We now use the well-known fact that H_kin, if separable, can be represented as a direct integral of Hilbert spaces...".
Right. But this seems to be just a review of the 'Direct Integral Method' which is not used any further in the paper.
Let me note that the physical Hilbert space in Thiemann's paper is indeed separable. But that's no surprise, the physical Hilbert space is by construction much 'smaller'.
In order to see clearly how this should be compared to the usual approach, consider this:
The ordinary Hilbert space of the OCQ (old covariant quantization) or BRST quantization of the (super-)string is a kinematical Hilbert space because it contains physical and non-physical states. The physical Hilbert space is only the subspace which is generated by acting with DDF operators (http://golem.ph.utexas.edu/string/archives/000301.html) on physical massless (or tachyonic states). Since there are no constraints, i.e. no equations of motion to be imposed on the physical Hilbert space it is really quite inessential whether it is separable or not (it might for instance be an uncountable product of separable superselection sectors). But I believe that it is the non-separability of the kinetic Hilbert space on which all the action with operators and constraints happens, which allows Thiemann's non-standard quantization. See this entry (http://golem.ph.utexas.edu/string/archives/000299.html#c000525) for more details.
I did a little seraching for literature on Hilbert spaces in LQG in general. The situation is pretty confusing for non-specialists, since there seem to be lots of different Hilbert space that were studied. But I think a general pattern is that the kinematical Hilbert spaces are non-separable, while the physical ones often have separable superselection sectors.
I don't see the point in arguing that the non-separability is 'only a gauge artefact'. Yes, sure it is, since the systems we are talking about have no dynamics except for those imposed by constraints.
The question that I consider crucial is whether the space on which the constraints are imposed as operator equations is separable or not. If it is not, apparently very unusual things can happen, as in Thiemann's paper.
Originally posted by Urs
I have just received email by Thiemann where he confirms that his kinematical Hilbert space in hep-th/0401172 (the 'LQG'-string) is indeed non-separable...
There seems to be a difference in what the two authors call the kinematical Hilbert space. I quoted Rovelli (page 173
secton 6.4.2) saying the "kinematical Hilbert space of LQG", the one he uses that is, is separable.
I wonder how far this difference between Rovelli and Thiemann's terminology extends, or if it could be only in the paper
you mentioned (the "LQG-string" one).
Hi Marcus,
I don't have Rovelli's book (anything available online?) but from what you wrote before it seems pretty clear what is going on:
The full kinematical Hilbert space of LQG is non-seperable.
After solving the spatial reparametrization constraints (but not yet the Hamiltonian constraint) we are left with something like an 'almost physical' Hilbert space. This is apparently what is called the 'kinematical Space' by Rovelli. It is separable - since lots of constraints have been solved.
But in Thiemann's paper of the 'LQG-string' there are only 2 constraints (at a given point of the string, of course) and he solves them both at the same time. Actually, because the theory splits into the left- and right-moving sectors, there is in a certain sense only one constraint (at a given point). So here it makes little sense to first solve some of the constraints and then the rest, getting an 'almost physical separable kinematical Hilbert space' as an intermediate object.
The full kinematical Hilbert space of the 'LQG-string' on which all of the original constraints are imposed is non-separable, and I guess that this is true for all LQG quantizations.
The point is that we want to compare the LQG quantization with ordinary quantizations. In the standard OCQ/BRST quantization of the string the Hilbert space if also 'fully kinematical' in the sense that all constraints have to be solved inside this space. No constrains have been dealt with before constructing this space. This means that the fact that the analogous space in Thiemann's construction is non-separable is a real difference to the standard approach, not just a gauge artifact or something like that.
Originally posted by Urs
Hi Marcus,
I don't have Rovelli's book (anything available online?)...
Yes,
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
His homepage is
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
and if you scroll down there's the link to the book PDF file
a fairly recent draft
its now at the press but I just have the draft
Originally posted by Urs
...but from what you wrote before it seems pretty clear what is going on:
... not just a gauge artifact or something like that.
thanks for the explanation[:)]
I'm still not completely clear about how Rovelli's approach differs but that can wait.
Rovelli gets a separable space before he imposes physical constraint
and he says specifically that non-separability is "a gauge artifact".
If you happen to look at that page in the draft of "Quantum Gravity" let me know.
It is curious. something like what you describe must be going on but I am not completely sure.
He gets a non-sep Hilbert space. Then he takes equivalence classes under the action of a certain group
(diffeo gauge group)
so in effect he mods out
identifies kinematic states that are equivalent modulo diffeomorphisms
then the Hilbert space is separable
----------------------
maybe modding out is morally equivalent to imposing a constraint[;)]
selfAdjoint
Jan30-04, 10:00 AM
Urs, your original development of the non separability argulment was this:
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.
Now I claim that every exp(Int_I Y), where I is a Borel interval is in the closure of a dense countable set of exp(Int_K Y), where K is an Oresme-Tschebyschev interval. Is this false? And if true, is this not separability?
The orthogonal side of it doesn't phase me so much because this is explicitly not a Hilbert space of physics states. So instead of "arrows at right angles" or whatever we just have a zero inner product.
Hi sA,
perhaps the separability is not the key issue
but rather *Algebra reps and GNS are more central
to Thiemann's actual paper
but even tho it may be a side issue, separability of
LQG kinematic state space is interesting to discuss.
I appreciate your going back to the General Topology
definition so to speak---the countable dense subset.
In the Rovelli context things tend to be simpler
(and is this necessarily a deceptive simplicity?)
For instance, as I understand it a separable H space is
just one with a countable basis.
So it is no big deal for rovelli to show that his kinematic
state space K_diff is separable.
It just comes out of the spin-network basis.
The spin-networks embedded in the manifold M span the
(non-sep)hilbertspace K
But the theory has to be diff-invariant so we were always
planning to take diff-equivalence classes of states.
This was in the cards. Diffeomorphisms are gauge.
Two elements of K which differ merely by a diffeo are
the same state
But when you look at equivalence classes of spin-networks
there are only a countable number of them
they are abstract knots---only distinguished by their topology, so to speak,
and abstract knots are something you can count combinatorially
So the hilbertspace of equivalence classes has a countable basis.
It is pretty simple, all there on page 173 and whatever that discussion refers to. Only technicality is that he uses
an extended set of diffeos----they can be unsmooth at a finite set of points so like they are "almost-everywhere" diffeomorphisms
this approach----seeing there is a countable basis to the vectorspace---seems more intuitive than the General Topology approach, tho
perhaps less fundamental. does it seem ok to you?
oh, rovelli seems to find the separable kinematic state space convenient for calculating. must be a consideration
Hi selfAdjoint,
in my original post about non-separability here on PF I mistakenly focused on disjoint Borel sets. But that's totally irrelevant because in Thiemann's Hilbert space the states associated with exp(Int_I Y) and exp(Int_J Y) are orthogonal iff I != J. Even if I and J overlap but are not identical the respective states are orthogonal. Therefore it if I is, for instance, the union of J1 and J2, then the states associated with I, J1, J2 are all different and mutually orthogonal.
Now I claim that every exp(Int_I Y), where I is a Borel interval is in the closure of a dense countable set of exp(Int_K Y), where K is an Oresme-Tschebyschev interval. Is this false? And if true, is this not separability?
In which sense do you want to take the closure of these states?
As far as I understand you are arguing that there is a countable set OT of Borel subsets of S^1 such that any arbitrary <Borel subset of S^1> can be written as a (possibly infinite but countable) union of elements in OT.
Even if this is true I don't see how it shows that there are countably many states associated with these sets. That's because a different state is associated with every different <Borel subset of S^1>. Even if the Borel subset I is the union of J1 and J2 the states associated with I,J1, and J2 are all different and mutually orthogonal. So you are right that there are countably many states associated with the Borel subsets in OT, but these are not all the states in the Hilbert space, nor can all the other states be written as linear combinations of the states in OT. That's because any <Borel subset of S^1> that is not an element of OT (even though it may be the union of sets in OT) defines a state which is orthogonal to all the states associated with elements in OT.
Phew! :-)
BTW did you see that over (http://golem.ph.utexas.edu/string/archives/000299.html#c000526) at the Coffee Table Jacques Distler is claiming that Thiemann (and I, for that matter :-( ) makes elementary technical mistakes in his paper? I don't think that his criticism is legitimate, but I guess such a discussion is worthwhile (even though I would rather not be the one disagreeing with Jacques...).
selfAdjoint
Jan30-04, 01:59 PM
Urs, thanks for the clarification. I do see if the states being orthogonal if the intervals differ at all sidetracks my idea.
No I didn't see that attack on the technical competence of the paper. I firmly agee with your approach, to take the paper seriously and see where it leads to different physics. BTW how much of the different physics could be due to the fact that this is a radically non perturbative theory?
Hi -
I don't think that non-pertubativity is any issue at all. The worldsheet CFT usually used to describe the string is also non-perturbatively defined - after all it can be solved exactly (for flat target space, as also in Themann's paper, at least)!
One must well distinguish between the worldsheet theory of the single non-interacting string and the spacetime theory. The latter is defined only perturbatively by summing over an infinite number of CFTs on various Riemann surfaces. It is this summing which makes string theory pertuabtive. Every single contribution to this sum is well defined and exactly defined (nonperturbative on the worldsheet).
selfAdjoint
Jan30-04, 02:07 PM
I think the answer to Distler is Rovelli, or else Ashtekar. We're not talking something that was made up yesterday by amateurs. The Ashtekar group, including Thiemann, has publications in this area going back to the 80's. And just now, Distler discovers elementary errors?
I still think the issue is a crappy inner product :)
They have this inner product that gives completely orthonormal basis elements.
[psi_I,psi_J] = delta_{IJ}
even if psi_I and psi_J overlap. This is analogous to what we found (sorry to keep bringing up our work, but I think its relevent). We corrected this problem by introducing a "g-modified" inner product
[A,B]_g = [A,gB]
for some self-adjoint operator (with respect to the original inner product) of grade zero that "mixes" up the components. I wonder what would happen if they introduced a similar g-modified inner product?
Eric
Originally posted by eforgy
...They have this inner product that gives completely orthonormal basis elements.
[psi_I,psi_J] = delta_{IJ}
even if psi_I and psi_J overlap.
...
...
...for some self-adjoint operator (with respect to the original inner product) of grade zero that "mixes" up the components. I wonder what would happen if they introduced a similar g-modified inner product?
Eric
You have got me curious. do you want to be more specific in describing the LQG inner product?
by psi_I and psi_J do you mean spin network states?
how about hinting how the innerproduct of two such things is defined?
perhaps someone should, just for intelligibility
Originally posted by marcus
You have got me curious. do you want to be more specific in describing the LQG inner product?
Hi Marcus,
I'm no expert for sure, but check out page 142 of
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf
5.3.6 Lattice scalar products, intertwiners, and spin network state.
Equation (5.160) is the culprit (in my opinion).
Eric
selfAdjoint wrote:
I think the answer to Distler is Rovelli, or else Ashtekar. We're not talking something that was made up yesterday by amateurs. The Ashtekar group, including Thiemann, has publications in this area going back to the 80's. And just now, Distler discovers elementary errors?
Yes, I think this is an indication of the fact that there has been very little serious interaction. I hope that string theorist who are convinced that LQG is flawed take the opportunity of the 'toy-example' that Thiemann has provided to precisely pinpoint which steps they do not accept. People have pointed out many physical oddities of LQGs which to many makes it look unacceptable as a theory of quantum gravity (e.g. the lack of semiclassical states so far, or the results of the BH entropy calculation - many string string theorists say that these results are not good at all). But I would very much like to understand the technical problems, if there are any. If it should turn out that there are consistent quantizations of the string worldsheet, for instance, which are inequivalent to the standard one, then I will want to understand this. Maybe my hope is just to understand why I can reject the non-standard quantizations. But to do so I first have to understand the details. That's my goal here.
I am trying hard to answer Distler's charges, see here (http://golem.ph.utexas.edu/string/archives/000299.html#c000530). He seems to be getting impatient with me. If anyone wants to chime in I'd appreciate it!
In my last reply (http://golem.ph.utexas.edu/string/archives/000299.html#c000532) to Distler I discovered that I don't fully understand the following point of Thiemann's paper:
Are the operators exp(i L_n), where L_n are the Virasoro generators, even represented on his Hilbert space?
I am asking because a priori only polynomials in the W have explicit representations. For the Pohlmeyer charges, which instead need the Y ~ ln W there are already lots of subtleties, discussed by Thiemann in section 6.5. Can something similar be done for the exp(i L_n)? Are even the subtleties for the Pohlmeyer charges fully resolved?
And BTW: Why don't we just use the classical DDf invariants (http://golem.ph.utexas.edu/string/archives/000301.html) instead of the Pohlmeyer invariants? They have a much nicer algeba, nicely describe the string's spectrum and have a generalization to the superstring.
selfAdjoint
Jan30-04, 04:21 PM
I think I know why Thiemann did not use the DDF operstors. In GSW they are derived in the light cone gauge, and in Polchinski by CFT methods. Thiemann's effort is to build a nonperturbative quantized string without either of those approaches. So if he wanted to use DDF operators he would have to construct them anew himself. Whereas the Pohlmeyer operators were ready made and available. But this is certainly an effort that could pay off in the future, remembering that they have to be defined within the Thiemann quantization, not by mixing it with pertubrative methods.
Hi -
there is absolutely no problem in copying the idea of the CFT DDF states and turn them into classical invariants. Just replace integrals over dz z^n by integrals over d sigma e^{in sigma}, replace partial X by something proportional to pi + X' (just as in the Thiemann paper, too) and so on. I have done that once. I'll provide the details tomorrow. I have to catch some sleep now (night over germany...).
Best,
Urs
selfAdjoint
Jan30-04, 07:49 PM
Urs, when you read this tomorrow, could you do this? Write up your Coffee Table desription of DDR states here, in LaTex? You just use your ordinary LaTex syntax, and put it between boxes [ tex ] and [ /tex ] without the blanks. If you need a style sheet there is both a quickie one and a more complete one available by link at the "Introducing LaTeX Math Typesetting" which is the second thread in the General Physics board up above here.
I have been trying to follow your Coffee Table description but my browser is IE and it comes out hard to understand.
Thanks.
Originally posted by selfAdjoint
I have been trying to follow your Coffee Table description but my browser is IE and it comes out hard to understand.
Download a copy of mozilla
http://www.mozilla.org/products/mozilla1.x/download/
Hi selfAdjoint -
as jeff says, since the Coffee Table uses MathML to display its math you need a browser which understands this standard. For Wintel that's currently only Mozilla, which is freely available. You might furthermore need to install a font, which is also available on the net for free. For more details see here (http://golem.ph.utexas.edu/~distler/blog/mathml.html).
But thanks for explaining me how to write pretty-printed math here on PF. However, instead of reproducing what I have written at the Coffee Table already I have opted for taking the time to write all this stuff down cleanly in a pdf file:
Urs Schreiber, DDF-like classical invariants of (super)string (http://www-stud.uni-essen.de/~sb0264/p5.pdf).
It is just a set of private notes. Let me know what you think! :-)
\exp(i\pi) + 1 = 0
selfAdjoint
Jan31-04, 02:17 PM
Thanks so much! I have printed it off and I am going to start studying it NOW! What a lot of work went into it.
I am really leery of downloading Mozilla, to fill up my hard drive with unknown software that has unknown interactions with my operating system. Does anyone else have Mozilla with Windows NT? Any war stories?
Hi selfAdjoint -
I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.
Originally posted by Urs
Hi selfAdjoint -
I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.
Urs, I see you are online at the moment
has anyone found any one equation in TT's paper that
you can put your finger on and say is wrong?
could you point me to some "equation number such and such on page
soandso" and say what is wrong with it
or is the controversy(if there is still some discussion) only about the verbal interpretations
and what conclusions to draw?
Hi Marcus -
1) Jacques Distler say that this step (http://golem.ph.utexas.edu/string/archives/000299.html#c000530) is not allowed. That would invalidate the second and third paragraph on p.20.
2) If the constraints themselves are not represented on H_kin then (5.3) to (5.5) would not make sense. I don't know if they are, see here (http://golem.ph.utexas.edu/string/archives/000299.html#c000532).
Originally posted by marcus
could you point me to some equation...and say what is wrong with it...or is the controversy only about the verbal interpretations and what conclusions to draw?
It was clear right away that the LQG-string is unphysical, and whether it's mathematically sound isn't a matter of interpretation.
Originally posted by Urs
Hi Marcus -
1) Jacques Distler say that this step http://golem.ph.utexas.edu/string/archives/000299.html#c000530
is not allowed. That would invalidate the second and third paragraph on p.20.
Urs, thanks for the reply! As I indicated I am looking for some
specific step in the argument or some equation, that is believed mathematically unsound.
I have a better idea now (Jacques says second and third paragraph
on page 20 are invalid)
but I still do not know what false step is supposed to make them invalid because
your link is to coffeetable and covers a lot of ground.
If not too much trouble could someone please point to which particular step
on some page in the TT paper
that Jacques says is wrong.
In the meanwhile I will have a look at those paragraphs on page 20.
does anyone else know what "step" Jacques thought
could be mathematically unsound?
I mean what page and what line on that page
where he says some definite thing that is invalid.
It isnt easy to read the coffeetable and
I dont see any specific reference there to
some equation number or some sentence that Urs
link points to.
what is the wrong "step" that makes second and third
paragraphs of page 20 invalid?
Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.
Here is one of those two paragraphs. What is wrong with it, can anybody say?
"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."
selfAdjoint
Feb1-04, 04:44 PM
For the benefit of others, here are those two paragraphs.
1. Second paragraph on p. 20_________________________
Next we construct from \mathfrak{P} bounded functions on \mathcal{M} which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on \mathfrak{P} be promoted to commutators divided by i\hbar and the adjoint respectively. Denote the resulting *-algebra by \mathfrak{A}.
____________________________
Notes on 1. \mathfrak{P} is a classical Poisson subalgebra constructed from the Pohlmayer charges. \mathcal{M} is the target manifold (background space).
2. Third paragraph on page 20._____
The automorphism groups \alpha^{\pm}_{\varphi}, \varphi \in Diff(S^1) generated by the Virasoro constraints s well as the Poincare automorphism group \alpha_{a,L} extend naturally from \mathfrak{P} to \mathfrak{A} simply by \alpha (W(Y_{\pm})) = W (\alpha,(Y_{\pm})). A general representation of \mathfrak{A} should now be such that the automorphism groups \alpha are represented by inner automorphisms, that is, by conjugation by unitary operators representing the corresponding group elements. Physically the representation property amounts to an anomaly-free implementation of both the local gauge group and the global symmetry group while unitarity implies that expectation values of gauge invariant or Poincare invariant observables does not depend on the gauge or frame of the measuring state. Finally, the representation should be irreducible or at least cyclic.
_______________________________
selfAdjoint
Feb1-04, 04:56 PM
Originally posted by marcus
Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.
Here is one of those two paragraphs. What is wrong with it, can anybody say?
"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."
Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.
Originally posted by selfAdjoint
Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.
selfAdjoint many thanks for this report. I am glad that you have
Mozilla running and can read coffeetable.
I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.
It is certainly possible that TT's paper is flawless mathematically, I should say, and that no one will be able to point to any specific line in it where TT makes a false move.
But as you know it is not uncommon either for math papers to need corrections when they are first circulated in draft and it would be helpful to TT if anyone can find some error or unclear point, which he could be told about so he could have a chance to fix it.
The overall conclusions certainly are interesting, are they not?
selfAdjoint
Feb1-04, 05:43 PM
The conclusions are strong in my opinion. BTW I have been trying to fit Urs' DDF operators into Thiemann's scheme, so far without success (I just don't heve this stuff sufficiently at the tip of my mind).
Here is Distler's first comment on THE LQG String:
Why don’t I just close my eyes, click my heels and wish away all anomalies?
What are the rules here?
It is well known that it is impossible to preserve all of the relations of the classical Poisson-bracket algebra as operator relations in the quantum theory.
What principle allows Thiemann to decide which relations will be carried over into the quantum theory?
Where does he discuss which relations fail to carry over?
To which the answer is, see the last five years of LQG theory, especially by the Ashtekar school.
And here is his second comment.
No, I don’t believe there’s any quantization scheme that takes the full Poisson-bracket algebra of the classical theory and carries it over — unaltered — into the operator algebra of the quantum theory.
Depending on the quantization scheme, you may be able to carry over some subalgebra (the prototypical example, being the CCRs).
And his third comment (getting down to some detail).
OK, so you (he) claim(s) that there is a quantization in which the commutation relations of X '(ó) , Ð (ó) , T + + (ó) and T - - (ó) are carried over from the classical Poisson-bracket algebra, unaltered (i.e., the commutators of the T ’s do not pick up a central term)?
Certainly, that’s not true if the T ’s lie in the universal enveloping algebra generated by X '(ó) , Ð (ó) — as is conventionally the case.
Without being sure, I suspect he's still working from inside string theory here.
And finally, the nuv of his argument.
You mean aside from the fact that none of the symbols are well-defined?
Look, this is elementary stuff.
We can expand everything in Fourier modes. If T + + is in the universal enveloping algebra of the Fourier modes of X ' and Ð , then its Fourier modes (conventionally called L n ) are some expressions quadratic in those modes.
Since the Fourier modes of X ' and Ð (the “oscillators”) don’t commute, you need to specify an ordering. I don’t care what ordering you choose, but I insist that you choose one.
Now compute the commutator of two L n ’s. Again, you will obtain something which is at most quadratic in oscillators (there will, in general, also be a piece 0 th -order in oscillators). And it must be re-ordered to agree with your original definition of the L n s.
Carrying out this computation, you obtain the central term in the Virasoro algebra, and I believe that it is a theorem that the result is independent of what ordering you chose for the L n s.
Note that I never mentioned what Hilbert space I hope to represent these operators on. So I don’t see where its separability (or lack thereof) enters into the considerations.
It's pretty clear here that the X' and \Pi he is talking about come out of the echt string context. "Oscillators"!
Originally posted by selfAdjoint
The conclusions are strong in my opinion.
Respectfully, what conclusions?
selfAdjoint
Feb1-04, 06:46 PM
Jeff, I meant the conclusions of the Thiemann paper, The LQG String.
Notice that I hold that criticism of it based on the techniques and constraints of string physics are by the point, or at least that they have to be explicitly shown to bear on what Thiemann is doing. Almost all of his paper, including the Hilbert space and operator algebra parts, is common to the developments of LQG over the last few years, and much of it is common to the work of mathematical physicists over the past several decades - the GNS construction, for example is truly classic.
I believe the weakest point of the paper is the Pohlmeyer charges, which Thiemann seems to have used as he found them, but I also think that Urs has provided the beginning of a fix for that in his generalized (un-string-ized) DDF charges. The problem with these as far as I understand is that he has provided a classical pre-quantum development of them, and Thiemann is set upon introducing his charges post-quantization. That difficulty is only temporary, I am sure.
BTW, thank you for the link to the Mozilla page. As I said above, I have installed it (browser only) and it seems to be working fine.
Originally posted by marcus
I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.
mozilla will not threaten your existing Internet Explorer installation. it is perfectly safe. the mozilla suite includes a web browser, email client, and html editor, all rolled into one. if you only need a web browser, you can get just the web browser component alone as an application called Mozilla Firebird (http://www.mozilla.org/products/firebird/).
marcus, i think you should give firebird a try. not only is it the only browser around that will let you read MathML, but pop up advertisements will become a thing of the past, and tabbed browsing is very useful.
furthermore, firebird doesn t "install" in your computer at all. you just download the application, unzip it, and double click. don t like it? got tired of it? just delete it, you don t even have to uninstall. it is completely safe and completely free, and there is a chance that you will like it so well, that you will never know why you stuck with IE.
Originally posted by lethe
...there is a chance that you will like it so well, that you will never know why you stuck with IE.
thanks Lethe, that is a persuasive recommendation
my resistance to trying it is weakening, must admit
also message to Meteor:
have been reading parts of the paper by Elias V.
(U. Barcelona) and the Iranian physicist Setare.
Get the impression that QNMs of anything besides
Schw. and ReissnerNordstrom holes are not well
understood at all, which
gives more potential for surprises.
have been reading parts of the paper by Elias V.
(U. Barcelona) and the Iranian physicist Setare.
Get the impression that QNMs of anything besides
Schw. and RessnerNordstrom holes are not well
understood at all, which
gives more potential for surprises.
Seems that the speciality of Mr. Vagenas are 2D black holes. Just see his list of publications.
I will call him tomorrow and will ask him about QNM. And will invite also him to join the forum!
Hi -
let me be more precise: Jacques Distler's point indeed pertains to Thiemann's paper. That's not a question of arguing 'from within string theory' or not and it has nothing to do with being perturbative or not. Thiemann claims in these paragraphs on p. 20 that he can construct a quantization of the classical Poisson algebra of the string which completely preserves the Virasoro algebra without adding a central extension. Distler says that that's not possible in general. This should not be a matter of approaches or opninions.
The point is that Thiemann in passing claims that he can find a quantization of the Poisson algebra such that
\alpha(W(Y_\pm)) = W(\alpha(Y_\pm))
holds true on the quantum level, where \alpha is supposed to be an inner automorphism of the algebra. (second sentence of third paragraph on p.20).
(Hm, something is wrong with the parsing engine. I'll try to continue in another post.)
Explicitly this means that for K some element of the algebra we set
\alpha(K) :=
\exp(i t^I V_I)
K
\exp(-i t^I V_I)
where V_I are the quantized Virasoro constraints and
t^I are some constants that parameterize
\alpha. This obviously has the advertized property of inducing group transformations only if the V_I generate a group as they do classically. Distler says that this is not possible - no matter what, because the anomaly will appear in any imaginable quantization.
So this discussion at the Coffee Table is directly relevant to Thiemann's paper. There I have provided two versions (the functional and the mode-basis one) of how I think Thiemann is quantizing the Virasoro constraints. Distler's point is that that's too naive, since it ignores a subtle issue. I haven't yet had time to fully solve the excercise that he told me to do, but I understand that his point is the following:
Consider what I wrote inthis comment (http://golem.ph.utexas.edu/string/archives/000299.html#c000507). Why does Distler say that's not well-defined? Because it is naively multiplying distributions, which is not a well defined operation. Since the Y(\sigma) are technically operator-valued distributions already their product Y(\sigma)Y(\sigma) is not well defined. This becomes more obvious when one ignores this and caclulates the commutator
[Y(\sigma)Y(\sigma),Y(\sigma^\prime)Y(\sigma^\prime )]
=
2 \delta^\prime(\sigma,\sigma^\prime)
\left(
Y(\sigma)Y(\sigma^\prime)
+
Y(\sigma^\prime)Y(\sigma)
\right)
=
4 \delta^\prime(\sigma,\sigma^\prime)
Y(\sigma)Y(\sigma^\prime)
-2
\delta^\prime(\sigma,\sigma^\prime)
\delta^\prime(\sigma,\sigma^\prime)
Here the first term is the one that gives the usual Virasoro algebra, while the second term comes from re-ordering and should be related to the anomaly. It is however not well defined when written this way. Depending on how you decide to deal with this term it might look like 0 when integrated over or like infinity.
So this does not make sense. The above commutator has be be regulated by introdicing appropriate smearing functions. After computing the regularized commutator these smearing functions can be taken to be delta-functions again. This is the functional version of what Distler proposed to do, namely to introduce a cutoff in the summation over modes in the Virasoro generators. And it should produce an honest and well defined term which is an anomaly.
So the claim is that Thiemann is using the naive quantization where you don't see the anomaly, even though it is really there.
Originally posted by meteor
Seems that the speciality of Mr. Vagenas are 2D black holes. Just see his list of publications.
I will call him tomorrow and will ask him about QNM. And will invite also him to join the forum!
Yes! I have a few papers on my computer which I cannot axcess(PDF-adobe fails).
It is very interesting that dimensionally BHs are not only holding Galaxies intact(Spacetime), but there is increasing evidence that they are Inter-Dimensional crossroads.
QNMs are where a certain dimensional energies meets another ..different dimensional Energy.
Originally posted by Urs
...Why does Distler say that's not well-defined? Because it is naively multiplying distributions, which is not a well defined operation. Since the Y(\sigma) are technically operator-valued distributions already their product Y(\sigma)Y(\sigma) is not well defined. This becomes more obvious when one ignores this and caclulates the commutator
[Y(\sigma)Y(\sigma),Y(\sigma^\prime)Y(\sigma^\prime )]
=
2 \delta^\prime(\sigma,\sigma^\prime)
\left(
Y(\sigma)Y(\sigma^\prime)
+
Y(\sigma^\prime)Y(\sigma)
\right)
=
4 \delta^\prime(\sigma,\sigma^\prime)
Y(\sigma)Y(\sigma^\prime)
-2
\delta^\prime(\sigma,\sigma^\prime)
\delta^\prime(\sigma,\sigma^\prime)
Here the first term is the one that gives the usual Virasoro algebra, while the second term comes from re-ordering and should be related to the anomaly. It is however not well defined when written this way. Depending on how you decide to deal with this term it might look like 0 when integrated over or like infinity.
...
Urs, thanks for taking the trouble to
move things over here to PF and write it out
in this level of detail. I much appreciate
and suspect that others will too!
Also this point about not being able to directly
multiply distributions---this is an understandable
specific criticism. At first sight I cannot agree
or disagree but it is the kind of error that
people (even good mathematicians) can easily make
now and then in a long paper. The distributions look
like ordinary (true) functions and one does not watch
and then one treats them as if they were!
I hope more specifics like this will come to PF
where I will get a chance to see. (still do not have
Mozilla)
Originally posted by Urs
...Thiemann claims in these paragraphs on p. 20 that he can construct a quantization of the classical Poisson algebra of the string which completely preserves the Virasoro algebra without adding a central extension. Distler says that that's not possible in general. This should not be a matter of approaches or opninions.
The point is that Thiemann in passing claims that he can find a quantization of the Poisson algebra such that
\alpha(W(Y_\pm)) = W(\alpha(Y_\pm))
holds true on the quantum level,...
I see where he says that (in passing) at the beginning of the third paragraph on page 20.
And then later in the same paragraph he says:
"Physically this representation property amounts to an anomaly-free implementation of both the local group and the global symmetry group..."
So (while I cannot yet form an independent judgement that takes everything into account) I can at least see where Jacques point connects to the paper. This is a real help!
Offhand I would say it sounds like the sort of thing TT would like to be told about, and I hope this happens.
selfAdjoint
Feb2-04, 10:50 AM
Urs, I can't believe Thiemann made a mistake about multiplying distributions. He's an authority in this field and has a book coming out on it. Marcus, in addition to that book, he cites Rovelli's forthcoming book Quantum Gravity to to support his construction. I'm going to dig into that, but as you're now the expert on Rovelli maybe you can look too to see how he does this algebra transformation.
Hi -
I have now checked the calculation that Jacques Disller told me to do. Indeed, when one regulates the non-normal-ordered Virasoro generators that Thomas Thiemann is using, computes their commutators and then removes the regulator afterwards, one picks up an anomaly. Jacque's point is that without doing this regularization one is dealing with ill-defined quantities. I think that's uncontroversial, I have discussed it in my previous message.
Thiemann is in contact with us. I am going to ask him about this issue.
Originally posted by Urs
Thiemann is in contact with us. I am going to ask him about this issue.
that sounds like a constructive thing to do!
(i'm not trying to guess if there is an error or not, either Distler or Thiemann could be mistaken I suppose, but asking about it is a helpful initiative)
I am glad to know you are in contact with Thiemann.
Thiemann says that he knows about the issue raised by Distler! But he also says that he thinks that it does not affect anything he did.
He says that he does not represent the constraints on the Hilbert space but just imposes the condition
U(T)W(Y) \Omega_\omega
=
W(\alpha(Y))\Omega_\omega
by hand.
I thought all along that this should follow from his quantization of the constraints, but apparently that's the definition of the action of the U-operators.
So the issue here is not one of mathematical correctness, after all.
But is it physically viable to define the anomaly away like this?
Thomas says he will join the Coffee Table discussion. Let's see what happens.
Originally posted by selfAdjoint
Marcus, in addition to that book, he cites Rovelli's forthcoming book Quantum Gravity to to support his construction...
it would save me some fumbling if you point me at a particular page of TT where he uses something from Rovelli's book.
concerning this fracas about the anomaly,
I cant second guess but I do know
that math papers when they first come out are checked over and
nitpicked by friends and colleagues, and even good people
can make errors and really benefit from other people
scrutinizing their work. I hope TT is getting a lot of this
feedback and that Distler is only one of many looking for nits.
Originally posted by Urs
Thiemann says that he knows about the issue raised by Distler! But he also says that he thinks that it does not affect anything he did.
...
Thomas says he will join the Coffee Table discussion. Let's see what happens.
great news! glad to hear it.
This (http://golem.ph.utexas.edu/string/archives/000299.html#c000550) lastes comment by Distler looks important to me.
Originally posted by Urs
Thiemann...just imposes the condition
U(T)W(Y) \Omega_\omega
=
W(\alpha(Y))\Omega_\omega
by hand.
I thought all along that this should follow from his quantization of the constraints, but apparently that's the definition of the action of the U-operators.
So the issue here is not one of mathematical correctness, after all.
Did I not try to warn you about this (albeit in retrospect using somewhat sloppy phrasing) much earlier in the thread?:
Originally posted by jeff
Urs,
Did you notice thiemann's declaration on p3 that the reps considered in the paper have been taken by definition to be anomaly-free right out of the box? Thus it may be more the choice of representation than the method of quantization that's at the heart of this.
Originally posted by Urs
But is it physically viable to define the anomaly away like this?
This is yet another example of the contrived approach to theory construction so characteristic of the whole LQG program.
selfAdjoint
Feb2-04, 05:04 PM
Jeff, I still say you are misinterpreting Thiemann's repeated use of "by definition" on page 3. I read him as meaning "By the definitions of our systems (which are valid in and of themselves) these results fall out". What he is certainly not saying is "We are just going to write in whatever we want".
Originally posted by selfAdjoint
Jeff, I still say you are misinterpreting Thiemann's repeated use of "by definition" on page 3. I read him as meaning "By the definitions of our systems (which are valid in and of themselves) these results fall out". What he is certainly not saying is "We are just going to write in whatever we want".
We disagree not about the interpretation, but rather about it's significance: thiemann is making these "definitions" without the further justification that is required by the unavoidability of operator ordering anomalies as jacques distler pointed out and as verified by urs'.
selfAdjoint
Feb2-04, 09:34 PM
Jeff, bear with me. The following is not intended at all as an insult to string physicists, a group of people I admire greatly.
Think of a cartoon where a group of mountaineers are laborously and bravely ascending a difficult peak. At last they reach the top and are stunned to see a bunch of people in festive clothes, lounging around with refreshments. How can this be? they shout. Easy, says the group, there's an escalator up the other side of the mountain!
Now imagine what use are the following arguments.
- Everybody knows you can't reach the top without crossing the ice field. They didn't cross the ice field so they couldn't have come up. They're a mirage!
- Everybody knows that you can't ascend this mountain without using pitons. If you don't place pitons, you will fall. They didn't place pitons, so they will shortly fall!
IF (big if!) Thiemann has discovered a way to come at string theory from another direction, using different but well tested methods, then repeating over and over that they aren't using YOUR methods or obeying YOUR constraints is just fruitless. And saying that the work is only mathematical not physical only repeats a canard; if they can predict things and be right, that's physics. Everything else is just different schools of math.
The way to falsify The LQG String is to search into it's methods, learn their basis, and show they are false, or that the string cannot be described from that direction. But remember: the string he claims to describe is the nonperturbative string, so telling us things about a perturbative string is not going to work. You may be an expert on snarks, but Thiemann is talking about a Boojum.
So it seems the basic issue has been resolved. We had long arguments about how Thomas Thiemann can avoid the anomaly and related issues. (I, for one, did learn something in the process. Thanks to everybody who participated in the discussion!) But in the end it turned out that all these arguments about technical issues of representing the string constraint algebra had no relevance to Thomas Thiemann's paper. That's because he does not represent the Virasoro constraints and does not deal with their quantum algebra. What he does is this:
He says that there exist operators U_\pm(\varphi) on his Hilbert space which act on the string oscillators just as classical Poisson brackets would. He declares that the quantum theory is solved when states are found which are invariant under these U_\pm(\varphi).
There is no doubt that these U_\pm(\varphi) do exist. They are defined by their desired action on the states.
The intensive discussion about how these U_\pm(\varphi) might be constructed in terms of the canonical variables \pi^\mu and X^{\prime\mu} was vain: They are not constructed or indeed constructible from these canonical operators.
As Jacques Distler says: It seems that we haved achieved nothing by pointing out that the U_\pm(\varphi) exist.
I believe that this point can be made quite clear by the following simple argument:
I can by straight analogy define similar operators U_\pm(\varphi) on the standard string Hilbert space, i.e. a 2-parameter family of operators that has the algebra of the classical conformal algebra in 2d. (Simply define these operators by their action on every single state.) If I were to declare that the quantum string is given by states invariant under these operators on the standard string Hilbert space I would see no anomaly here, either!
Thomas Thiemann appeals to Dirac's quantization procedure. But this procedure says that the classical first class constraints should be implemented as operator constraint equations in the quantum theory. However, the U_\pm(\varphi) are not the exponentiations of the quantized classical first order constraints. They are not expressible in terms of these quantized constraints even in principle.
The reason why no quantum effects are seen in Thomas Thiemann's paper now is seen to be the result of the fact that the quantum constraints are not imposed, but an auxiliary set of constraints which is modeled after the classical theory but does not follow from any standard quantization procedure.
As an example, consider what it means that L_0 is not represented on Thomas Thiemann's Hilbert space: It means that he does not get the Klein-Gordon equation, which is nothing but L_0|\psi\rangle = a|\psi\rangle for string states of a given mass. This is not even representable in his approach. So not only relatively esoteric issues like the anomaly are lost, but even the well known behaviour of point particles of a given mass is not describeable.
I am afraid that I have to second Jeff's point.
(I also want to repeat that it is not true that Thomas Thiemann's approach is distinguished from the standard quantization by being non-perturbative. It is, but the standard quantization is, too. One must make sure not to confuse what the non-perturbativity of string theory is referring to, namely to the physics of target space, not that on the worldsheet. The standard quantization of the Nambu-Goto or the Polyakov action can be completely and exactly solved, the space of solutions being spanned by the DDF states. If a theory is completely solved it is non-perturbatively understood, obviously. The perturbation expansion in string theory appears instead when we compute target space amplitudes by summing CFT correlators for worldsheets of arbitrary genus. Every single such correlator (an expectation value for a CFT on a surface of given topology) is nonperturbatively and even rigorously defined. What is not really defined is the infinite sum of all these correlators which is supposed to give the target space amplitude.)
With this insight into the LQG quantization of a 1+1 dimensional rep invariant field theory it would now be highly interesting to have a second look at the quantization procedures used in LQG for the quantization of 1+3d gravity.
Originally posted by Urs
With this insight into the LQG quantization of a 1+1 dimensional rep invariant field theory it would now be highly interesting to have a second look at the quantization procedures used in LQG for the quantization of 1+3d gravity.
Urs, it would be great if you and others would examine the approach to quantizing 1+3-dimensional gravity taken in TT's book, which is expected to come out this year.
Also if you wish, compare the approach to LQG in Rovelli's book, which is quite different:
it does not use such abstract algebraic and functional-analytic machinery as the GNS construction nor does it force certain conditions to be met by algebraic "fiat".
Rovelli's construction of LQG is far more "down-to-earth". But I don't want to suggest that one version is more valid than the other---although the contrast is stark (as Rovelli points out in his Preface).
(As I noted earlier, Rovelli's kinematical state space is separable but AFAIK that difference may be merely superficial and arise from the order in which things are defined.)
Originally posted by selfAdjoint
... You may be an expert on snarks, but Thiemann is talking about a Boojum.
[:D]
selfAdjoint, does TT's strategy remind you of how some professors construct the complex numbers as
R[x] modulo an ideal generated by the polynomial x2+1.
They make a polynomial ring and factor it down by the ideal denoted by (x2+1).
So the complex numbers are written
R[x]/(x2+1)
And thus they force the equation x2= -1
to be true by fiat.
and then there remain things to be proved about that object
(you do not save work you just change the order in which the work
must be done, but it has a kind of "mod" elegance)
the Boojum method
Hi Marcus -
it will be hard to find the spare time to look into the LQG constructions for 1+3d gravity in full detail. But I have a rough idea of what they are doing (from review papers, talks, and discussions with LQGists).
But maybe, now that I/we know what we have to be looking for we can simply make the experts tell us. Over at the Coffee Table Thomas has joined the discussion (http://golem.ph.utexas.edu/string/archives/000299.html#c000554) and I am beginning to ask him about 1+3d gravity.
He already told me that there the spatial diffeo constraints are imposed and solved by the same method as in the LQG-string, while the Hamiltonian constraint is imposed the ordinary way. (Does that resonate with what you see in Rovelli's book?)
That's, now that we have understood the LQG-string, a very valuable and interesting information. Maybe the whole diagreement about the viability of LQG can be condensed to the question whether it is really allowed to call constraints of the form used in the 'LQG-string' a quantization.
I think that's precisely where the problem lies.
Originally posted by Urs
I think that's precisely where the problem lies.
Hi Urs,
Is this as potentially deadly as it sounds? It almost seems like LQG is one proof away from going up in smoke.
Eric
Originally posted by selfAdjoint
Jeff, bear with me. The following is not intended at all as an insult to string physicists...
However they initially strike me, my assumption that your posts aren't meant as insults is automatic so you really don't need to worry about that.
Originally posted by selfAdjoint
You may be an expert on snarks, but Thiemann is talking about a Boojum.
I looked up "snark" and "boojum" in my dictionary (shorter oxford). Are you saying I'm expert on elusive truths while thiemann is expert on dangerous imaginary theories?
Anyway, you already understand the main ideas of LQG well enough. Maybe it was the breadth of urs' perspective that allowed him to have so much fun with this.
selfAdjoint
Feb3-04, 01:32 PM
I did really enjoy Urs' development of the classical DDF states. That was educational too. And who knows, ther may be life in the old LQG yet, I don't think we have penetrated to the bottom of what Thiemann is about yet.
By Snark and Boojum I only meant that the mysterious and deadly Boojum was an as yet undescribed variety of the commonplace and innocuous Snark. Undescribed in Carrol's poem because it killed everyone who ever saw it. But it was just two varieties of a single thing with very different characterisics that I brought it up.
I hope the gamekeeper is around! :-) I have read only the last Harry Potter in the original English version, so I guessed Snarks and Boojums might have appeared in Hagrid's class in the years before... ;-)
Anyway, Eric writes:
Is this as potentially deadly as it sounds? It almost seems like LQG is one proof away from going up in smoke.
Let me focus on technical issues which have a definite answer:
I know understand something quite important which was not apparent to me before:
Not all of the constraints used in LQG are represented as operators on some Hilbert space.
This has just been confirmed by Thomas at the Coffee Table. There he writes
I understand that the question whether constraint quantization can be done with the group or the algebra is controversial. The uneasy feeling may come from your experience with Fock spaces of which perturbative path integral quantization is just another version. In those representations one usually deals with the algebra, however, notice that one can work as well with the group. So you question my procedure by using an example where both approaches work. I would say
that there is no evidence for concern. For instance in LQG we have a similar phenomenon with respect to the spatial diffeomorphism group. We can only quantize the group, not its algebra. Yet the solution space consists of states which are supported on generalized knot classes which sounds completely right. There are other examples where the group treatment, also known as group averaging or refined algebraic quantization produces precisely the correct answer.
See for instance [23] and references therein.
So is the question: "Group or algebra?"?. I am not sure. To me the problem rather seems to be that the group constraints that Thomas uses are built by hand, modeled after the classical group action. But I think we would rather want the constraints drop out of the quantization process by a quantization mechanism (compatibility to the path integral woudn't hurt). This mechanism gives us the quantized first class constraints, i.e. the quantized constraint algebra. Is it ok to simply ignore it and construct different operators and using them as constraints? I'd say the LQG-string shows that this is not ok.
But at this point we are bitten by the fact that we are physicists, not mathematicians. One can always claim that the new, modified, quantization is what really describes nature. Maybe it would not even help if we could see the LQG quantization of some system that can actually be tested experimentally. If the approach failed to comply with experiment one could still claim that this ordinary system is not described by LQG quantization, but that quantum gravity is!
On the other hand, quantum gravity can be tested, right now. The 0th order approximation is classical gravity. LQG could still be tested by showing that it can, or cannot, reproduce smooth, locally flat space, gravitons, etc. As you know, so far this has not been done.
But, if there are gravitons to be found in the theory, it must of course have some description in terms of a path integral, at least in the appropriate limit. It looks quite problematic then that the fundamental theory cannot be described by a path integral.
Ok, I am rambling. To answer your question a little bit clearer: Personally I am more sceptic about LQG after having seen the LQG-string then I was before.
Originally posted by Urs
Ok, I am rambling. To answer your question a little bit clearer: Personally I am more sceptic about LQG after having seen the LQG-string then I was before.
Hi Urs,
This is interesting. I guess a more constructive question would be to ask if something can be done to fix the situation? You seemed to be suggesting an alternative in the String blog. Is there anything to that line of thought? I hope all this ends up bringing about more discussion among the two camps.
Of course every theory of today is sick is some way or another and could use improvement.
Eric
Why should it necessarily have a description in terms of path integrals? Functional integrals as we all know, are not well defined.
In fact its not even that they are not defined, its the fact that they can be WRONG! Its a well known fact that canonical quantization and path integration need not produce the same results, past the classic solutions; often a clever mathematician has to tinker with them to get something that looks realistic.
Physicists that naively expect it to produce 100% accuracy, in every context are 1 step shy of being delusional.
The path integral should in correct a future lorentz invariant quantum field heory, yield something new, more general and more mathematically sound 'quoth A. Zee'
However, the last part is of course valid. LQG must produce flat space as a limit at some point for it to be considered more than a mathematical excursion.
selfAdjoint
Feb3-04, 08:06 PM
Where did you get all this about functional integrals not being well defined? It's bushwa.
Originally posted by selfAdjoint
Where did you get all this about functional integrals not being well defined? It's bushwa.
i m not exactly sure what he is talking about, but it is probably the complaint heard from mathematicians all the time that there is no invariant measure with which to do path integrals.
Functional analysis is the branch of mathematics that should incorporate path integrals, unfortunately so far its never been able too. There is no precise definition analogous to say a Lebesgue measure, or an epsilon delta proof for analysis.
In short, its not well defined, much like say the Dirac delta wasn't defined until distribution theory came along.
Its somewhat painful for physicists to accept this, and is curiously not common knowledge the exact extent of the discrepancy. Feynmann, was very well aware of the problems, in fact some of his first papers include appendixes describing them. Many physicists of the time, felt that it was simply a calculational scheme, and that canonical quantization was the real heart of *real* QFT.
For instance, in most books on Quantum field theory, they'll show a proof of the duality between canonical quantization and path integrals (pionered by Dyson). Unfortunately, its wrong, valid only in certain circumstances (say when the action is nice and positive quadratic) or when it approaches regular quantum mechanies (the Wiener measure)! Mathematicians, will point out that in general there are serious operator ordering issues, and considerable abuse of complex structure. Wicks theorem is completely abused for instance, all throughout field theory.
In short, if you go through the details, you'll find that in general path integration must be treated on a case by case basis with considerable care and often adhoc assumptions to fix the problems. Branches of field theory have tried to make all this rigorous (one approach is the algebraic/axiomatic QFT people) but have been so far unsuccessful in making everything work and mantaining the nice experimental success that functional integrals enjoy.
So I mantain, that its premature and silly from a logical and mathematical point of view, to blindly excpect that all this works in quantum gravity. It might, but then there is reason to think that it might not!
It is true that path integrals in general, or the conclusions that are drawn by 'using' them, are not rigorously defined - as is true for QFT in general. In his introduction to the Yang-Mills problem in the Clay Millenium Prize questions Witten emphasizes that mathematicians should try to come to terms with QFT in the future. He knows that many beautiful results are hidden there.
Still, many results in QFT are obtained by using path integrals in a semi-heuristic way and somehow it works. Most of modern quantum field theory is in fact defined in terms of perturbative path integral calculations.
Anyway, for special cases path integrals are rigorously defineable. This is in particular true for Gaussian ones describing free field theories. The wordlsheet theory of the string (in trivial flat background) happens to be free and the path integral is well defined. The Wick rotation is also unproblematic (as discussed in Polchinski) and one can work on the Euclidean worldsheet. The path integral quantization of the string gives of course the 'standard' result, including the anomaly etc.
If we ignore the fact that we are dealing with a string which is supposed to be embedded in some target space and simply regard the theory as an example of a free quantum field theory in 1+1 dimensions, sort of as a toy example for interacting field theories in 3+1 dimensions, then it is sort of disconcerting that the 'LQG-string' does not reproduce the path integral quantization result.
But it is no suprise that it does not: The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using.
Originally posted by selfAdjoint
"We" is Thiemann. Marcus is quoting from the abstract as you will see if you check the link.
Sorry the pedantic mode, but "we" in a scientific text means "the author and all the people who is following the reasonment"
Originally posted by arivero
Sorry the pedantic mode, but "we" in a scientific text means "the author and all the people who is following the reasonment"
It seems likely that this is what Thiemann meant when he said "we" in his abstract. (My post consisted of a quote of the abstract, essentially without comment, to start a thread of discussion.)
BTW Alejandro, am I right that you have discussed DSR in past threads? I seem to recall your mentioning Amelino-Camelia, maybe also Magueijo. Might you contribute your current opinions if we began a thread on DSR?
selfAdjoint
Feb4-04, 09:16 AM
Urs, you wrote
If we ignore the fact that we are dealing with a string which is supposed to be embedded in some target space and simply regard the theory as an example of a free quantum field theory in 1+1 dimensions, sort of as a toy example for interacting field theories in 3+1 dimensions, then it is sort of disconcerting that the 'LQG-string' does not reproduce the path integral quantization result.
But it is no suprise that it does not: The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using.
I have a couple of questions. How do you go about quantizing the 1+1 toy model? And where does the "by hand" come into Thiemann's derivation? Is it imported with group averaging? I guess what I am asking here is whether the fault you see goes back to what we might call the Potsdam school of algebraic quantization or is specific to Thiemann's use of that material.
One other thing (actually this is to anyone reading this) on the Coffee Table site, Distler recommended a paper:
Alvarez-Gaumé and Nelson, “Hamiltonian Interpretation Of Anomalies,” Commun. Math. Phys. 99 (1985) 103.
The online issues of that journal don't go back to 1985, and are behind a Springer pay wall anyway. I live in a small Wisconsin town hundreds of miles from any university that might carry such a journal. Do you know of any other source that I might find the same information in? A morning of scrounging around in the cites of Thiemann's paper and of the group averaging papers he does cite haven't brought me any enlightenment on the anomaly question in the general modern algebraic quantization of Hamiltonian systems with constraints. Neither did arxiv search on hamiltonian quantization anomaly. Couldn't find anything on it in the math phys books I have, Haag and Baez-Siegel-Zhou, either.
Hi selfAdjoint -
well what do you mean by asking how to quantize the toy model? Begin by deriving the constraints from either the Nambu-Goto action or the Polyakov action and impose them as usual to get the usual spectrum of the string=1+1d toy model.
The "by hand" in Thomas Thiemann's approach comes in at the point where he ignores the quantized constraints (the quantum Virasoro generators) and chooses to call physical states those which are invariant under his operators U_\pm(\varphi). These do not follow from the quantization procedure but are constructed by hand so that they have the desired group algebra. This is where his approach differs from the standard approach.
I don't really quite know what the definition of 'algebraic quantization' is and if that's related to the way Thiemann chooses the constraints. If you do, please let me know. I guess that algebraic quantization refers to taking a classical Poisson algebra and trying to deform it into a quantum commutator algebra. Thiemann is doing that, too, in his paper, but that's a different issue.
I understand this better in terms of the classical DDF invariants. When you impose the constraints the way Thiemann does ('by hand') then the classical DDF invariants are literally the same when quantized. But when the proper quantization is used then these logarithmic correction terms have to be included that I discuss in my draft (http://golem.ph.utexas.edu/string/archives/000300.html#c000562). These logarithmic correction terms are related to the anomaly and stuff, which is missing in Thiemann's approach. I haven't checked yet, but the logarithmic terms should be essential for making the 'longitudinal' DDF states null, as it should be to avoid negative norm states.
I have ordered a pdf copy of the article that Distler mentioned. As soon as I obtain it (might take a couple of days) I'll send you a copy.
Best,
Urs
Originally posted by marcus
BTW Alejandro, am I right that you have discussed DSR in past threads? I seem to recall your mentioning Amelino-Camelia, maybe also Magueijo. Might you contribute your current opinions if we began a thread on DSR?
Yes I also remember to take a look to the papers on doubly special relativity a year ago but I do not remember which my opinion was. So I guess I was not very excited. Still it could be interesting to open a thread on it. I will keep an eye on the forum.
Hi Alejandro -
you are an expert on NCG: Do you have any experience with using the spectral action principle for getting gravity and gauge theory form a spectral triple?
I am asking because Eric Forgy and myself are planning to insert our NCG on discrete spaces (http://www-stud.uni-essen.de/~sb0264/p4a.pdf) into the spectral action principle in order to get a theory of discrete gravity. Any help by experts is appreciated! :-)
In general, I would be interested in hearing your comments on the paper at the above link. The introduction has a brief overview over the central ideas.
All the best,
Urs
Originally posted by selfAdjoint
How do you go about quantizing the 1+1 toy model?
I'm not sure what's puzzling you here.
Originally posted by selfAdjoint...where does the "by hand" [i.e., urs: "The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using"] come into Thiemann's derivation?
I don't want to put words in urs's mouth, but I think he meant that thiemann defines his theory so that the structure of the classical lie algebra of the gauge group is preserved in the quantum theory. See the paragraph of equation (5.2) for the general idea.
Hi Jeff -
yes, exactly, that's what I meant. The point is that what Thiemann does is find some operators which represent the classical gauge group on his Hilbert space. These operators, the U_\pm(\varphi) don't come from quantiizing the classical first class constraints. They come from using analogy with the classical theory.
Note that on a large enough Hilbert space it is possible to find operators that represent all kinds of groups. (Essentially because one can think of these operators as being large matrices and almost everything can be expressed in terms of large matrices.)
selfAdjoint
Feb4-04, 10:39 AM
OK, now I am baffled. Group averaging is a method of quantizing a hamiltonian system with constraints.
Thiemann says:
Alternatively in rare cases it is possible to quantize the finite gauge transformations generated by the classical constraints by the classical constraints provided they exponentiate to a group.
It seems to be this exponentiation that you regard as put in by hand. I have been trying to track down the source of Thiemann's statement above but so far have been unable to find it.
Once he starts on group averaging he has available the theorem in the Giulini and Marolf paper he cites as [23], if you can meet the hypotheses of their group averaging construction, then the quantization is essentially unique, at least there is a unique "rigging function" from the kinematical Hilbert Space to the physical one. So if there is a quantization at all, and G.A. applies, then the G.A. quantization is that existing quantization.
So it doesn't look like G.A. takes us away from the familiar quantization with the anomalies; it must be the exponentiation that does it.
So is it your thesis that the exponentialion of the constraint operators is arbitrary? Thiemann seems to be saying he has support for doing it, and it's not just an arbitrary procedure.
Hi selfAdjoint -
good that you insist on clarifications about this point. It is THE most crucial point, apparently.
Yes, the group averaging method as such is most probably completely uncontroversial as a mathematical technique.
But note that Thiemann writes "provided the CLASSICAL constraints exponentiate to a group". So he checks if the classical constraints exponentiate to a group and then he constructs an operator representation U(t) of that group on a Hilbert space. But these U(t) are not the exponentiated QUANTUM constraints (if there is an anomaly)! They are just some operators that represent the classical symmetry group of the system.
So this way the classical symmetry is put in 'by hand'. Distler says, an I think that he is right, that this is not how QFT works. We should not just try to represent the classical symmetry on a Hilbert space.
Even though mathematically this can be done, it is not related (or so the claim of the critics of this approach is) to the physical process called quantization. Quantization rather requires that you represent the classical constraints C as operators \pi(C) on a Hilber space and demand that
\langle \psi|\pi(C)|\psi\rangle = 0. In Thiemann's approach the classical constraints themselves are not even representable on his Hilbert space.
Maybe this should be compared to the 'quantization' of a simple mechanical system where the Hamiltonian itself is not representable on the Hilbert space but where the claim is that the time evolution operator U(t) is. Surely this would be very different from the ordinary notion of 'quantization'.
Of course Thiemann's procedure is no really 'arbitrary'. After all he models his quantum symmetry after the classical sysmmetry. The point is, however, that this is a choice of procedure different from the ordinary rules.
Originally posted by selfAdjoint
...is it your [urs's] thesis that the exponentialion of the constraint operators is arbitrary?
Adding a bit to what urs said, let's look at the paragraph of definition (5.2) I mentioned. Thiemann, under the assumptions mentioned above (5.2), takes (5.2) to be valid. He then infers from this definition that the classical lie algebra carries over to the quantum one appearing just below (5.2). The problem is that this inference is mistaken since gauge symmetries of the classical theory will pick up anomalies upon quantization so that the quantum algebra as given below (5.2) is simply wrong and thus the definition (5.2) makes no sense.
True, this isn't even self-consistently formulated, because in this section he is explicitly dealing with quantum Cs. Later on (as Thomas has also emphasized again in the Coffee Table discussion) however he defines the action of the U as in equation (6.25)
U_\omega(g)\pi_\omega(b)\Omega_\omega
=
\pi_\omega(\alpha_g(b))\Omega_\omega
.
This is what I am referring to all along as implementing the classical symmetry by hand on the Hilbert space. But you are right, this does not follow from (5.2) and this is what apparently confused me for quite a while, because all this discussion about non-separability of the Hilbert space etc. was essentially an attempt to understand how both (5.2) and (6.25) can be true. But (5.2) doesn't make sense at all in his quantization of the string, because
\pi(C_I) does not even exist! As Thiemann has emphasized in the Coffee Table discussion, the Virasoro constraints $C_I$ are not representable on his Hilbert space!
Hm...
selfAdjoint
Feb4-04, 12:42 PM
So it all turns on that "Alternatively.." phrase up above. That's the cop-out from the requirement that the \pi(C_{\mathcal{I}}) be densely defined "on a suitable domain of \mathcal H_{Kin}". Wish I knew where he got that, and just what its hypotheses are.
BTW, Distler just indirectly said at the Coffee Table that for 1+3d gravity the \pi(C_I) cannot exist even in principle (anomaly or not) because their commutators cannot make sense.
selfAdjoint
Feb4-04, 01:31 PM
I have just been rereading the Giulini-Marolf paper (gr-qc/9902045) and find that in their RAQ scenario, the constraints have to be defined in the auxiliary (resp. kinematic) Hilbert space. So if I read them right, you can't even do the group averaging, or at least rely on it being unique, if you don't have them. If they have them they then exponentiate them to get unitary operators to work with, and they map the constraint equations into the fact of the unitary transformations leaving the corresponding thing invariant.
Do they say anything about the case when the \pi(C_I) don't exponentiate to a group, due to an anomaly in their commutators?
selfAdjoint
Feb4-04, 02:25 PM
They don't mention anomalies explicitly. Here is what they do say:
Refined Algebraic Quantization is a framework for the implementation of the Dirac constrained quantization procedure which begins by first considering an 'unconstrained' quantum system in which even gauge dependent quantum operators act on an auxiliary Hilbert space \mathcal H_{aux} . On this auxiliary space are defined self-adjoint constraint operators C_i The commutator algebra of these quantum constraints is assumed to close and form a Lie algebra. Exponentiation of the operators will then yield a unitary representation of the corresponding Lie group. We will choose to formulate refined algebraic quantization entirely in terms of this unitary representation U in order to avoid dealing with unbounded operators.
As with any version of the Dirac procedure, physical states must be identified which in some sense solve the quantum constraints C_{\mathcal I}. Physically the same requirement is given* by the statement that the unitary operators U(g) (the exponentiated raw operators sA) should act trivially on the physical states for any g in the gauge group. Now, as the discrete spectrum of the unitary operator need not contain one, the auxiliary Hilbert space will in general not contain any such solutions. However by choosing some subspace \Phi \subset H_{aux} of 'test states' we can seek solutions in the algebraic dual \Phi^* of \Phi.... Solutions of the constraints are then elements f \in \Phi^* for which U(g)f = f for all g.
In RAQ , observables together with their adjoints are required to include \Phi in their domain and to map \Phi into itself....'Gauge invariance' of such an operator \mathcal O then means that \mathcal O commutes with the G-action on the domain of \Phi .
* at least for unimodular groups. See appendix A and B for a discussion of the non-unimodular groups]
Urs,
Get thiemann's paper entitled Introduction to Modern Canonical Quantum General Relativity,
http://xxx.lanl.gov/abs/gr-qc/0110034
and look on page 41 at equations (II.2.1.2b) & (II.2.1.3b) and at the top of page 42.
Take a minute, and marvel at the whole situation. From what I can see, Thiemann has essentially outputed (by definition mostly) a new quantization scheme. It bothers me considerably, that I can't see a good way to disprove it, even though it flies in the face of what we are taught about quantum anomalies, the promotion of classical algebras to proper quantum commutators, etc etc
Why? B/c its so damned hard to find precise mathematically well defined theorems on any of this. I read paper after paper, where they basically tell you whats right and whats wrong, but I can't find any formal proof of uniqueness. Instead, (and people here do a much better job of seeing it than I), we are forced to look for self consistency measures in his own scheme.
I almost want to do a handwaving physicists proof, and start with a simple example of a system with an anomaly (say from the Standard model), and then apply the scheme and show it violates experiment. But then, I can't think of an example that would be sufficiently applicable.
Here is the paper that Jacques Distler had mentioned:
P. Nelson and L. Alvarez-Gaume, Hamiltonian Interpretation of Anomalies (http://www-stud.uni-essen.de/~sb0264/HamiltonianInterpretationOfAnomalies.pdf).
selfAdjoint cites Giulini and Marolf about RAQ:
On this auxiliary space are defined self-adjoint constraint operators The commutator algebra of these quantum constraints is assumed to close and form a Lie algebra.
This is where the anomaly comes into the game. The commutator algebra of the quantum Virasoro constraints does not close to form a Lie algebra.
In a Lie algebra every commutator of two elements must be an element of the algebra again. But for the quantized Virasoro generators we instead find
[L_n,L_m] = (n-m)L_{n+m} + \delta_{n,-m}A(m)
where A(m) is the anomaly, a number and hence not an element of the Lie algebra.
Therefore Giulini and Marolf exclude the quantization of the string by means of RAQ.
Jeff,
sorry, I cannot find the formulas that you are referring to. (?) The copy of http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0110/0110034.pdf that I am looking at has no numbered formulas on p. 41 and I cannot see any formula labeled (II.2.1.2b).
Originally posted by Urs
I cannot find the formulas that you are referring to. (?) The copy of http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0110/0110034.pdf that I am looking at has no numbered formulas on p. 41 and I cannot see any formula labeled (II.2.1.2b).
Oops. The correct link is,
http://xxx.lanl.gov/abs/gr-qc/0210094.
Again, look on page 41 at equations (II.2.1.2b) & (II.2.1.3b) and at the top of page 42.
Search the previously mentioned much more detailed paper under "anomaly" and "anomalies". Sorry about this.
Thiemann says that it maybe "useful to remember" that he "treated the constraints EXACTLY the same as one quantizes the Poincare group of ordinary QFT." But in that case (as well as in his treatment of the closed bosonic string) the poincare symmetry is a global symmetry and picks up no anomaly upon quantization, unlike the local gauge symmetry Diff(S1).
Originally posted by Haelfix
I almost want to do a handwaving physicists proof, and start with a simple example of a system with an anomaly (say from the Standard model), and then apply the scheme and show it violates experiment. But then, I can't think of an example that would be sufficiently applicable.
Hi Haelfix,
This thread has brought up the issue of anomalies in quantization. I admit I never understood anomalies and I've only learned introductory QFT from Sakurai (eons ago). I just grabbed that paper from Urs' web site (before copyright lawyers turn up on his doorstep :)) and plan to take a look at it, but in the meantime I was wondering if there are in fact experimental results that REQUIRE anomalies for their explanation and what those experiments are? Basically, I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.
Thanks,
Eric
PS: I am a bit paranoid about being criticized for being off topic so let me say that the reason I bring this up here is that the issue Distler has with Thiemann's paper is that Distler thinks the anomalies are unavoidable while Thiemann disagrees.
selfAdjoint
Feb5-04, 09:01 AM
Originally posted by Urs
Here is the paper that Jacques Distler had mentioned:
P. Nelson and L. Alvarez-Gaume, Hamiltonian Interpretation of Anomalies (http://www-stud.uni-essen.de/~sb0264/HamiltonianInterpretationOfAnomalies.pdf).
selfAdjoint cites Giulini and Marolf about RAQ:
This is where the anomaly comes into the game. The commutator algebra of the quantum Virasoro constraints does not close to form a Lie algebra.
In a Lie algebra every commutator of two elements must be an element of the algebra again. But for the quantized Virasoro generators we instead find
[L_n,L_m] = (n-m)L_{n+m} + \delta_{n,-m}A(m)
where A(m) is the anomaly, a number and hence not an element of the Lie algebra.
Therefore Giulini and Marolf exclude the quantization of the string by means of RAQ.
Boy, unless Thiemann has a good answer for this, it sure shoots down his derivation! I didn't know enough to finger the algebraic closure myself, but I suspected it. I ran through some more of these RAQ papers yesterday and it looked like they all make that assumption. They only study the nice case where no anomalies interfere.
selfAdjoint
Feb5-04, 09:23 AM
BTW Urs, thanks for the Hamiltonian anomaly paper. Like Haelfix I'm going to study it. Since this incident has introduced some of us to anomalies and quantization we might as well learn from it. I just went over Polchinski's introduction to the central charge and it sucks. Calculate-calculate and gee! Look here! The energy isn't a tensor! See, it has this extra term! Actually I was by this development in an online study group a couple of years ago, but it sure didn't prepare me to couple to this discussion.
Originally posted by eforgy
I am a bit paranoid about being criticized for being off topic
My criticism of your post as being OT was quite unfair so don't worry about going a bit OT. Really really sorry about that.
Originally posted by eforgy
...I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.
Anomalies are symmetry violators left behind by regulators when they're removed. While gauge theories must be anomaly free to be consistent - a fact which can be used to constrain them - global symmetries can be violated without causing problems. In fact - and this is typically the first example of anomaly one comes across in QFT courses - the appearance of an anomaly breaking a global symmetry of the strong interaction solved the so-called "π0 decay problem" of explaining the observed rate of the dominent decay mode π0 → 2γ.
Originally posted by selfAdjoint
...the nice case where no anomalies interfere.
Perhaps nice from some purely mathematical standpoint, but as I mentioned above, the requirement that anomalies violating gauge symmetries cancel can be used to make theories more predictive, which is nice since uniqueness in fundamental theories is highly desirable for obvious reasons.
For example, thiemann advertised the LQG-string as working in any number of spacetime dimensions as if it's not being able to explain why there must be some unique number of spacetime dimensions is a good thing. In the bosonic string theory based on the polyakov action on the other hand the requirement that the weyl anomaly vanish gives us a definite answer, and this is much more satisfying I think.
Originally posted by selfAdjoint
...this incident...
"Incident"? [:)]
I happened to see a 4 February post on SPR that might be of general interest.
Baez mentioned that in his Quantum Gravity seminar they were
quantizing the open string---the same basic venture that Thiemann has embarked on (but he treats the closed string and takes the revolutionary approach of attempting it within LQG). Urs noticed Baez remark, and he posted the following a propos questions:
--------quote from Urs--------
"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
news:bvbu6e$cgb$1@glue.ucr.edu...
> Almost time for the quantum gravity seminar. Today we're quantizing
> the open string with Dirichlet boundary conditions! And with any
> luck, we'll make a *movie* of what it looks like! Gotta go!"
Just out of curiosity, since we are currently discussing this with Thomas Thiemann (see http://golem.ph.utexas.edu/string/archives/000299.html#c000554): Are you quantizing the the open string with D-boundaries the standard way as for instance described in
V. Schomerus, Lectures on Branes in Curved Backgrounds, hep-th/0209241
or by adapting the 'non-standard' way described for closed strings in
T. Thiemann, The LQG-String I., hep-th/0401172
to open strings?
What do you think about this non-standard way and the objections that have been brought forward (as for instance in
http://golem.ph.utexas.edu/string/archives/000299.html#c000560)?
-------end quote--------
I cannot think of any reason to suppose that Baez seminar would, in fact, be embarked on a similar venture to Thiemann (unless it is a conspiracy[8)] !!!) but I guess it is (as Urs suggests) a possibility and I hope Baez will reply soon and lay the question to rest.
Originally posted by marcus
...but [Thiemann] treats the closed string and takes the revolutionary approach of attempting it within LQG...
If by "revolutionary" you mean wrong or useless.
Eric wrote:
Basically, I'm wondering if the appearance of anomalies is a requirement of experiment or a result of academic inertia.
As Jeff has said, anomalies are 'very physical' and by no means just a formal artefact. In the standard model the effect of a global anomaly can be observed, experimentally. The effect of the local gauge anomaly can also be observed, sort of, because if it would not vanish then the standard model were inconsistent, which it apparently isn't because it is being observed! :-)
In string theory, too, there is a lot of physical information in the central charge (the prefactor of the anomaly, essentially). It fixes the number of spacetime dimensions and controls the partition function of the string, for instance.
So anomalies are not something that theorists haven't figured out how to get rid of but which should be absent. Instead, it took people quite a while to realize the role of anomalies in the standard and the physics related to that. Anomalies are a quantum effect which is just as real as any other quantum effect.
selfAdjoint wrote:
I just went over Polchinski's introduction to the central charge and it sucks. Calculate-calculate and gee! Look here!
If you want to use CFT methods then see Polchinski's equation (2.6.18) which again follows from (2.2.11). This is short, easy and straightforward.
If you are more into Fock space oscillators then see the derivation in equation (2.2.31) of Green&Schwarz&Witten. Also pretty easy, but needs some algebraic input.
If you are more a canonical kind of guy :-) you might want to look at my derivation (http://golem.ph.utexas.edu/string/archives/000299.html#c000572) at the Coffee Table, which uses canonical functional notation a la Thiemann, regulated appropriately.
Hi,
The following quote from the Nelson/Alvarez-Gaume abstract is troubling.
This provides a hamiltonian interpretation of anomalies: in the affected theories Gauss' law cannot be implemented.
What?!?!?
Sorry, I am guilty of not reading much more than the abstract so far, but how on earth can Gauss' law not be implemented? Stokes' theorem (Gauss' law being a special case) is what I have thought of as being the most profound statement in all of mathematical physics. Are Nelson et al saying that d^2 != 0??
*panic* :)
Eric
selfAdjoint
Feb5-04, 06:47 PM
They're saying you can't implement Gauss' law because the topology of configuration space won't let you. What they say the anomaly does is put a "twist" in the topology, a la the Moebius band (which is actually their first example, though you have to read down before they admit it). Thus you can't shrink n-spheres and that shrinkability is at the heart of the generalized Gauss law. It's what topologists call an obstruction.
selfAdjoint
Feb5-04, 09:44 PM
Urs, I have been following the discussion between you, Thiemann and Distler on the Cofee Table site.
It seems to me that Thiemann is saying "Ignore everything in sections 1 through 5, ignore group averaging and all of that. Here in section 6.1 is what I am really doing." And indeed if we look at 6.1, it does seem to be independent of what has gone before.
What he does is take the Borel intervals on the circle (which he did remark in your discussion are orthogonal if they differ anywhere - as you pointed out to me earlier!). He smears them in a particular special way with functions fk and asserts that the "handed" smeared functions Yk close to a Poisson *-algebra.
Is this true so far?
Then he introduces the Weyl elements W = exp(iYk), and invokes the Baker-Campbell-Hausdorff formula to get a value for their product and concludes from this that the W's for right handed and left handed Y's commute.
Any problems yet?
He then deduces from the general intersection geometry of intervals on the circle that "a general element of A (that is, a Weyl element W) can be written as a finite, complex linear combination of elements of the form
W_+(I)W_-(J), where W_{\pm}(I) = exp(i\sum Y_{\pm}^k(I)) for some finite collection of non-empty non-overlapping intervals, i.e. they intersect at most in boundary points.
Then he states a definition. A [i] momentum network s is a pair (\gamma,k) is a finite colllection of nonoverlapping intervals as above and k is an assignment of momentum to each interval. It now appears that the assignment of k agrees with the index k on his smearing function. So a momentum network operator is defined to be one of those W defined by the linear combination of exponentiated Yk, where k is now the momentum assigned to the interval Y comes from.
All this is reminiscent of the cylinder functions in LQG.
He then defines the analogs of holonomies and fluxes in terms of the intervals and their momentum operators. He asserts that both "holonomies" and "fluxes" close to a maximal abelian subalgebra of A, the algebra of W's.
He now defines the gauge group to by two copies of the diffeomorphism group of the circle plus the Poincare group, and notes that the diffs act only on the intervals of his net, while the Poincare group Lorentz transforms act only on the momenta and its translations leave the W unchanged because they only depend on the coordinate derivatives.
At this point he is ready for his GNS consideration. And I ask, is there any anomaly visible to you in this work? Is there any reson why the GNS will not work?
I don't understand how Thomas expects anomalies to be representation dependant.
Again, we're sorta taught to look at the induced topology that quantization outputs. The anomaly is just an indication of nontriviality in this context. There are many examples, and branches of physics that make use of this. Incluing the Witten anomaly, applications in twisted SUSY, etc etc
However, where it starts getting hazy, is the actual assumptions that go into this. For instance, its assumed that there is some notion of a continous metric where the guage bundles can live. LQG and other techniques does away with this, so in a sense the fine points of the usual structural geometry we are used to need not be the same (or at least, I don't know if it needs to be the same a priori).
Having said that, it seems like he is insisting that all guage anomalies found so far in the literature are now put into question. Thats quite a grandiose claim, and obviously subject to extreme scrutiny.
Originally posted by selfAdjoint
They're saying you can't implement Gauss' law because the topology of configuration space won't let you. What they say the anomaly does is put a "twist" in the topology, a la the Moebius band (which is actually their first example, though you have to read down before they admit it). Thus you can't shrink n-spheres and that shrinkability is at the heart of the generalized Gauss law. It's what topologists call an obstruction.
Hi selfAdjoint,
Again being guilty of not reading the paper (which will probably go over my head anyway), I don't understand how topology , in the sense you mention, has anything to do with generalized Gauss' law. The expression
int_M dA = int_@M A
is valid in extremely general circumstances. My friend Jenny Harrison has done this for fractals even. It is certainly valid for n-spheres and any other noncontractible manifolds. It is at the heart of Urs' and my paper. It almost seems like our work will not be valid for quantum theory because we have d^2 = 0. It is hard for me to belive this. Please tell me I am misunderstanding something simple (which is usually the case).
Best regards,
Eric
Hi eforgy,
In topological terms, field configurations satisfying gauss's law are contractible, or equivalently, the charges generating the fields are pointlike. (For comparison, it's worth noting that in ordinary maxwellian electrodynamics magnetic fields are divergence free so that, unlike with electric charges, there are no magnetic monopoles, at least according to maxwell). So in these terms, we can say that anomalies give rise to topologically non-trivial field configurations.
Originally posted by jeff
Hi eforgy,
In topological terms, field configurations satisfying gauss's law are contractible, or equivalently, the charges generating the fields are pointlike. (For comparison, it's worth noting that in ordinary maxwellian electrodynamics magnetic fields are divergence free so that, unlike with electric charges, there are no magnetic monopoles, at least according to maxwell). So in these terms, we can say that anomalies give rise to topologically non-trivial field configurations.
Hi Jeff,
Thanks. I tried to read through the paper. I can't say that I understand it (yet), but I do see that what he means by Gauss' law is not the same as what I mean by Gauss' law. To me (and most geometers I would think), Gauss' law is just an incarnation of the generalized Stokes theorem. The generalized Stokes theorem is valid in general. I'll have to make more effort to understand their meaning of Gauss' law. Thanks. I'm making progress.
Eric
Originally posted by eforgy
I'll have to make more effort to understand their meaning of Gauss' law.
I haven't studied the paper, but the point of my previous post was that - their precise mathematical formulation notwithstanding - I'm pretty sure that at bottom, by gauss's law they really mean contractible fields. They'd then classify anomalies in terms of field topologies.
Hi selfAdjoint -
you wrote:
It seems to me that Thiemann is saying "Ignore everything in sections 1 through 5, ignore group averaging and all of that. Here in section 6.1 is what I am really doing." And indeed if we look at 6.1, it does seem to be independent of what has gone before.
It kind of looks this way, yes. The most recent discussion at the Coffee Table shows, though, that Thomas might, after all, have made the same mistake that I did in the beginning, namely assuming that there is a quantization of the Virasoro algebra without an anomaly.
What he does is take the Borel intervals on the circle (which he did remark in your discussion are orthogonal if they differ anywhere - as you pointed out to me earlier!). He smears them in a particular special way with functions fk and asserts that the "handed" smeared functions Yk close to a Poisson *-algebra.
Yes, that's totally uncontoversial. It is, after all, nothing but an exotic reformulation of the fact that the usual worldsheet oscillators form a Poisson algebra.
Then he introduces the Weyl elements W = exp(iYk), and invokes the Baker-Campbell-Hausdorff formula to get a value for their product and concludes from this that the W's for right handed and left handed Y's commute.
Here a certain problem is beginning to show, which, at least for me, is a general one in this paper: It is not clear what, at this point, is assumption, definition and derivation.
The problem is that the Ys themselves are not represented as operators on Thomas Thiemann's Hilbert space. So how can we apply BCH to them, if they are not even operators? Of course we know that the Ys could be easily represented on some Hilbert space and we could compute their commutator there and it is the one that Thomas is using in the exponent of the BCH formula. But that's no real help either, because on Hilbert spaces where the Ys are represented (such as the usual Fock Hilbert space) their exponentiations are not unambiguously defined, unless we specify some rule of normal ordering. This gives, in the usual treatment, rise to the peculiar conformal dimension of such exponentiated operators, that you can see for instance in equation (2.4.17) of Polchinski. Therefore, whichever way I try to look at Thomas' equation (6.7) as something derived from previous input it makes me feel uneasy. I can accept (6.7) as a definition of the algebra of the Ws, though. But, just as with the definition of the Us by fiat, this is, while mathematically consistent, not manifestly related to physics-as-we-know-it, I think.
He then deduces from the general intersection geometry of intervals on the circle that "a general element of A (that is, a Weyl element W) can be written as a finite, complex linear combination of elements of the form [...]
Ok, given the algebra of the Ws, somehow, this follows without doubt.
He now defines the gauge group to by two copies of the diffeomorphism group of the circle plus the Poincare group
This is the point that we have been discussing in some detail with Thomas over at the Coffee Table. This way of defining the quantum gauge group means to simply copy the classical gauge group. That's mathematically possible, but not related to any standard quantization procedures. Jacques Distler has today given a further example (http://golem.ph.utexas.edu/string/archives/000299.html#c000579) for why this procedure is usually unphysical.
is there any anomaly visible to you in this work? Is there any reson why the GNS will not work?
No, the anomaly is indeed not there in this approach. But the reason is that by definition Thiemann is using a rep of the classical symmetry group on his Hilbert space. This is not the usual quantization procedure. There is no standard quantum anomaly because there is also no standard quantization.
The GNS theorem will work fine for the algebra of the Ws. The problem is that it is not clear what this algebra has to do with the standard quantization of the system at hand.
I can see that you are trying hard to escape the conclusion that is beginning to force itself upon us. I very much appreciate it. In a way I am delighted that the LQG-string is doing exactly what Nicolai has intended it to do: To show in terms of a simple example what is really going on in LQG. As long as we are dealing with 3+1d nonperturbative quantum gravity nobodoy knows what to expect and hence criticism of new proposals is very difficult. But now we are dealing with a case where we know much better what to expect and it has been possible to spot a very crucial difference of the LQG quantization approach to the standard procdedure:
LQG does not attempt to canonically quantize all the first-class constraints.
Actually, this is hardly a suprprise because, as Jacques has kindly reminded me, the ADM constraints of gravity simply cannot, even in priciple, be canonically quantized. LQG apparently circumvents this by not representing the constraints themselves on some Hilbert space but instead representing the symmetry group generated classically by them (at least for the spatial diffeos).
But this means breaking with a fundamental principle of quantum mechanics and can, at best, be addressed as an alternative quantization procedure. There are many people who are proposing alternatives to standard quantization, for various reasons. I am open-minded and willing to consider all alternatives to standard physics as potentially interesting. But one should be fully aware of what is standard physics and what is a radically new and speculative proposal.
In fact, I am currently thinking about asking Ashtekar, or someone similar, if it is really technically correct to say that LQG is about canonical quantization.
selfAdjoint
Feb6-04, 04:12 PM
Urs, just on this one point:
The problem is that the Ys themselves are not represented as operators on Thomas Thiemann's Hilbert space. So how can we apply BCH to them, if they are not even operators?
I looked up the BCH theorem on google. There are various definitions, but some of them do not require the elements to be operators on a linear space, just members of a Lie algebra. Well, Thiemann has the Y's as the members of a Poisson algebra, so I'll be he can quote chapter and verse to defend this transition.
Being bone ignorant, I just am not as sensitive to the awful non-standardness of Thiemann's work, but I am sensitive to things that just don't work. My problem right now is that all that section 6 material I quoted does sound to me like a string! Borel intervals, momentumful smearing, yes, I can see it. And I've read enough in LQG literature to recognize what he does with this. When I thought he couldn't rigorously apply GNS or group averaging I was ready to give up on him, but rereading this later material brings me back to the table.
I still have doubts like this: His string is all by itself. As I remember it, Virasoro comes out of string interaction. You have the circle where the other world tube joins this one, and you "projectively" represent that tube as a punctured disc, and develop a Laurent series, and th coefficients of that generate the Virasoro algebra, up to ordering. So can his representation, his quantization, do interaction?
I do know that if you say Foch space he will not agree; he thinks of this work as freeing physics from Foch space arguments.
selfAdjoint
Feb6-04, 04:17 PM
Originally posted by eforgy
Hi Jeff,
Thanks. I tried to read through the paper. I can't say that I understand it (yet), but I do see that what he means by Gauss' law is not the same as what I mean by Gauss' law. To me (and most geometers I would think), Gauss' law is just an incarnation of the generalized Stokes theorem. The generalized Stokes theorem is valid in general. I'll have to make more effort to understand their meaning of Gauss' law. Thanks. I'm making progress.
Eric
Eric, I think your version of Gauss law is LOCAL. The problem is to extend it over the whole parameter space, to a GLOBAL law. And the twist obstructs that extension.
Originally posted by selfAdjoint
Eric, I think your version of Gauss law is LOCAL. The problem is to extend it over the whole parameter space, to a GLOBAL law. And the twist obstructs that extension.
Hi selfAdjoint,
The problem is not with the locality of my version. I assure you that generalized Stokes theorem is not a local theorem. It is defined globally. I still didn't put my finger on exactly what the issue is, but based on Jeff's comment, I am thinking that it might be related to the existence of a Hodge star. Gauss' law on an n-manifold usually refers to (n-1)-forms A with
int_M dA = int_@M A.
It we want to think of this (n-1)-form as coming from vector field X, we need to convert this vector field to a 1-form alpha via the metric. Then we convert this 1-form alpha to an (n-1) "pseudo" form A = *alpha. I think the paper probably refers to some vector field appearing in the integrand as Gauss' law. I could accept this. So is it that when you quantize, the configuration space has no Hodge star, which means you can't define Gauss' law, which in turn gives you anomalies?
Eric
Now that we have had a chance to get used to the idea of the "LQG-string" what conclusions, if any, do you think could be incorrect?
The thread is long, with posts apparently containing criticisms that were later dropped. Perhaps it would not be a good time to sum up the main points---so that the busy reader does not have to sift through these many (often contradictory) posts.
The core of the paper is section 6. (pages 19-40).
I gather that a critical reading of pages 19-40 did not produce
any conclusive finding of error. There was plenty of it that some of us, especially string theorists, did not like or found unfamiliar, and Urs said he might ask Abhay Ashtekar about something. (That sounds like a good idea, hopefully he has done this already.)
But after listening to the critiques one was not left with the certainty that anything was actually wrong with Thiemann's math.
(If not some overlooked detail which he could correct and still sustain his conclusions.)
So now the question is which of the conclusions does anyone wish to challenge?
Thiemann concluded for instance that string theory does not, after all, require 11 dimensions, or 26 dimensions. There is no critical dimension, after all, that it must have in order to work, because he models it in LQG in all dimensions including ordinary 4D spacetime.
He also concluded that string theory does not, after all, require supersymmetry. Nor, when modeled in a LQG context, does it have the
undesirable "ghosts" and "tachyons".
So as to recall what is the topic of this thread, I will give the link to Thiemann's paper again, and quote the abstract:
http://arxiv.org/hep-th/0401172
"The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space"
"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."
selfAdjoint
Feb7-04, 09:14 PM
Marcus, here is where I am.
I do not believe the string people have made their case because Thiemann's technique bypasses everything they know and of course they can only argue on the basis of what they know. So we see Thiemann and Distler going at each other on the Coffe Table site, talking past each other until it's almost "'Tis so" - "'Tis not".
Distler's final shot is that it's mathematically inconsistent to get a string theory without an anomaly - he means that every mathematical technique he or Urs has tried infallibly produces the anomaly. Thiemann's retort is that all those things are just partial views and products of the way they go about quantizing the string. So there. Thiemann points out that there is no rigorous development of all this, so to talk about mathematical consistency is a bit rich.
That said, I am uneasy about Thiemann's theory. The paper, as we have discovered, is hastily slapped together. What we all thought were logical trains of thought, weren't. So for me there's smoke. I can't find any fire. We'll have to wait for bigger guns than we've seen so far. Probably at that Mexico meeting.
Originally posted by selfAdjoint
...We'll have to wait for bigger guns than we've seen so far....
I tend to agree. BTW great quote from H the V.
"Once more unto the breach, dear friends, once more!"
also delighted by that reference to the "awful non-standardness".
Yes but now its not clear to me, why Thiemanns method can't be used in other contexts (Distler picks the Y-M eqns).
eigenguy
Feb8-04, 08:14 AM
Originally posted by marcus
Now that we have had a chance to get used to the idea of the "LQG-string" what conclusions, if any, do you think could be incorrect?
I'm not sure who you are asking. What specific conclusions do you think could be incorrect? Since you asked the question, I don't think it is unfair to ask you your opinion. Or maybe you are asking some more knowledgable member. If so, who would this be?
Originally posted by marcus
I gather that a critical reading of pages 19-40 did not produce
any conclusive finding of error. But after listening to the critiques one was not left with the certainty that anything was actually wrong with Thiemann's math.
Would you mind substantiating this a bit for the other members? I just don't think these kinds of broad superficial comments are fair given the difficulty of these issues. I guess what I'm asking is if you would mind explaining your feelings the same way you do with other topics which you know well. Again, I think these are fair questions given your post and the nature of the topic.
Why did you stay out of the technical discussion when it moved into full swing? If you don't feel qualified to comment, I'm not sure why you would feel comfortable posting this, or at least without some clear qualification.
Originally posted by marcus
We'll have to wait for bigger guns than we've seen so far.
Jacques distler, the guy who was arguing with thomas, is one of the worlds most brilliant theorists, even more so than ashtekar et al. So it is quite safe to take distler's point of view seriously.
Originally posted by selfAdjoint
Marcus, here is where I am.
I do not believe the string people have made their case because Thiemann's technique bypasses everything they know and of course they can only argue on the basis of what they know. So we see Thiemann and Distler going at each other on the Coffe Table site, talking past each other until it's almost "'Tis so" - "'Tis not".
Distler's final shot is that it's mathematically inconsistent to get a string theory without an anomaly - he means that every mathematical technique he or Urs has tried infallibly produces the anomaly. Thiemann's retort is that all those things are just partial views and products of the way they go about quantizing the string. So there. Thiemann points out that there is no rigorous development of all this, so to talk about mathematical consistency is a bit rich.
That said, I am uneasy about Thiemann's theory. The paper, as we have discovered, is hastily slapped together. What we all thought were logical trains of thought, weren't. So for me there's smoke. I can't find any fire. We'll have to wait for bigger guns than we've seen so far. Probably at that Mexico meeting.
this seems like a fair-minded summation and as I said before I tend to agree with your "waiting for bigger guns" comment.
the context of a conference is a good arena for probing the soundness and implications of new work and some of that probably did go on
at the Mexico meeting---I only have a secondhand report from nonunitary though.
At the May conference in Marseille Thiemann will give the main talk
at the "dynamics and low-energy limit" session. I have posted the program on the surrogate sticky.
So he will be discussing latest developments with LQG Hamiltonian.
I should imagine he will be asked to discuss this paper as well.
But what I personally think would constitute "bigger guns" would be
more in MPI-Potsdam. Hermann Nicolai's institute trains both string and loop theorists, and appears to me to have expert people in both lines of research.
In case anyone's interested here's the program for the May
conference where Thiemann will be doing the Hamiltonian and low-energy limit talk:
http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/
http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/programmeprovisoire.html
"Tentative list of morning talks.
Loop Quantum Gravity:
Abhay Ashtekar (quantum geometry)
Thomas Thiemann (dynamics and low energy)
Lee Smolin (overall results)
Ted Jacobson (devil's advocate)
Applications: .....
......
...... etc."
selfAdjoint
Feb8-04, 09:22 AM
Sorry, I just misremembered Mexico for Marseiles. My idea of a good critique of Thiemann's paper would be someone who is a real expert on string quantization issues, and who will couple to Thiemann's argument on its own terms. This is exactly what Distler did not do. If Thiemann's paper is mathematically inconsistent, as Distler claims, then where is the inconsistency? Urs, who is fair-minded found physical reasons he couldn't accept the work, but didn't find any inconsistency.
eigenguy
Feb8-04, 09:29 AM
Marcus, I now have no doubt that you are an unbelievable phony and the fact that you have that physics of the year award thing even if it is just for fun disgraces this site for people who unike you are in rational and knowledgable. SelfAdjoint isn't a phoney, but I think he's out of his depth here as well. Someoen should start a new thread to let people know what actually happened with thomas and jacques distler.
Originally posted by selfAdjoint
Sorry, I just misremembered Mexico for Marseiles. My idea of a good critique of Thiemann's paper would be someone who is a real expert on string quantization issues, and who will couple to Thiemann's argument on its own terms. This is exactly what Distler did not do. If Thiemann's paper is mathematically inconsistent, as Distler claims, then where is the inconsistency? Urs, who is fair-minded found physical reasons he couldn't accept the work, but didn't find any inconsistency.
Yes! If I can hazard an opinion, the importance of Thiemann's paper is as a straw in the wind. If his method extends, or if other methods can extend his results, then it seems to have major consequences. This is how I read his introduction that I quoted 5 or 6 posts back:
"...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.
...
...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 full quote is in
http://www.physicsforums.com/showthread.php?s=&postid=1431999#post143199
5 or 6 of my posts back, on the preceding page.
eigenguy
Feb8-04, 11:20 AM
You know marcus, the site guidelines require one responds to any fair question about the claims they make.
In fact, I think anyone would rather prove someone wrong than sticking their head in the ground and hope no one notices. It then stands to reason that if you won't respond, people here will simply believe you can't back up your claims.
Originally posted by eigenguy
Marcus, I now have no doubt that you are an unbelievable phony and the fact that you have that physics of the year award thing even if it is just for fun disgraces this site for people who unike you are in rational and knowledgable. SelfAdjoint isn't a phoney, but I think he's out of his depth here as well...
no comment
eigenguy
Feb8-04, 12:32 PM
Originally posted by marcus
no comment
Hey, your the one who refuses to answer fair questions about comments you made about physics. This is a physics forum you know. So what's behind all your bluster. What was about the "critiques" that left you with the impression that nothing was resolved? You said it, I didn't. I'm just asking what you are talking about because my reading of it is that thiemann was shown quite clearly that his paper made neither physical nor mathematical sense. Just look at the thread-ending final exchange between him and distler.
selfAdjoint
Feb8-04, 02:50 PM
You may believe your last three posts are fair comments deserving responses, but they look to me a lot like intemperate ad hominem slurs. Any comment on that?
eigenguy
Feb8-04, 03:39 PM
Originally posted by selfAdjoint
You may believe your last three posts are fair comments deserving responses, but they look to me a lot like intemperate ad hominem slurs. Any comment on that?
You could say exactly the same thing about the exchanges at the string coffee table, so this is complete baloney. Even if I was rude, my questions are valid and are owed an answer. In a similar position, I would have simply backed up what I said by answering the questions directly and ignored the rudeness, that is, if I had the answers, of course.
I think you know quite well selfadjoint that there is no way anyone could frame the basic question I asked marcus so that he wouldn't find some way to weasle out of it. He did the same thing the one and only other time I was here. At that time I asked why LQG was not taken seriously by physicists. You should review those first exchanges between marcus and me and tell me who was rude. Just search under my name.
Anyway, the fact that marcus would put you in a position were you felt you had to fight his batttles for him should make you wonder about his character, but not his physics because I think you know he pretends to know much more than he actually does, a fact which while monitoring the thread I saw demonstrated quite clearly by the exchanges between marcus and both lubos and urs, after which marcus left the thread and came back only after he thought the "coast was clear". The only reason marcus always turns to you is that he knows he can trust you not to challenge him in a way that would show him up. I think the choice you've made to help marcus keep the wool over everyone's eyes is questionable to say the least. Specifically, after urs made a tremendous effort to explain why the LQG-string is senseless, and it is senseless, marcus makes a completely false pronouncement on what actually happened, summarily dismissing by implication what one of the worlds leading physicists said. Talk about arrogance!
I'm not surprised by the fact that people like jeff, urs and lubos motl don't participate very much around here. You guys are so ignorant that you don't even understand that you don't understand. For example, I notice that when it comes to complicated physics, you aren't able to actually put your finger on the relevant issue in a paper. Instead you just go through the methodology figuring that this way, at least what your saying is probably not wrong.
I think just as in the physics community, we need more people here to be tough and keep the membership honest about the physics and I don't think that anyone can rely on you to do that.
selfAdjoint
Feb8-04, 04:59 PM
Well, my own lack of technical expertise is no secret, but I am not stupid, either, and I believe I have an insight here that all you experts, Lubos, Distler, Jeff, yourself, and even Urs haven't faced up to. There is something Thiemann has found, distinct from dumb mistakes, that leads him to make his assertions. Perhhaps we should take his hint and blame the Polyakov action. If you didn't make the worldsheet the center of your original analysis, and didn't derive the conformal and Weyl invariances on it, what would your physics be like? If you weren't able to deflect criticism by reference to school excercises, what then?
Generally, your nasty tone, Lubos' fury, and Distler's sarcasm only suggest to me that you are all suffering from mauvais foix - that you fiercely assert this must be, because if it weren't you would all be at sea without a compass.
eigenguy
Feb8-04, 07:07 PM
Originally posted by selfAdjoint
There is something Thiemann has found, distinct from dumb mistakes, that leads him to make his assertions. Perhhaps we should take his hint and blame the Polyakov action. If you didn't make the worldsheet the center of your original analysis, and didn't derive the conformal and Weyl invariances on it, what would your physics be like?
Okay, so let's talk about this. Firstly, you should take what I say about the physics with a large grain of salt because I know little about LQG. But I have been studying polchinski volume I which covers the string related issues thiemann raises.
In a nutshell, here's what I think are the crucial parts of the exchange at the "string coffee table":
Thiemann claimed to have shown that the existence in the quantized closed bosonic string of a critical dimension, a virasoro anomaly, and a tachyon state which requires supersymmetry to remove, was simply a result of the representation used by string theory guys, the one that follows from the polyakov action. In particular, thiemann claimed his rep to be anomaly free.
Distler pointed out that urs's calculation of the virasoro anomaly depended only on the canonical commutation relations, the point being that these are essentially the same for any quantization of the closed bosonic string.
So thiemann tried to salvage his paper's main results by arguing that whether the virasoro algebra has an anomaly is irrelevant since he was quantizing the group elements directly and not their lie algebra generators. But then distler made the obvious point that one only has to consider group elements near the identity to see that this argument also fails.
What do you think?
selfAdjoint
Feb8-04, 07:43 PM
eigenguy, these are good questions. Give me till tomorrow and I will try to answer them. I have an idea about the neighborhood of the identity objection but I want to think it over and check it out before I expose it.
Otherwise I saw the thread with Distler on the Coffee Table site as falling into two segments. In the first, Distler convinces Urs that Thiemann is not doing what Urs believes, but is really outside the bounds of proper string theory. In the second, Distler and Thiemann trade high level counterarguments.
I'll do my best to respond to your questions, and I suggest we adopt Jeff's sig line and Keep It About the Physica.
eigenguy
Feb8-04, 11:15 PM
Originally posted by selfAdjoint
eigenguy, these are good questions. Give me till tomorrow and I will try to answer them. I have an idea about the neighborhood of the identity objection but I want to think it over and check it out before I expose it.
Otherwise I saw the thread with Distler on the Coffee Table site as falling into two segments. In the first, Distler convinces Urs that Thiemann is not doing what Urs believes, but is really outside the bounds of proper string theory. In the second, Distler and Thiemann trade high level counterarguments.
I'll do my best to respond to your questions, and I suggest we adopt Jeff's sig line and Keep It About the Physica.
At the very end of the thread distler has given this link (http://golem.ph.utexas.edu/~distler/blog/archives/000307.html) to his summation of the discussion which as it turns out is for the most part pretty much the same as mine.
Let me rephrase again what the conclusion of the discussion is - and Thomas Thiemann did agree about this point, just not about it's relevance:
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.
In his last message Thomas confirmed that he hence thinks that the issue is one that has to be resolved by experiment. Certainly existing experiments strongly confirm standard quantum theory, which is Jacques Disler's point, quoting YM theory.
So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra) it is speculative physics.
In particular, the method used by Thomas is not "canonical quantization" as usually understood. It is not Dirac quantization of first-class constraints.
Often LQG is advertised as a very 'conservative' approach to quantum gravity. I no longer see how this can be claimed. Modifying the basis of quantum theory is hardly a conservative approach. There is so far no hint that the LQG way to impose the constraints is realized in nature.
Originally posted by Urs
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.
Fine :-D . Just I take the work, against my own desire, of pointing out a hint of the relationship between area quantization and standard quantum theory, and it seems that the whole congress has concluded on the contrary ! This is a real sincronicity.
Originally posted by Urs
In particular, the method used by Thomas is not "canonical quantization" as usually understood. It is not Dirac quantization of first-class constraints.
Often LQG is advertised as a very 'conservative' approach to quantum gravity. I no longer see how this can be claimed. Modifying the basis of quantum theory is hardly a conservative approach. There is so far no hint that the LQG way to impose the constraints is realized in nature.
Urs, you are making a blanket statement about LQG.
Please have a look at Rovelli's book "Quantum Gravity"
(which Thiemann cites in his references) and tell us if you see
anything which you would like to declare non-standard.
It would be extremely interesting if you would point out a section of
the book where the quantum theory is not kosher according to you.
Originally posted by Urs
...
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.
...
So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra) it is speculative physics.
Urs, I appreciate the fact that you have just taken part in a lively discussion at what I take to be Jacques Distler's message board. I'm glad to hear from you what you believe can be concluded from that discussion!
Please tell me at what point in the "LQG-String" paper does TT use a procedure that is not related to standard quantum theory. I assume this has nothing specifically to do with String (which is not so-far "standard quantum theory") but is a LQG procedure which you find non-standard. I would very much like to know what this is and have the paper printed out here. So if you tell me a page number and quote some lines, I will be closer to understanding what this non-standardness is, or at least be able to ask for clarification.
I also have Thiemann's "Lectures on Loop Quantum Gravity" which I gather Springer Verlag published last year---a kind of textbook on LQG. It is available, as you know, online (gr-qc/0210094) and is less than 100 pages long. It would be great if you could find the non-standard procedure in "Lectures" and explain it in that context. That way the issues would be kept separate from string theory, making it easier to judge what is speculative and what is not speculative.
Thanks in advance
eigenguy
Feb9-04, 07:57 AM
Originally posted by Urs
So, yes, while there are no mathematical inconsistencies in Thiemann's paper (once we allow for the fact that he does not mean to imply that group averaging is applicable to the Virasoro algebra)
Then what's the significance of distler's remark that thiemann's approach of quantizing Diff(S^1) directly can't avoid the virasoro anomaly issue?
Originally posted by Urs
Often LQG is advertised as a very 'conservative' approach to quantum gravity. Modifying the basis of quantum theory is hardly a conservative approach.
So the LQG-string framework is in fact that of standard LQG?
eigenguy
Feb9-04, 08:02 AM
Originally posted by marcus
String (which is not so-far "standard quantum theory")
Yes it is standard quantum theory, but applied to strings. Keep in mind that in the low energy limit ST reduces to ordinary QFT.
selfAdjoint
Feb9-04, 08:22 AM
Urs, let's discuss this once more
The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.
Every step that Thiemann takes is based on some previous result, mostly from classical mathematical physics. His use of the GNS construction is exactly as in Haag's book Local Quantum Physics, his quantization is per the Giulini-Marolf paper. Maybe this isn't the way string, or particle - physicists go about things, but it's a valid way within mathematical physics.
I do have a question, in that the symmetry group in all those prior theorems* is assumed to be locally compact, which pretty much much means finitely generated, and Diff(S1) isn't. I think that in earlier LQG papers we saw GNS extended to infinitely generated groups (Marcus, help me out here!), but if not, then his work is invalid. But then that would make his work mathematically wrong, not physically meaningless.
* I'm thinking here especially of Corollary 4.1 where a "G-invariant state" is introduced out of the blue. The parallel discussion in Haag has attention paid to the nature of G, which is assumed to be locally compact.
Originally posted by selfAdjoint
... I think that in earlier LQG papers we saw GNS extended to infinitely generated groups (Marcus, help me out here!),...
selfAdjoint,
for starters I will put out some arxiv numbers of papers which
we looked at or discussed at PF some months back. then I will
have a look-see if any of these fill the bill
Okolow and Lewandowski
"Diffeomorphism covariant representations of the holonomy-flux *-algebra"
http://arxiv.org/gr-qc/0302059
this was Jerzy Lewandowski's reaction to the work of Hanno Sahlmann, then at AEI-Potsdam with Thiemann. Then there were some papers of Sahlmann and of Thiemann/Sahlmann. Here are a couple, which would have references to others.
Hanno Sahlmann
"Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity"
http://arxiv.org/gr-qc/0207111
Sahlmann and Thiemann
"Irreducibility of the Ashtekar-Isham-Lewandowski Representation"
http://arxiv.org/gr-qc/0303074
Sometime while we were discussing these and related papers I recall
getting out my old copy of Naimark's book "Normed Rings" and
studying up on the Gelfand-Naimark construction. Or I guess one calls it the "GNS" for Gelfand-Naimark-Segal.
I think the role of GNS in Loop Gravity goes back to much earlier work---by Ashtekar, Lewandowski and others. I am responding too quickly perhaps, not sure if this is to the point.
But I will have a look at some of these papers and see if I can reply better.
selfAdjoint
Feb9-04, 10:16 AM
-The only restriction on G is that it be locally compact, and I now think that DIFF(S1) is, because the circle is compact. Take a neighborhood of the identity - diffeomorphisms that don't move any point as much as some small \epsilon, then inside that we can lift pointwise convergence to diffeomorphism convergence.
-GNS seems to have been introduced into LQG in a 1992 paper by Ashtekar and Isham, hep-th/9202053.
Notice also that Thiemann never claims to have a genral representation theory of G; he says that will have to be a topic of further research, and offers instead just about the simplest example you could think of, one that takes the value 1 on all his momentumized networks and is zero only on the empty network (6.20).
I am now digging into the details of his implementation of the Pohlmayer charges, and the development of the algebraic representations, sections 6.5 and 6.6. I should have done this in the first place.
You beat me by 2 years
you found a 1992 paper and I just came back with a 1994 paper
by Ashtekar, Lewandowski, Don Marolf, Jose Mourao, and Thomas Thiemann
It is called
"Coherent State Transforms for Spaces of Connections"
http://arxiv.org/gr-qc/9412014
page 9, for instance, uses the Gelfand-Naimark construction
This will give some of the flavor of the 1994 paper that Thiemann co-authored with Ashtekar, Lewandowski, Marolf and Mourao.
I cannot easily reproduce the symbols from their gothic and script fonts. I will leave off the overbar on A/G and write mu for the greek mu and so on. this is just a exerpt to give a feel for how the Gelfand-Naimark theory was used at around that time.
-----quote from page 9 of gr-qc/9412014------
"The classical configuration space is then the space A/G of orbits in A generated by the action of the group G of smooth vertical automorphisms of P. In quantum mechanics, the domain space of quantum states coincides with the classical configuration space. In quantum field theories, on the other hand, the domain spaces are typically larger; indeed the classical configuration spaces generally form a set of zero measure. In gauge theories, therefore, one is led to the problem of finding suitable extensions of A/G. The problem is somewhat involved because A/G is a rather complicated, non-linear space.
One avenue [6] towards the resolution of this problem is offered by the Gel'fand-Naimark theory of commutative C*-algebras. Since traces of holonomies of connections around closed loops are gauge invariant, one can use them to construct a certain Abelian C*-algebra with identity, called the holonomy algebra. Elements of this algebra separate points of A/G, whence, A/G is densely embedded in the spectrum of the algebra. The spectrum is therefore denoted by [A/G bar, can't make the symbol]. This extension of A/G can be taken to be the domain space of quantum states.
Indeed, in every cyclic representation of the holonomy algebra, states can be identified as elements of L2(A/G; mu) for some regular Borel measure mu on A/G. One can characterize the space A/G purely algebraically [6, 7] as the space of all homomorphisms from a certain group (formed out of piecewise analytic, based loops in Sigma) to the structure group G. Another {and, for the present paper more convenient) characterization can be given using certain projective limit techniques [10, 14]: A/G with the Gel'fand topology is homeomorphic to the projective limit, with Tychonov topology, of an appropriate projective family of finite dimensional compact spaces.
This result simplifies the analysis of the structure of A/G considerably. Furthermore, it provides an extension of A/G also in the case when the structure group G is non-compact.
Projective techniques were first used in [10, 14] for measure theoretic purposes and then extended in [13] to introduce
"differential geometry" on A/G
The first example of a non-trivial measure on A/G was constructed in
[7] using the Haar measure on the structure group G. This is a natural
measure in that it does not require any additional input; it is also faithful and invariant under the induced action of the diffeomorphism group of Sigma.
Baez [8] then proved that every measure on A/G is given by a suitably consistent family of measures on the projective family..."
eigenguy
Feb9-04, 11:07 AM
Originally posted by selfAdjoint
Every step that Thiemann takes is based on some previous result
Originally posted by urs The conclusion is that the LQG-string uses a procedure that is not related to standard quantum theory.
but it's a valid way within mathematical physics.
I like this post because it touches on my own feeling that much of the friction between the LQG and ST camps is due to their belief that the other's opinion about what constitutes genuine physical research is wrong.
My own opinion is that neither the mathematical consistency of, nor the presence within a theory of analogs or generalizations of ideas whose physical validity has been proven or otherwise generally accepted, is sufficient cause to view it as physicsally viable or valid: theorists should be guided by plausibility rather than mere logical possibility.
What would cause you to abandon your interest in LQG?
Marcus,
have you followed the disucssion over at the Coffee Table? Thomas Thiemann himself confirmed that in LQG the spatial diffeo constraints are imposed in the same way that he imposes the Virasoro constraints in his 'LQG-string' paper. This is precisely the step which is non-standard, as Distler has made quite clear, because it does follow neither from path intergal nor from canonical Dirac quantization but instead conjures up a new principle which says that it is fine to find any rep of the classical symmetry group on the quantum theory's Hilbert space and demand that physical states be invariant under this group.
selfAdjoint,
you write
his quantization is per the Giulini-Marolf paper.
No, it is not. Giulini-Marolf require a rep of the quantum first class constraints which is anomaly free. Thiemann has no rep at all of the first class constraints and cannot even in principle get one that is anomaly free. Instead of Giulini-Marolf what he does is group averaging with a group of operators that does not follow from standard quantization in any way.
This is not controversial, I think, because Thiemann himself confirmed repeatedly at the Coffee Table that this is what he is doing. What is controversial is only whether this 'new' method could have something to do with physics.
Thomas Thiemann says at the Coffee Table that he thinks that only experiment can tell whether his form of quantization is correct or the standard one. I can accept this, but we then have to be quite clear on what this means: This means that Thomas Thiemann is proposing a modification of the quantum principle (at the Planck scale). This means that LQG is not canonical quantization, but a new kind of quantization.
I am the last one to embrace this conclusion, but it is what Thomas Thiemann is saying.
Originally posted by eigenguy
What would cause you to abandon your interest in LQG?
Eating too much of it, like chocolate (if I may be allowed to reply [:)]
Interest or non-interest in a developing line of research is a matter of personal taste.
I do not ask you Eigenguy and/or Jeff to justify NOT being interested.
Perhaps you are interested in String---well, I do not ask you to explain this (although I am not interested in String myself)
this is diverting a physics discussion to argument about personality issues
"keep it about the physics"
eigenguy
Feb9-04, 11:56 AM
I have a question for urs.
What is your feeling about the view that any attempt to quantize GR directly is naive because the assumption that the einstein-hilbert action isn't just the leading term of a more general effective theory is naive?
Originally posted by Urs
Marcus,
have you followed the disucssion over at the Coffee Table? Thomas Thiemann himself confirmed that in LQG the spatial diffeo constraints are imposed in the same way that he imposes ...
Urs, so nice to hear from you! I am glad you are concerned with an issue that is purely about Loop Gravity, in isolation from String.
That is, you fear something might be wrong in the development of LQG proper, not just in this particular analysis of a string within a LQG model by Thiemann.
The thing to do, I feel sure, is to learn what is exactly that we are talking about.
Regardless of what you understood Thiemann to have said in some discussion, we should find in his "Lectures on LQG" where what you are worried about happens. Or in some other textbook.
It is the old idea of actually looking in the horses mouth to count the teeth.
Early in the thread I gave you a reference to a page in Rovelli textbook where the spatio diffeomorphisms are imposed.
Two network states are made equivalent if they differ by a diffeo.
Thus the states become "equivalence classes" and equivalence classes of network states are knot states. It is a common algebraic procedure to factor something down to equivalence classes. This is all familiar to you! Anyway, I referred to that part of Rovelli very early on in the thread. Unless I misunderstand your question, you can see how it is done there (I think around pages 170-173) and see if you like it or not!!!
I would be delighted to know if you do not like how Rovelli takes care of invariance under spatial diffeo! This would be a choice topic of discussion.
Also it seems to me very clean and easy to understand. He does it quickly without much notation and trouble--then you can say this is kosher or not-kosher, traditional or not-traditional, according to how you think.
Since Rovelli is one of the main Loop Gravity textbooks that would
be reasonable basis for general statements about how things are done.
If you think it is bad----or if I misunderstand your question--I would very much like to know.
BTW you asked if I followed TT and JD on the other board, no because I dont want to change browsers and its very hard to read with the Microsoft browser (no symbols, fine print, as we discussed). But this issue is much broader----how diffeomorphism invariance (a basic feature of General Relativity) is handled in LQG---in particular how LQG handles spatial diffeos. We should be clear about whether or not it's kosher quantum theory by your standards.
eigenguy
Feb9-04, 12:51 PM
Originally posted by marcus
I do not ask you Eigenguy and/or Jeff
Just for the record, I don't necessarily agree with everything jeff thinks and I'm perfectly capable of forming my own ideas without anyone else's help.
selfAdjoint
Feb9-04, 01:06 PM
I believe I have some surpising information about this issue:
No, it is not. Giulini-Marolf require a rep of the quantum first class constraints which is anomaly free. Thiemann has no rep at all of the first class constraints and cannot even in principle get one that is anomaly free. Instead of Giulini-Marolf what he does is group averaging with a group of operators that does not follow from standard quantization in any way.
It is this. Thiemann does not use Giulini-Marolf or group averaging in his actual construction in this paper! He does use GNS intensively. But nearly everything he does in his specific example (which is the only quantization he does, as opposed to talks about) is careful manipulation of Hilbert space issues. For example he does not actually exponentiate the Pohlmayer charges; he regulates them and develops a specific expression in terms of the regulator that show the Pohlmayer charges as functions of the W's.
Maybe his further reaserches on the representation theory of his algebra will involve these issues, but the construction of a string quantization which he actually exhibits in this paper does not.
Marcus,
the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.
Hi selfAdjoint -
yes, he does a couple of things on this Hilbert space which are probably all fine and dandy. But what he does not do is impose the constraints in the usual way. This is particular implies that the Pohlmeyer charges do not commute with the usual constraints. They are merely invariant under the classical symmetry group action that Thomas Thiemann is using.
I have mentioned a way around this problem: Use the classical DDF invariants instead of the Pohlmeyer charges. Then quantize correctly, find the anomaly in the longitudinaly DDF invariants, include the logarithmic counter term to cancel these and - voila - one is left with the standard string! :-)
Hi eigenguy,
in the discussion with Jacques Distlet I was reminded of a simple fact which I apparently did not sufficiently appreciate before: There is no canonical quantization in principle of the ADM constraints of the EH action. LQG only avoids/ignores this no-go-fact by quantizing only the Hamiltonian constraint and imposing the classical diffeo constraints by hand. So if I were to believe that gravity has to have a canonical quantization, then I would hope that EH is only a leading order term, because otherwise I'd have to give up immediately.
To me this insight is a completely new perspective on the old discussion about what is conservative about LQG and about strings.
But, personally, I don't know if I hope that gravity can be quantized canonically. I feel much more comfortable with quantizing really small things than really big ones! :-) I find it much more trustworthy to apply quantization to a tiny string than to the entire universe. We are more likely to get the former right, I'd say.
On the other hand, string theory of course has the promise of giving us tools to quantize the entire universe in one stroke by means of Matrix Theory.
Originally posted by Urs
Marcus,
the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.
Great! Thanks for looking it up in a standard LQG source Urs. I have that article printed out in a pile of papers
by my desk and I will look it up and see what you mean.
Marcus -
the analogous problem in Rovelli's book http://www.cpt.univ-mrs.fr/~rovelli/book.pdf is indeed on pp. 170. Consider a diffeomorphism \phi that leaves orientation and ordering of links of some graph \Gamma invariant. Then according to the first in-line equation in section 6.4 Rovelli sets
U_\phi|\Gamma\rangle
=
|\phi\Gamma\rangle
\,.
This is the precise analogue of equation (6.25) in Thiemann's paper. And this is the problem, because this relation only holds because the U_\phi are constructed in a way to satisfy precisely this relations. That's certainly possible, the operators U_\phi undoubtly exist. What is problematic is that nothing in the world so far tells us that we should demand quantum states to be invariant under the classical gauge group induced by these U_\phi, which is however the content of equation (6.43) in Rovelli's book.
The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation \langle \psi|\pi(C)|\psi\ranfgle = 0.
In the last paragraph on page 34 of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf Rovelli seems to claim that the latter is possible. This is in contradiction to what Thomas Thiemann said at the Coffee Table.
eigenguy
Feb9-04, 02:30 PM
Originally posted by Urs
There is no canonical quantization in principle of the ADM constraints of the EH action.
Whoa! Is this widely known?
Originally posted by Urs
LQG only avoids/ignores this no-go-fact by quantizing only the Hamiltonian constraint...
...which seems to be virtually impossible to solve, while the constraints that have been solved are imposed...
Originally posted by Urs
... by hand.
There is a message in here somewhere.
selfAdjoint
Feb9-04, 02:48 PM
Urs,
Thiemann's equation 6.25,
U_{\omega}(g)\pi_{\omega}(b)\Omega_{\omega} = \pi_{\omega}(\alpha_g(b))\Omega_{\omega}
Comes directly from his corollary 4.1 (which I commented on above):
U_{\omega}\pi_{\omega} := \pi_{\omega}(\alpha_g (a))\Omega
Which is, he claims, given to him by GNS, and modulo my doubts about his handling of the group, this is true according to Haag. He may have pulled part of Corollary 4.1 out of the blue but that is not true of this unitary relationship.
If his GNS is kosher, then this U can be assumed to exist as part of the construction. In that case to reject it as not proper quantum mechanics is to reject GNS and the whole enterprise of algebraic quantum field theory too.
Now Urs has said what he thinks is wrong with LQG and what, in his view, invalidates the paper under discussion. And he refers me to what are, for me, standard texts of LQG (rovelli 1998 livingreviews and rovelli 2004 "Quantum Gravity" book)
I am very content with this. I dont have to try to say whether Urs is wrong or right or whether Rovelli is right or wrong. The important thing is Urs has said what he thinks is wrong and I can study it and give it the appropriate consideration. This is a big benefit and improvement!
So some thanks are due to both of you selfAdjoint and Urs for steering the rowboat of this conversation thru the rough waters
of unfriendly argument and finally into some calm understanding!
I am impressed with the patience shown by both of you! It is even
surprising me that we didnt tip over and all sink at some point.
eigenguy
Feb9-04, 03:42 PM
Originally posted by selfAdjoint
Urs,
Thiemann's equation 6.25,
U_{\omega}(g)\pi_{\omega}(b)\Omega_{\omega} = \pi_{\omega}(\alpha_g(b))\Omega_{\omega}
Comes directly from his corollary 4.1 (which I commented on above):
U_{\omega}\pi_{\omega} := \pi_{\omega}(\alpha_g (a))\Omega
Which is, he claims, given to him by GNS, and modulo my doubts about his handling of the group, this is true according to Haag. He may have pulled part of Corollary 4.1 out of the blue but that is not true of this unitary relationship.
If his GNS is kosher, then this U can be assumed to exist as part of the construction. In that case to reject it as not proper quantum mechanics is to reject GNS and the whole enterprise of algebraic quantum field theory too.
Maybe GNS allows more than the usual quantum theories, with some being more viable physically than others. In thiemann's implementation of it, the quantum states are assumed, in urs's words, "invariant under the classical gauge group induced by these U_{\omega}.
selfAdjoint,
yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.
Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.
The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!
Originally posted by Urs
Marcus,
the problem is in equation (33) of http://relativity.livingreviews.org/Articles/lrr-1998-1/download/lrr-1998-1.pdf . This is essentially the equivalent to (6.25) in Thiemann's paper and says that the classical group is acting by fiat on the quantum states and that physical states are those invariant under this classical group action. This step does not follow from any standard quantization procedure.
Incredibly enough this (33) was the equation I was trying to tell you about early in the thread, and now we have come round to it again.
To be a little finicky about language it does not say to "fillet" or take out the invariant states.
It says to take the quotient Hilbert space by a certain eqivalence relation.
The vectors in the new vector space are sets of vectors from the old space.
the vectors in the new space (of physical states or HDiff)
are equivalence classes of old vectors, under the operation of the group Diff(M).
We have all met this in algebra countless times, including for some even the first time they met the complex numbers---which some books define as a quotient of a polynomial ring.
In the new book "Quantum Gravity" Rovelli uses an extended Diff group and gets a reduced quotient that happens to be separable. That was why I was talking about separable earlier in thread. But this does not matter. I dont want to talk about that again!!!!
I am just glad we have finally met at this (equivalence class quotient) algebraic definition of the state space.
I will think more about your objection to it.
when you take diffeo equivalence classes of networks you get abstract knots
so Urs, on the page 34 you pointed me to, I see
"The second reason [that diffeo invariance is good for the theory] is that HDiff turns out to have a natural basis labeled by knots.
...an equivalence class of spin networks under diffeomorphism...
..is characterized by its "abstract" graph (defined only by the adjacency relations between links and nodes), by the coloring, and by its knotting and linking properties, as in knot theory.
Thus, the physical quantum gravity states of the gravitational field turn out to be essentially classified by knot theory"
think how heart-warming this could sound to a mathematician.
So the spatial diffeo invariance "has" to be handled this way because of the nice topological and algebraic outcome that the states of the grav. field are a hilbertspace of knots.
and quantum superpositions of knots.
But dont give up on a real red-blooded constraint too!!!
There is still a diffeomorphism constraint coming later.
We only dealt with the spatial diffeo invariance. there is still more. so a constraint will be imposed later-----gauss, diffeo, hamiltonian. Three of them.
I am not saying you should agree or be happy. I am just kind of sketching the outlines of how I see your objections.
You point me to (33) which defines
HDiff
and you dont like it. I will try to digest and understand this.
Again thanks, and in advance for any more explanation of what you find nonstandard!
Originally posted by marcus
Now Urs has said what he thinks is wrong with LQG and what, in his view, invalidates the paper under discussion. And he refers me to what are, for me, standard texts of LQG (rovelli 1998 livingreviews and rovelli 2004 "Quantum Gravity" book)
I am very content with this. I dont have to try to say whether Urs is wrong or right or whether Rovelli is right or wrong. The important thing is Urs has said what he thinks is wrong and I can study it and give it the appropriate consideration. This is a big benefit and improvement!
So some thanks are due to both of you selfAdjoint and Urs for steering the rowboat of this conversation thru the rough waters
of unfriendly argument and finally into some calm understanding!
I am impressed with the patience shown by both of you! It is even
surprising me that we didnt tip over and all sink at some point.
Marcus a paper by Marolf and Rovelli from sometime ago may have a baring on this thread:http://uk.arxiv.org/abs/gr-qc?0203056
Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights [a)]
selfAdjoint
Feb9-04, 05:34 PM
Originally posted by Urs
selfAdjoint,
yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.
Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.
The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!
Urs, I'm going to quit this discussion because we are talking past each other. Thiemann has two things, after the dust settles: he has a very persuasive model of the string, laid out in his section 6.2, and he has the classic results of "local quantum physics" as Haag puts it. His achievement is to apply the latter to the former. Now you say this is not what you are told to do by standard quantum theory. So much the worse for standard quantum theory. Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.
So I can't convince you and I'm afraid you can't convince me.
Originally posted by ranyart
Marcus a paper by Marolf and Rovelli from sometime ago may have a bearing on this thread:http://uk.arxiv.org/abs/gr-qc?0203056
Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights [a)]
you know ranyart though I dont have the right to judge I have to say I think Rovelli's thoughts about quantum theory are among the most perceptive and sophisticated--especially in connection with relativity. he thinks about situations and measurments in an extremely concrete fashion.
I keep seeing Marolf's name, maybe he is another one who really thinks instead of just operating at a symbolic level.
Rovelli has a section, pages 62-68 in the book, where he talks about
"Physical coordinates and GPS observables"
it uses the Global Positioning Satellite system to illustrate something about general relativity. I havent grasped it. have you looked at it?
Anyway thanks for the link.
what it means to me relative to this thread is the article you give is further evidence that Rovelli does not just quantize by rote, or by accepted procedures. He is one of the more philosophically astute people in knowing what is going on when he quantizes something. (IMHO of course :))
arivero
Feb10-04, 05:18 AM
Originally posted by Urs
The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation \langle \psi|\pi(C)|\psi\ranfgle = 0.
Let me to notice the historical remark in Rovelli Living Review:
The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin [182, 163, 183, 184] to ``change basis in the Hilbert space of the theory''
...The immediate results of the loop representation were two: The diffeomorphism constraint was completely solved by knot states (loop functionals that depend only on the knotting of the loops), making earlier suggestions by Smolin on the role of knot theory in quantum gravity [195] concrete; and (suitable [184, 196] extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.
It seems there are so sure of his technique that the review articles already forget to relate it to the constrains.
On other hand, Thiemann Hamiltonian constrain is a later development, dated 1996.
Yes, it's kind of strange. The quantum constraints are not even mentioned anymore when it comes to 'solving' diffeo-invariance in LQG reviews. I believe this is a trap. At least people should be well aware that at this point standard canonical quantization is abandoned. Luckiliy, this has become clear now in the simpler example of quantization of the Nambu-Goto action by Thomas Thiemann.
eigenguy
Feb10-04, 10:06 AM
I know that the following is implicit in what urs said, but it's worth pointing out that on the issue of whether gravity should be quantized, dirac said that it would be hard to see how a theory that treats gravity classically and other interactions quantumly could be consistent. For the same reason, it seems reasonable that gravity should be quantized in the same way as other interactions as well, making LQG seem even less plausible.
I have now contacted A. Ashtekar and H. Nicolai. Let's see if they have something to say about the LQG-string.
Originally posted by Urs
I have now contacted A. Ashtekar and H. Nicolai. Let's see if they have something to say about the LQG-string.
Bravo Urs! This is a great thread, we are getting our money's worth, so to speak. must again express thanks to you for your carefulness, open-mindedness, patience etc.
whatever they may say, it is only to the good that they answer---but I do hope they respond in timely manner!
Hi Marcus,
yes, but they might answer at the Coffee Table! :-) So get a copy of Mozilla. It's free, it's easy, it does not not interfere with anything and Mozilla is more politically correct than MSIE, anyway. ;-)
BTW, anyone who is interested in following the discussion at the Coffee Table but wants to be informed automatically about new comments should download an "RSS News Aggregator" such as Sharp Reader (http://www.sharpreader.net/). This allows you to read the Coffee Table just like any usenet newsgroup, plus some extras. Just download, install, and then drag-and-drop the boxes that sit under the headline "Syndicate" at the Coffee Table entry page into the SharpReader window.
eigenguy
Feb10-04, 02:21 PM
Hi urs,
You need to edit the link to sharp reader.
Originally posted by Urs
BTW, anyone who is interested in following the discussion at the Coffee Table but wants to be informed automatically about new comments should download an "RSS News Aggregator" such as Sharp Reader (http://http://www.sharpreader.net/). This allows you to read the Coffee Table just like any usenet newsgroup, plus some extras.
Urs-
is there a RSS news reader that supports MathML? i didn't try sharpreader, since i don't run windows. does it support MathML?
No, unfortunately I couldn't make SharpReader display MathML. I use the reader to stay in touch with new comments and switch to Mozilla when I need to read equations. That's not the way it should be, of course.
There should be "RSS News Aggregators" for all kinds of operating systems. I'll ask Jacques Distler. He himself is using MacOS.
eigenguy
Feb10-04, 06:07 PM
Originally posted by selfAdjoint
Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.
From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.
selfAdjoint
Feb10-04, 06:47 PM
Originally posted by eigenguy
From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.
And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.
eigenguy
Feb10-04, 07:31 PM
Originally posted by selfAdjoint
And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.
I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.
Originally posted by eigenguy
I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.
um, actually, it does. the claymath problem is specifically about QFT.
eigenguy
Feb10-04, 07:52 PM
Something just occurred to me. Suppose it turns out that LQG is wrong for the reasons that urs discovered. Wouldn't it stand to reason that if other physicists had given LQG a serious look, they would have seen this a long time ago? I believe that feynman said the physicists job is to prove themselves wrong as quickly as possible (Of course, from this point of view, the LQG camp would deserve most of the blame).
eigenguy
Feb10-04, 08:01 PM
Originally posted by lethe
um, actually, it does. the claymath problem is specifically about QFT.
You will find the following description here (http://www.claymath.org/millennium/Yang-Mills_Theory/):
Yang-Mills and Mass Gap
The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory [note the word "field" is not used here or anywhere else in this paragraph] is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.
Clearly, no assumption has been, nor should be, made about what the solution will look like.
Originally posted by eigenguy
Clearly, no assumption has been, nor should be, made about what the solution will look like.
ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!
eigenguy
Feb10-04, 08:35 PM
Originally posted by lethe
ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.
Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.
OK, fine, string theory allows nonabelian gauge theories. but Yang-Mills theory is not string theory, it is a quantum field theory. The positive mass gap conjecture is not about string theory or some other as-yet-undetermined theory, it is about Yang-Mills theory.
selfAdjoint
Feb10-04, 09:42 PM
Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.
Eigen, I am afraid you've got your foot in by your tonsils. The words Yang-Mills, followed by the word theory, refer to a class of Quantum Field Theories. If you want to refer to Yang-Mills symmetry, you say Yang-Mills symmetry. See for example
Peskin & schoeder section 15.2, the field theory associated with a non-commuting local symmetry is termed a non-Abelian gauge theory.
Ryder, section 3.5 The Yang-Mills field.
Both P & S and Ryder have in their indices, Yang-Mills theory, see non-Abelian gauge theory.
Yang-Mills theory was quantized by Veltzmann & 'tHooft, becoming thereby a Quantum Field Theory. It is this theory that is usually referred to as Y-M theory.
eigenguy
Feb10-04, 10:53 PM
Originally posted by selfAdjoint
Eigen, I am afraid you've got your foot in by your tonsils. The words Yang-Mills, followed by the word theory, refer to a class of Quantum Field Theories. If you want to refer to Yang-Mills symmetry, you say Yang-Mills symmetry. See for example
Peskin & schoeder section 15.2, the field theory associated with a non-commuting local symmetry is termed a non-Abelian gauge theory.
Ryder, section 3.5 The Yang-Mills field.
Both P & S and Ryder have in their indices, Yang-Mills theory, see non-Abelian gauge theory.
Yang-Mills theory was quantized by Veltzmann & 'tHooft, becoming thereby a Quantum Field Theory. It is this theory that is usually referred to as Y-M theory.
Obviously if you look in a QFT book the definitions you find will be made in terms of QFT. But ST came after QFT, and in ST you will find discussions of non-abelian gauge theory in which the term yang-mills is used without reference to QFT.
However none of this matters. My point was that the status of the problem of finding a rigorous formulation of QFT is tied to the question of whether QFT is just an approximation to some more fundamental theory. If it is, then we shouldn't be upset or surprised if in fact QFT can never be formulated in a completely rigorous way. Note that the basic legitimacy of my point really does not depend on the definition of yang-mills.
Btw, I got this from weinberg, so go argue with him.
eigenguy
Feb10-04, 10:55 PM
By the way selfadjoint,
I pm'ed you again asking what you thought of haag's book. I'd really like to know, especially about what it says about the GNS construction.
Originally posted by eigenguy
Obviously if you look in a QFT book the definitions you find will be made in terms of QFT.
actually, if you look in any book where the author knows what he is talking about, you will find yang-mills defined as a field theory. any properly trained physicist knows this, and would not say otherwise. of course this includes the authors of many popular quantum field theory books, but also many other books, including string theory books.
But ST came after QFT, and in ST you will find discussions of non-abelian gauge theory in which the term yang-mills is used without reference to QFT.
can you please provide a reference to a string theory book which refers to Yang-Mills theory without meaning it as a field theory?
However none of this matters. My point was that the status of the problem of finding a rigorous formulation of QFT is tied to the question of whether QFT is just an approximation to some more fundamental theory. If it is, then we shouldn't be upset or surprised if in fact QFT can never be formulated in a completely rigorous way. Note that the basic legitimacy of my point really does not depend on the definition of yang-mills.
even if Yang-Mills theory is only a low energy effective theory, it still makes sense to ask questions about its properties. whether it is a consistent theory. the is a derivation in Peskin and Schroeder that shows that no matter what the high energy theory, there should be a renormalizable low energy effective quantum field theory describing it at some energy scale.
the Claymath problem asks for some properties of this theory to be put on a firm mathematical basis.
Btw, I got this from weinberg, so go argue with him.
if your point is that quantum field theory is irrelevant, then i would say that you have misinterpreted Weinberg, he would almost certainly say no such thing. so i will argue with you.
of course, you can prove me wrong quite easily: provide references. papers? page numbers? just show me where Weinberg thinks that solving nonperturbative Yang-Mills would be irrelevant.
eigenguy
Feb11-04, 01:21 AM
Originally posted by lethe
if your point is that quantum field theory is irrelevant
Of course thats not my point and of course thats not what weinberg says.
However, I believe I am wrong and you and selfadjoint are right about the expression "yang-mills theory" meaning QFT. But I also think one can refer to a non-abelian gauge symmetry as a yang-mills type symmetry in theories that aren't field theories.
But I was initially responding to the following post
Originally posted by selfAdjoint
Urs, I'm going to quit this discussion because we are talking past each other. Thiemann has two things, after the dust settles: he has a very persuasive model of the string, laid out in his section 6.2, and he has the classic results of "local quantum physics" as Haag puts it. His achievement is to apply the latter to the former. Now you say this is not what you are told to do by standard quantum theory. So much the worse for standard quantum theory. Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.
So I can't convince you and I'm afraid you can't convince me.
My point was that it may be that if QFT is an approximation to a more fundamental theory, we should not be upset or surprised if it turns out there is no rigorous way to formulate QFT so that the algebraic or any other approach to doing so may be destined to fail. But of course QFT is valid in it's domain of applicability.
Originally posted by eigenguy
My point was that it may be that if QFT is an approximation to a more fundamental theory, we should not be upset or surprised if it turns out there is no rigorous way to formulate QFT so that the algebraic or any other approach to doing so may be destined to fail. But of course QFT is valid in it's domain of applicability.
you do know that string theory is a 2D quantum field theory, right?
eigenguy
Feb11-04, 07:07 AM
Originally posted by lethe
string theory is a 2D quantum field theory, right?
I didn't say the new theory must be ST, and ST is not a theory of fields, but reduces to one in the low energy limit.
Originally posted by eigenguy
A theory of strings on the world-sheet is not the same as an ordinary 2D theory of fields, the difference being due to the extended nature of strings. ST is not a theory of fields, but reduces to one in the low energy limit.
you are not correct.
eigenguy
Feb11-04, 07:16 AM
Originally posted by lethe
you are not correct.
So you are saying that ST can be understood completely in ordinary field theoretic terms?
Originally posted by eigenguy
So you are saying that ST can be understood completely in ordinary field theoretic terms?
i am only saying what you see me saying. do not put words in my mouth that you did not see me say.
eigenguy
Feb11-04, 07:27 AM
Originally posted by lethe
i am only saying what you see me saying. do not put words in my mouth that you did not see me say.
I'm certainly not trying to put words in your mouth. Could you help me understand your view of the relation between ST and QFT. Keep in mind, I'm no expert and do not claim to be and am quite happy to admit I'm wrong the moment I believe that I am. I should point out that weinberg explains that he wrote his QFT books to address the possibility that a final theory does not necessarily have to be a field theory, and uses the example of ST to make his point. So basically, I've just been giving my best understanding of weinberg's views. Also, I'm pretty sure the mathematics of ST goes well beyond that of QFT and this is what is most germaine to my argument about whether we should expect there be a way to rigorously formulate it.
Originally posted by eigenguy
I'm certainly not trying to put words in your mouth. Could you help me understand your view of the relation between ST and QFT.
physically, string theory is not a spacetime quantum field theory, since in spacetime, it has strings instead of points. however, mathematically, it is simply a 2D quantum field theory. it uses all the standard techniques of conformal field theory.
in short: if quantum field theory is broken for some reason, then so is string theory.
Keep in mind, I'm no expert and do not claim to be and am quite happy to admit I'm wrong the moment I believe that I am.
i will keep that in mind.
arivero
Feb11-04, 08:05 AM
Hmm 2D Conformal Field Theory is surely simpler than 4D QFT. All these 1+1 theories enjoy a symmetry group a lot more restricted than 3+1 Lorentz.
eigenguy
Feb11-04, 08:14 AM
Originally posted by lethe
physically, string theory is not a spacetime quantum field theory, since in spacetime, it has strings instead of points. however, mathematically, it is simply a 2D quantum field theory. it uses all the standard techniques of conformal field theory.
But ST includes D-branes, what about them?
Originally posted by eigenguy
But ST includes D-branes, what about them?
what about them?
eigenguy
Feb11-04, 08:51 AM
Originally posted by lethe
what about them?
Can D-branes be understood purely in terms of the world-sheet theory? Btw, are you a ST expert? I'm sure urs can clear this up.
selfAdjoint
Feb11-04, 08:53 AM
Originally posted by eigenguy
By the way selfadjoint,
I pm'ed you again asking what you thought of haag's book. I'd really like to know, especially about what it says about the GNS construction.
And I replied, didn't you get it?
eigenguy
Feb11-04, 09:05 AM
Originally posted by selfAdjoint
And I replied, didn't you get it?
Oops! I just checked and yes I got it. Sorry about that. I think I will order a copy though since it sounds like it will be useful and I know it has many very worthwhile insights. I'll let you know how I'm doing with it after I've had a chance to peruse it a bit. Thanks a bunch!
Originally posted by eigenguy
Can D-branes be understood purely in terms of the world-sheet theory?
yes. of course, they can also be understood in target space theory, but that is a good thing.
Btw, are you a ST expert?
i prefer to stay anonymous.
Haelfix
Feb11-04, 11:32 AM
We;re deviating too much from the original thread, but indeed ST is intimately linked to quantum field theory and the mathematical machinery behind it.
If QFT has a mathematical error at some point, (for instance Fadeev Popov quantization) then its nearly guarenteed to pop out in perturbative ST as well.
Presumably, the nonperturbative sector of ST is something different and new, but no one knows what that is either.
eigenguy
Feb11-04, 11:41 AM
Originally posted by lethe
i prefer to stay anonymous.
As you should, and I would never ask you to compromise that. It's just that I'm studying polchinski volume I now and if you are ahead of me that would be good to know, assuming you like talking about it, which you seem to.
Originally posted by lethe
yes. of course, they can also be understood in target space theory, but that is a good thing.
Okay, so let me comment in on this specifically in terms of what I've read in polchinski. Then I want to take a closer look at the basic issue you raised about ST really being a 2D QFT because if you are right and I'm not getting this, then I really have to reexamine things. I'm going to state things in a matter of fact way, so don't assume I'm pointing something out because I think you don't already know it.
So first target space "theory". The polyakov action is an example of a so-called non-linear sigma model embedding the world-sheet in a target space which here is spacetime. On the other hand, the dynamics of D-branes is governed by the born-infeld action whose relation to 2D CFT is not entirely clear to me. It seems there must be some connection though because D-branes arise by T-duality from open strings on backgrounds involving wilson lines. Perhaps you can explain this further. But I haven't heard of a target space "theory". I guess you probably meant what I just said anyway.
Now on the ST-QFT connection. I guess what you are saying is that in some very real sense ST can be broken down to or understood in terms of what could be legitimately viewed in some sense as field theory. I don't think the basic configuration variables X^\mu are fields in the sense of QFT. For example, string rest mass is not equal to the square of their 4-momentum, but includes contributions from it's internal oscillations as well. In fact the mode oscillators give rise to infinite dimensional algebras that (I think) are missing from ordinary field theory. Maybe we are using different definitions of field?
Originally posted by Haelfix
ST is intimately linked to quantum field theory
Yes, in that it appears in it's low energy limit. But I don't think inconsistencies in QFT necessarily imply inconsistencies in whatever M-theory turns out to actually be.
Originally posted by eigenguy
As you should, and I would never ask you to compromise that. It's just that I'm studying polchinski volume I now and if you are ahead of me that would be good to know, assuming you like talking about it, which you seem to.
well, i have seen you launch character assaults on people on this forum based on your impression of their knowledge. for this reason, i prefer it when your impression of my knowledge is very vague.
But I haven't heard of a target space "theory". I guess you probably meant what I just said anyway.
by that i just meant the low energy effective field theory in the target space. the details depend on which string theory you are looking at.
Now on the ST-QFT connection. I guess what you are saying is that in some very real sense ST can be broken down to or understood in terms of what could be legitimately viewed in some sense as field theory. I don't think the basic configuration variables X^\mu are fields in the sense of QFT.
X is a bosonic field.
For example, string rest mass is not equal to the square of their 4-momentum, but includes contributions from it's internal oscillations as well.
certainly it is not the center of mass momentum squared, but that is silly. the mass of a stringy excitation is indeed its momentum squared.
In fact the mode oscillators give rise to infinite dimensional algebras that (I think) are missing from ordinary field theory. Maybe we are using different definitions of field?
perhaps....
Yes, in that it appears in it's low energy limit. But I don't think inconsistencies in QFT necessarily imply inconsistencies in whatever M-theory turns out to actually be.
perhaps.
eigenguy
Feb11-04, 01:17 PM
Originally posted by lethe
X is a bosonic field.
Well, X is bosonic anyway: It is both a spacetime and world-sheet boson. The kind of fields I'm talking about are defined as such because of their "point-likeness":, i.e., they have no internal degrees of freedom.
Originally posted by lethe
certainly it is not the center of mass momentum squared, but that is silly. the mass of a stringy excitation is indeed its momentum squared.
The mass squared of an open bosonic string state is a sum of a zero mode term which is the as you say the "centre of mass" momentum squared, and terms involving higher modes. But I don't recall coming across attributions of spacetime momentum to internal excitations. I'll have a careful look at this, since what you are saying seems intuitively true, but from the standpoint of my above comment, this would not help you.
What about my question about D-branes?
Originally posted by lethe
well, i have seen you launch character assaults on people on this forum based on your impression of their knowledge. for this reason, i prefer it when your impression of my knowledge is very vague.
It's not their knowledge, it's their motives and tactics. Pointing out that despite the authoritive air that always accompanies their comments (especially when criticizing mainstream views which they admit they don't pay attention to) touching on the subject of their "religion" (which they also never really understood, something that can be easily seen by looking at their rather curt exchanges with urs in this very thread.) they really don't know what the hell they are talking about, is just one way of preventing members from being suckered into turning away from reality and joining their irrational fanaticism. Make no mistake, when it comes to these guys, it's all about egos, their interest in physics is really just incidental and would have played out the same way whatever the subject was.
Originally posted by eigenguy
What about my question about D-branes?
it is not at all clear to me what your question is, but i think we have been off-topic on this thread for far too long. i was enjoying this thread a lot, and i only stepped in to defend selfAdjoint from your false impressions about Yang-Mills theory.
eigenguy
Feb11-04, 02:16 PM
Originally posted by lethe
it is not at all clear to me what your question is, but i think we have been off-topic on this thread for far too long.
Fine, but for what its worth, my question about D-branes was how do you reconcile the fact that they are described by the born-infeld action with your statement that string theory is really just a 2D QFT.
We need to scroll back to page 20 of this thread to reconnect with the main substance of the discussion. Urs took exception to the fact that in LQG a hilbert space of (spin-labeled) knots serves to embody the states of the gravitational field.
Since the states start out embodied as spin networks, i.e. embedded spin-labeled graphs, to get abstract knots one has to take diffeomorphism equivalence classes. It is a common algebraic proceedure---factoring down by an equivalence relation----sometimes used in constructing, for example, the complex numbers.
Two spin-networks are to be considered equivalent if one can be smoothly deformed into the other.
That is, one mapped into the other by a diffeomorphism, or (if you like special effect movies) one network "morphed" into the other.
Only abstract knot-type info remains when the networks are grouped into diffeo-equivalence classes.
Urs argued that this algebraic way of realizing spatial
diffeo-invariance was not kosher quantum theory. Perhaps it invalidated the whole of LQG? SelfAdjoint mentioned that the proceedure was used in Algebraic Quantum Field Theory (AQFT).
Around this point, on page 20 of the thread, Urs said he had
contacted two authorities, Abhay Ashtekar and Hermann Nicolai,
about this.
This interesting issue arose because Thomas Thiemann did something analogous (implementing a certain relation algebraically) in his paper. An objection to the special case (in TT's paper) implied a general-case fundamental objection to the construction of the state space in LQG.
I happened to be online around noon Germany time when Urs checked in to this forum. But he just looked and went away. I think it is too bad the last 3 pages have been so off topic. So, in hopes of restoring a connection to the main thread, I will quote from page 20.
The first post here is from selfAdjoint.
=================================================
quote by selfAdjoint of something by Urs:
---------------------------------------------
Originally posted by Urs
selfAdjoint,
yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.
Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.
The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!
-------------------------------------------------------------
selfAdjoint:
Urs, I'm going to quit this discussion because we are talking past each other. Thiemann has two things, after the dust settles: he has a very persuasive model of the string, laid out in his section 6.2, and he has the classic results of "local quantum physics" as Haag puts it. His achievement is to apply the latter to the former. Now you say this is not what you are told to do by standard quantum theory. So much the worse for standard quantum theory. Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.
So I can't convince you and I'm afraid you can't convince me.
02-09-2004 03:34 PM
=================================================
marcus:
quote by me, of something by ranyart
--------------------------------------------------------------
Originally posted by ranyart
Marcus a paper by Marolf and Rovelli from sometime ago may have a bearing on this thread:http://uk.arxiv.org/abs/gr-qc?0203056
Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights
----------------------------------------------------------------
you know ranyart though I dont have the right to judge I have to say I think Rovelli's thoughts about quantum theory are among the most perceptive and sophisticated--especially in connection with relativity. he thinks about situations and measurments in an extremely concrete fashion.
I keep seeing Marolf's name, maybe he is another one who really thinks instead of just operating at a symbolic level.
Rovelli has a section, pages 62-68 in the book, where he talks about
"Physical coordinates and GPS observables"
it uses the Global Positioning Satellite system to illustrate something about general relativity. I havent grasped it. have you looked at it?
Anyway thanks for the link.
what it means to me relative to this thread is the article you give is further evidence that Rovelli does not just quantize by rote, or by accepted procedures. He is one of the more philosophically astute people in knowing what is going on when he quantizes something. (IMHO of course :))
02-09-2004 04:08 PM
===========================================
arivero:
invariance of diathige and trope
quote:
--------------------------------------------------------------------------------
Originally posted by Urs
The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation .
-----------------------------
Let me to notice the historical remark in Rovelli Living Review:
quote by arivero of something by Rovelli:
----------------------------------------------
The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin [182, 163, 183, 184] to ``change basis in the Hilbert space of the theory''
...The immediate results of the loop representation were two: The diffeomorphism constraint was completely solved by knot states (loop functionals that depend only on the knotting of the loops), making earlier suggestions by Smolin on the role of knot theory in quantum gravity [195] concrete; and (suitable [184, 196] extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.
-------------------------------
It seems there are so sure of his technique that the review articles already forget to relate it to the constrains.
On other hand, Thiemann Hamiltonian constrain is a later development, dated 1996.
02-10-2004 03:18 AM
====================================
Urs:
Yes, it's kind of strange. The quantum constraints are not even mentioned anymore when it comes to 'solving' diffeo-invariance in LQG reviews. I believe this is a trap. At least people should be well aware that at this point standard canonical quantization is abandoned. Luckiliy, this has become clear now in the simpler example of quantization of the Nambu-Goto action by Thomas Thiemann.
====================================
It was soon after this that Urs reported he had written to both
Abhay Ashtekar and Hermann Nicolai about this perceived "non-standardness" of LQG.
I hope very much their replies can be forwarded to PF and are not
relegated solely to Jacques Distler's message board!
selfAdjoint
Feb12-04, 11:23 AM
I guess I should put in something here. Of all the things in Thiemann's paper that are called arbitrary, I can find only one that I think really is arbitrary, and that is his representation of his Weyl algebra, display 6.20:
\omega_{\pm}(W_{\pm}(s)) := \delta_{s,0}
He adopts this wild and crazy representation (=1 on all his momentum networks, 0 on "the empty network") in default of being able to develop a real representation theory. The only authority for it he cites is his prior experience with LQG developments.
Now I am not able to show this myself, but it seems plausible that if you went with a representation that couldn't distinguish one momentum distribution from another, you might get a theory that couldn't recognized anomalies.
selfAdjoint, yes I remember that eqn. 6.20
It is in section "6.3 A specific example"
He warns us early on, as I recall, that he is opening up
a broad problem of finding all the representations and, in this
paper, only taking an initial "baby step" so to speak of
offering one representation, which IIRC he notes is not very interesting.
"In this paper we will content ourselves with giving just one
non-trivial example. Here it is:"
eigenguy
Feb12-04, 11:43 AM
I thought that in quantum theory poincare picks up no anomaly. So maybe momentum distributions aren't relevant (unless you are talking about a different kind of momentum).
The conclusions in TT's paper are phrased in a cautious fashion, as if to say "if we extend this and it checks out then so-and-so"
so in the abstract:
"While we do not solve the...representation theory completely...we present [one solution]...
The existence of this stable solution is...exciting because raises the hope [that by looking further for more complicated solutions]...
we find stable, phenomenonologically acceptable ones in lower dimensional target spaces..."
So this first solution, which you point to in equation 6.20
is a drop in the bucket----he cautions us up front, reasonably enough.
I guess the point is that, as he says, even that one rather artificial unphysical case is indeed exciting. Because we did seem to get excited whether over at Distler's board or here at PF.
But realistically it has to be followed by substantially more research to mean anything, or?
eigenguy
Feb12-04, 12:01 PM
selfadjoint, do you agree that the reason that thiemann gets no anomaly is that he doesn't use the original formulation of refined algebraic quantization?
selfAdjoint
Feb12-04, 01:37 PM
I have been digging into his quantization - BTW, all his claims of no anomaly come from the sections of his paper BEFORE quantization. That is, they are about his classical theory. But all the criticisms are that quantization brings in the central charge. It is true that Polchinski brings in the central charge at the classical level, but apparently that isn't required, and a theory that was classically clean but had the c.c. as a quantum anomaly would pass muster with the string physicists.
He uses the methods from the paper Quantization of diffeomeorphism invariant theories of connections with local degrees of freedom, gr-qc/9504018, by Ashtekar, Lewandowski, Marolf, Mourao, and Thiemann. Notice that Marolf is coauthor of the 1999 paper on group averaging (Giulini & Marolf, A uniqueness theorem for Constraint Quantization, gr-qc/9902045).
The 1995 paper also employs group averaging, but I am concerned that they explicitly DON'T do the Hamiltonian constraint (in the LQG context). They do diffeomeorphism constraints. I have been trying to work out what influence that might have on the Virasoro constraints in the string problem. It seems to be a spacelike/timelike thing.
eigenguy
Feb12-04, 02:49 PM
Distler just posted this,
Baby & Bathwater
So now, the party line is that Thiemann’s quantization is some clever new method of quantization, completely unrelated to canonical quantization, that no one has thought of before.
This is not only my interpretation, but Thomas Thiemann himself says that the procedure, sketched above, for dealing with the constraints, should be compared to experiment to see if nature favors it over standard Dirac/Gupta-Bleuler quantization.
It is well-known that if one is willing to abandon locality, one has great lattitude to “cancel” the anomalies which arise in local QFT. A charitable interpretation of Thiemann’s procedure is that it correponds precisely to such a nonlocal modification of local field theory.
There are reasons to reject nonlocal modification of the worldsheet theory of the bosonic string — to do with getting consistent string interaction, a problem on which Thiemann is clueless, as he has, at best, made a failed attempt to construct the free bosonic string.
However, it is quite clear why Thiemann does not wish to apply his methods to Quantum Field Theories people care about, like Yang-Mills Theory. There, we know quite clearly whose side Mother Nature has come down on.
selfAdjoint
Feb12-04, 04:03 PM
Thiemann's quantization procedure from the 1995 paper I cited is specifically stated to be local. The problem with it is that it may not be available in case the constraints don't close. Giulini and Marolf explicitly assume that, but what their paper provides, I now see, is just the fact that the "rigging map" is unique. So there is a possible freedom there, that the quantization is not unique. But I'm still digging.
Aside from the nonlocal suggestion, I don't see what this post by Distler accomplishes. Just some more of his sarcasm, and the continuation of his phoney issue with Yang-Mills theory.
eigenguy
Feb12-04, 04:42 PM
Originally posted by selfAdjoint
phoney issue with Yang-Mills theory
I guess he believes that all interactions should be quantized using the same procedure so if it doesn't work for YM, it's wrong.
Hi everybody -
Distler's point is that anomalies are real and important and that defining them away in testable theories like the standard model leads to conflict with experiment.
I have received an answer by Hermann Nicolai, who says he is going to have a look at this issue.
So far he only confirmed my ideas about the DDF invariants, saying that D. Bahns once gave a talk at his institute about the Pohlmeyer stuff. Afterwards he had asked her if these invariants should not be expressible in terms of DDF invariants, because that would seem to be the only possibility. Apparently she said no, but Nicolai tells me that he is sceptical about this answer.
BTW, I think it is interesting that the Pohlmeyer invariants are Wilson loops along the string for large matrix valued constant connections. This is precisely the same construction as used in the IIB/IKKT Matrix Model, e.g. see equation (2.7) of hep-th/9908038.
Regarding the string/field theory question:
The analogues of Feynman diagrams in string theory are computed using field theory on the worldsheet. This gives rise to effective theories on spacetime and on branes, which are also field theories. That's no surprise, even the theory of a single particle can be regarded as a field theory, one in 1+0 dimensions. Field theory is a pretty versatile thing.
Even string field theory reduces, when all the parts in the action that involve computations in the string's Hilbert space are integrated out, to a field theory, albeit one with infinitely many fields (one for each excitation of the string).
Usually, if somebody says "Yang-Mills theory" people will think of the respective field theory. The interesting thing in modern string theory is that apparently all supersymmetric YM theories are dual to string theory, i.e. they describe the effective theory on some brane which is embedded in a bulk in which closed string propagate.
The most prominent example is N=4 SYM which is believed to be equivalent to strings on AdS5 x S5.
eigenguy
Feb13-04, 10:51 AM
Hi urs,
Originally posted by Urs
Regarding the string/field theory question:
The analogues of Feynman diagrams in string theory are computed using field theory on the worldsheet. This gives rise to effective theories on spacetime and on branes, which are also field theories. That's no surprise, even the theory of a single particle can be regarded as a field theory, one in 1+0 dimensions. Field theory is a pretty versatile thing.
Even string field theory reduces, when all the parts in the action that involve computations in the string's Hilbert space are integrated out, to a field theory, albeit one with infinitely many fields (one for each excitation of the string).
So was lethe correct in saying that string theory is really just 2D QFT? Or maybe it's correct to say that perturbative string theory is just 2D QFT?
Originally posted by Urs
Usually, if somebody says "Yang-Mills theory" people will think of the respective field theory. The interesting thing in modern string theory is that apparently all supersymmetric YM theories are dual to string theory, i.e. they describe the effective theory on some brane which is embedded in a bulk in which closed string propagate.
The most prominent example is N=4 SYM which is believed to be equivalent to strings on AdS5 x S5.
Was lethe also correct in saying that it is incorrect to think of yang-mills theory as meaning anything other than a QFT and that the solution to the mass gap problem is necessarily a QFT solution?
Was lethe correct in attributing spacetime momentum to the non-zero modes of a string?
Was my basic point - that there may be no rigorous way to formulate QFT, and if it is just an approximation to a more fundamental theory that we shouldn't be upset or surprised by this - correct?
Thanks a bunch.
Originally posted by Urs
... D. Bahns once gave a talk at his institute about the Pohlmeyer stuff. Afterwards he had asked her...
the Dorothea Bahns that TT mentions in his acknowledgements section, I guess. Do you know where she is? at Potsdam or?
Originally posted by eigenguy
Was lethe correct in attributing spacetime momentum to the non-zero modes of a string?
see, for example, Polchinski Vol I, eqs 4.3.20-4.3.22
Originally posted by lethe
see, for example, Polchinski Vol I, eqs 4.3.20-4.3.22
Hi Lethe, would it be possible for you to start a separate
thread about these questions (D-brane, Yang-Mills) and let us
have this thread for discussion of Thomas Thiemann's paper
and the response to it by those whom Urs has contacted?
Urs too!
it would be fine if you would care to start a thread
to discuss purely string topics
and let this one be more focused on the TT paper topic
BTW thanks for sharing Nicolai's initial response with
us, hope you hear from him again soon, and AA as well
Hi eigenguy -
I don't think it helps to say that string theory is "just 2d QFT". Calculating any given order of the string perturbation series indeed involves just 2d QFT. But the fact that you need to sum up the contributions from just-2d-QFT calculations for QFT on surfaces of different genus, i.e. for different QFTs really, is something that itself is not captured by any QFT.
It is captured by string field theory, though, which can be rewritten as an ordinary field theory with infinitely many fields and interactions.
But the crucial insight is that maybe all of string theory can be rewritten in terms of some QFT with finitely many fields, anyway. Still, string theory is not a QFT, but it may be "equivalent" to one in a certain sense, this is the Maldacena conjecture.
If you read the introduction to the Clay Millenium Prize Questiion on YM and the mass gap you'll note that Witten roughly argues as follows:
Nature is described by quantum YM theory. So YM QFT is important and we need to understand it.
But YM QFT is hard. So let's maybe first understand the problem in an easier setup. Let's search the space of all possible QFTs for nice ones that are not quite as hard as YM. It turns out that supersymmetric QFTs have many properties that make them much easier to study than the non-susy ones. They sort of sit at exceptional points in the space of all QFTs.
Therefore we should be interested in SUSY QFTs, even if nature is not susy. Ok. So which susy QFT? It turns out that of these nice theories one of the nicest ist N=4 supersymmetric Yang-Mills. So let's try to understand that one first, not forgetting that we are really interested in ordeinary non-sus YM.
But now a miracle happens: SYM and N=4 SYM for the U(N) gauge group with large N in particular seems to be closely related to string theory! Maldacenas AdS/CFT conjecture says that it is indeed dual to string theory. This would confirm the old intuition by t'Hooft, who long ago argued that large N gauge theories are dominated by planar Feynman diagrams that look like string worldsheets interacting. Even apart from that, Matrix Theory tells us that maybe SYM dimensionally reduced to a line (BFSS) or even to a single point (IKKT/IIB) for N taken to infinity is the nonperturbative description of M-Theory!
So even if nature is neither susy nor stringy, there is a relation to string theory. The old hope that strings would give us the elementary particle masses uniquely seems to have vanished. The more fascinating aspects of modern string theory are its intimate relations to all kinds of gauge theories. Nobody today can be interested as a theoretician in gauge theories without coming across some string theory. String theorists are the leading figures in field theory, too. See Seiberg-Witten Theory or indeed most of what Witten has done. Witten's latest paper is about how N=4 SYM can be described by a topological string even wiothout taking N to infinity. Witten's whole work is really related to understanding ordinary YM theory, in a sense.
So will the solution to the mass-gap problem be a QFT solution? I don't know! Maybe the crucial clues will come from string theory, that's quite plausible. But of course, due to the duality, it will then also be understandable in a pure QFT kind of way.
It is not clear yet that string theory is related to the experimentally accessible universe. But it is already clear that string theory is discovering new continents in our universe of theories.
Haelfix
Feb13-04, 12:30 PM
Urs,
thats the most concise explanation I've ever read on why String theory should be important. to theoreticians, even if you are an abject skeptic.
In my opinion, Wittens work on the subject of the space of connections modulo guage transformations is his primary accomplishments as a physicist. Subsequent development of topological field theory, its applications to Morse theory and Gromow-Witten invariants gave him the fields medal, (which he richly deserved).
Back to the topic though.
Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories. I suppose you could argue that the central charge present has a different topological character, and hence inapplicable to say YM.
But I don't see a good reason a priori to restrict this scheme to only live in quantum gravity scenarios.
Originally posted by Haelfix
...Back to the topic though.
Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories...
Haelfix, thanks for redirecting the discussion back on topic!
To anyone with other string questions (not directly connected with TT's paper) it would be great if you would start a thread for them---we could use a good string thread.
eigenguy
Feb13-04, 12:53 PM
Originally posted by Urs
I don't think it helps to say that string theory is "just 2d QFT"...
Fantastic. Thanks!
selfAdjoint
Feb13-04, 01:01 PM
Originally posted by Haelfix
Self Adjoint,
Why do you think its bogus to apply that quantization to various other theories. I suppose you could argue that the central charge present has a different topological character, and hence inapplicable to say YM.
But I don't see a good reason a priori to restrict this scheme to only live in quantum gravity scenarios.
I didn't say it was bogus to apply the quantiszation to Y-M or other theories, I said Distler's harping on quantizing Y-M was bogus because he refused to look directly at TT's theory. Perhaps I was over excited.
I just found a good paper on the background of the TT quantization on the arxiv. It's hep-th/0402097, Lectures on Integrable Hierarchies and Vertex Operators, by A. A. Vladimirov. It is intended for undergraduates, and after scanning it over I am truly impressed by those Russian undergraduates!
eigenguy
Feb13-04, 04:55 PM
What's the simplest system one could play with in which the same basic issues being discussed here arise?
ranyart
Feb13-04, 08:09 PM
Originally posted by eigenguy
What's the simplest system one could play with in which the same basic issues being discussed here arise?
Probably a single Kodama State embedded into a Zero Dimensional Phase(Zero-Worldsheet).
eigenguy
Feb13-04, 08:29 PM
Originally posted by ranyart
a Kodama State embedded in a Zero Dimensional Phase...
A rabbi and priest in a rowboat...
eigenguy wrote:
What's the simplest system one could play with in which the same basic issues being discussed here arise?
Ok, here is a simple exercise that everybody who has followed our discussion should be able to solve:
1) Consider the Nambu-Goto action in 1+0 dimensions, which describes the free relativistic particle in Minkowski space (alternatively, for those who enjoy a bigger challenge: the charged particle in curved space with an electromagnetic field turned on)
2) Compute the single constraint of the theory.
3) Do a Dirac quantization by promoting this single constraint to an operator equation. Discuss the resulting quantum equation.
4) Now subject this system instead to the method used in Thomas Thiemann's paper. Discuss the ambiguity that one encounters and the differences and/or similarities to 3).
ranyart
Feb13-04, 09:09 PM
Originally posted by eigenguy
A rabbi and priest in a rowboat...
Indeed two 'Ed's' are better than one?[;)]
There is another simple example:
Look at the LQG-like quantization of the 1d non-relativistic point is
http://xxx.uni-augsburg.de/abs/gr-qc/0207106 . Then note equation (IV.5) and the one above it. In these equations care is taken that the usual quantum algebra is carried over to the LQG-like quantization scheme. If one were to remove the term \exp(-\alpha^2)/2 one would get instead the classical algebra, which corresponds to the way Thomas Thiemann 'quantizes' the LQG-string.
selfAdjoint
Feb17-04, 02:23 PM
Urs, I was interested in Rehren's email that you posted on the Coffee Table, with its presentation of the various things that can happen, and the origin of the different levels of anomalies, central charges, and broken invariances. It was very educational. Do you have any comments on it? Does it have any bearing on these examples you give us from other quantizations?
selfAdjoint -
maybe we should carry this discussion to the Coffee Table where Rehren can see it.
Rehren argues that what Thiemann does is formally a form of 'quantization'. But let's not argue about words. The paper by Ashtekar, Fairhurst and Willis shows that with such a general notion of quantization one does not obtain the correct results even for the nonrelativistic 1d particle. The same holds true when applying this 'generalized' notion of quantization to the Maxwell field or the free relativistic particle. In each of these cases the results differ drastically from known physics (even if the large ambiguity in how to impose the exponentiated operator equations is used in a way that closely follows the usual quantization instead of following the classical theory).
This is not disputed by Ashtekar, Fairhurst and Willis. Their argument is the same as that by Thiemann: Maybe this drastically different notion of quantization is the correct one for gravity. Right, maybe it is. But maybe not. What I would like to see is some sort of motivation for why a radical departure from usual quantization is the right thing to do in quantum gravity. It is certainly not the physically right thing to do in the other systems that have been studied by this method.
Originally posted by Urs
Ok, here is a simple exercise that everybody who has followed our discussion should be able to solve:
lemme see..
1) Consider the Nambu-Goto action in 1+0 dimensions, which describes the free relativistic particle in Minkowski space (alternatively, for those who enjoy a bigger challenge: the charged particle in curved space with an electromagnetic field turned on)
start with Minkowski space, no background field:
S=m\int ds=m\int\sqrt{\eta_{\mu\nu}\dot{x}^\mu\dot{x}^\nu} d\tau
canonical momentum:
p_\mu=\frac{\delta L}{\delta\dot{x}^\mu}=\frac{m\dot{x}_\mu}{\sqrt{\d ot{x}^\nu\dot{x}_\nu}}
2) Compute the single constraint of the theory.
from the expression for the canonical momentum, we see that
p^2=m^2
this is a first class constraint, since the equations of motion were not invoked.
3) Do a Dirac quantization by promoting this single constraint to an operator equation. Discuss the resulting quantum equation.
given a state |\psi>, look at its Fourier transform in Minkowski space
|\psi>=\int d^dk e^{ikx}|k>
since this is a generic Fourier transform over d-dimensional Minkowski space, the components of k are independent.
i guess i want to say something about enforcing only the expectation value of the constraint here, instead of the operator version of the constraint.
but i don't see why i have to do that here. lemme see...
Hi lethe -
you can make a Fourier transformation, of course, but probably that's not necessary to make the point.
You have derived the classical constraint. Quantize it. Then impose Dirac quantization of constraints. Alternatively, impose LQG quantization of constraints. What do you get?
By the way:
First-class constraints are those whose Poisson bracket closes on the set of constraints, i.e. is a linear combination of any of the constraints of the theory. Second class constraints are those whose Poisson brackte does not give another constraint.
In other words, the Poisson-bracket of 1st class constraints vanishes "weakly" or "on shell". The Poisson-bracket of second-class constraints does not.
Obviously only 1st class constraints have a chance to give a consistent set of operator constraint equations when quantized. Second class constraints must be eliminated by introducing "Dirac-brackets". This is a deformation of the usual Poisson bracket engineered in just such a way that all constraints become first class with respect to this new bracket. Dirac quantization really consists of replacing Dirac-brackets by commutators and sending the classical constraints to operator equations.
In any case, since for the relativistic particle there is only a single constraint it is trivially 1st class and we can quantize it immediately without worrying about 2nd class subtleties.
See for instance http://www.math.ias.edu/QFT/fall/faddeev4.ps
Originally posted by Urs
You have derived the classical constraint. Quantize it. Then impose Dirac quantization of constraints. Alternatively, impose LQG quantization of constraints. What do you get?
Thanks lethe for starting to solve this exercise. It's been so long since I've looked at QM that I wouldn't have had a chance to get as far as you did.
But picking up where you left off, I think Urs simply wants you to look at the equation
p^2|\psi\rangle = m^2|\psi\rangle
and note that this is the Klein-Gordon equation.
Best regards,
Eric
PS: By the way, why do you call this a "constraint"? Remember, it's been eons :)
PPS: I'd have no idea how to loop quantize this and I HAVE been following this thread :)
Originally posted by eforgy
PPS: I'd have no idea how to loop quantize this and I HAVE been following this thread :)
I can do a little better than this. Looking back at the earlier posts here, I'd write down
e^{p^2-m^2}|\psi\rangle = |\psi\rangle.
Since p^2 commutes with m^2, I guess we could write this as
e^{p^2} |\psi\rangle = e^{m^2} |\psi\rangle,
but I don't know if this buys us anything. Would it make sense to look at
e^{\epsilon(p^2 - m^2)}|\psi\rangle
= |\psi\rangle
and combine terms of the same order in \epsilon? If we did this, I think we'd end up with
(p^2 - m^2)^n |\psi\rangle = 0
for all n> 0. Otherwise, we'd end up with a mess.
Eric
Hi Eric -
thanks for chiming in.
Ok, let me give it away:
As you said, the operator version of the single constraint of the free relativistic particle is obtained by the usual correspondence rule
p^\mu \to \hat p^\mu = -i \hbar \frac{\partial}{\partial x^\mu} and yields nothing but the Klein-Gordon equation
\partial^\mu \partial_\mu \phi = -m^2 \phi
(up to factors of c,\hbar). I am calling this a constraint because, as lethe has derived, it is the operator version of the classical constraint \varphi = p^2 + m^2 = 0 of the action of the free relativistic particle. The classical free relativistic particle cannot move in all of its phase space, but has to stick to the subspace given by this equation, which is incidentally called the mass shell.
Ordinary quantization rules, found by Dirac and others, tell us that the quantum theory is given by demanding that this holds as an operator equation, which is nothing but the Klein-Gordon equation. This equation, together with its fermionic cousin, the Dirac equation, is very well tested experimentally, since it is the very basis on which all of QFT that can be measured in any accelerator is built.
Now let's see how LQG tells us to quantize the free relativistic particle:
There we are told not to consider the constraint \varphi itself but the group which is generated by it by means of Poisson brackets. I.e. we are supposed to look at the group elements
U(\tau) = \exp\left([\phi,\cdot]_\mathrm{PB}\right)
where [\cdot,\cdot]_\mathrm{PB}$ is the Poisson bracket and this guy is supposed to act on classical observables, i.e. functions on phase space.
But because there is just a single constraint this group is nothing but the group of real numbers under addition:
U(\tau_1)\circ U(\tau_2) = U(\tau_1 + \tau_2)
\,.
So LQG-like quantization consists of finding an operator representation $\hat U(\tau)$ of the group of real numbers on some Hilbert space so that the operator product satisfies
\hat U(\tau_1) \hat U(\tau_2) = \hatU(\tau_1 + \tau_2)
\,.
Of course, due to the simpliciy of this example, we could just choose the usual Hilbert space of the free relativistic particle and set
\hat U(\tau) := \exp(i \tau (\hat p^\mu \hat p_\mu + m^2))
\,.
But we could also choose something very different. This is the great ambiguity that I was referring to. For instance, if we followed the tretament by Ashtekar, Fairhurst and Willis of the LQG-like quantization of the 1d nonrelativistic particle, than we'd want to use a nonseparable Hilbert space on which the momentum operator $\hat p$ is not representable. In this case, which is the precise analog of what Thomas Thiemann does in the 'LQG-string' the above choice for \hat U is not an option.
But of course, due to the great simplicity of this toy example, if you pick any (hermitian) operator \hat O on any Hilbert space, you can set
\hat U(\tau) := \exp(\tau i \hat O)
and get a representation of the group of real numbers. You can furthermore choose operators that don't come from exponentiating other operators, of course.
LQG tells us that physical states are those invariant under the action of this group. Clearly, by choosing \hat U appropriately you can find an enormous number of states that satisfy the 'equation of motion'
\hat U(\tau)|\psi\rangle = |\psi\rangle
\,.
Almost anything goes.
For instance I could choose \hat O to be any projector. Than every state that is projected out by \hat O is physical, according to the LQG-like quantization prescription.
Or I could use strange Hilbert spaces, like, say, the 2-dimensional Hilbert space C^2.
The weirdest things are possible. The reason is, that by ignoring the form of the constraint \phi and just looking at the abstract classical group that it generates, we are loosing a lot of crucial physical information.
Thomas Thiemann similarly ignores the form of the Virasoro constraints and just cares about the classical group they generate. The result is no less strange than the above toy example. In fact, the above toy example is a subset of the full string quantization, namely that corresponding to strings that have no wiggly excitations.
I have pointed out the place at which this huge ambiguity appears in the Ashtekar, Fairhurst and Willis papers. There, too, one could choose almost anything else and get weird results. AF&W do choose to use a representation of the group operators that is very close to the usual one, but instead they choose a very strange Hilbert space. Other choices are possible. Nowehere in their paper is it explained why one choice should be preferred over the other.
Hi Eric -
yes, in your second post you demonstrate that exponentiating the KG constraint and demanding invariance yields the same thing as usual. The point is that LQG-like quantization allows you to exponentiate anything else, on any other Hilbert space and call it a quantization of the relativistic particle.
Originally posted by Urs
Hi Eric -
yes, in your second post you demonstrate that exponentiating the KG constraint and demanding invariance yields the same thing as usual. The point is that LQG-like quantization allows you to exponentiate anything else, on any other Hilbert space and call it a quantization of the relativistic particle.
Hi Urs,
Let me state in my words how I am beginning to understand this. I am sure I am just repeating what you've been saying all along (assuming what I say is correct that is).
In the "usual" approach, you take your constraints and "quantize" them directly as operators on some hilbert space
C_I|\psi\rangle = 0.
Now if you take this and exponentiate it, you get something like
e^{C_I} |\psi\rangle = |\psi\rangle,
but this is still just an operator on the same Hilbert space we started with. These exponentiated operators form a group I suppose? An algebra?
Then, are you saying that the loop quantization procedure then looks for any group or algebra of operators that satisfy the same rules on any old Hilbert space and call this a quantization?
I'm sure I don't have that right.
Eric
Hi Eric -
yes, that's the idea - almost! :-)
There is a crucial subtlety:
Let me write C_I for the classical constraints, i.e. these are functions on phase space and the system is restricted to be at points in phase space on which C_I = 0.
The C_I are, already classically, generators of the gauge transformations of the system and physical observables must be gauge invariant and hence Poisson-commute with the constraints. We can write either
[C_I,A]_\mathrm{PB} = 0
or
\exp([r^I C_I,\cdot]_\mathrm{PB})A = A
\,.
Classically this is both well defined and equivalent. The latter form makes us notice that the exponentiated constraints form a group (since they are 1st class)
\exp([r^I C_I,\cdot]_\mathrm{PB})
\exp([s^I C_I,\cdot]_\mathrm{PB})
=
\exp([(r \times s)^I C_I,\cdot]_\mathrm{PB})
\,.
This group is the gauge group of the system with elements
U(r^I) := \exp([r^I C_I,\cdot]_\mathrm{PB})
\,.
In standard Dirac quantization the classical constraints C_I are promoted to operators \hat C_I and the equations of motion become
\langle \psi | \hat C_I |\psi \rangle = 0
\,.
In simple cases this is equivalent to
\hat C_I |\psi \rangle = 0
\,.
In LQG-like quantization people like to choose a Hilbert space on which the C_I are not representable as operators. Then they claim that they can still 'quantize' the equation
\exp([r^I C_I,\cdot]_\mathrm{PB})A = A
by doing the following:
- Find operators
\hat U(r^I)
such that
\hat U(r^I)
\hat U(s^I)
=
\hat U((r \times s)^I)
\,.
- Demand that physical states satisfy
\hat U(r^I)|\psi \rangle = 0
\,.
In previous posts I have mentioned that this presription introduces a large arbitrariness in the choice of the \hat U(r^I).
The example of the KG particle show that the physical information contained only in the abstract structure of the group generated by the classical constraints is way too little to reconstruct the physical system that one started with
selfAdjoint
Feb20-04, 08:17 PM
And do the LQG people ever exhibit such a pretend quantisation of the Klein Gordon particle? I've never seen any but I'm a long way from completely famliar with their literature. It seems to me that you have to ask in each case, as Reheren does in his email, what are the degrees of freedom of the various things you are working with. Proving that the particle is too, umm, sparse to be quantized their way doesn't prove that their way is false in other cases.
And has anyone ever seen a KG particle, quantized or otherwise?
Originally posted by selfAdjoint
And has anyone ever seen a KG particle, quantized or otherwise?
yeah, i think they saw the pion already in the 30s
selfAdjoint -
I haven't seen the KG particle discussed by LQG-like methods. But whart I said about this quantization is exactly what Thomas Thiemann told me to do in general. Also note that, as I have said before, the 'LQG-string' contains the KG particle as a subcase. Nameley the 0-mode of the Virasoro constraints is nothing but the KG equation for the string, where the mass is given by internal oscialltions. So Thiemann dos not get the KG equation for the string.
Please note, as lethe has said, that all bosons that are found in nature are described by the KG equation and all fermions by the Dirac equation (in first quantized form). We could do exactly the same discussion for the Dirac particle and get exactly the same conclusions.
Finally note the example provided by LQG-people themselves: There are LQG-like papers on the 1d nonrelativistic particle as well as on the quantized EM field. In neither case is are the usual results obtained.
the discussion of Thiemann's "Loop-String" paper continues at
SPR (sci.physics.research). Today Thomas Larsson posted the following. Comment? Any explications would be most welcome!
---------Larsson's post---------
... expanded version of a post to the string coffee table,
http://golem.ph.utexas.edu/string/archives/000300.html . It is in response to a post by K-H Rehren, ...
<EDIT: I DECIDED TO MOVE LARSSON'S POST TO A SEPARATE THREAD,
SO FOR BREVITY THE MAIN PART IS SNIPPED OUT HERE>
This is the main algebraic difference between the LQG "lowest-A-number" reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
--------------end of post-----------------
ranyart
Feb22-04, 05:35 AM
Originally posted by marcus
This is the main algebraic difference between the LQG "lowest-A-number"
reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
--------------end of post-----------------
There are a number of problems as I see it with commuters with respect to inter-dimensional transports, commutations.
The starting point is where one makes the initial 'start-point' for the equations in question. If one is within a 3-Dimensional Space, where Geometric Structures are whole, then reduce the structures down through compacted Dimensions of 3-D>> 2-D>> 1D, you are 'breaking' Geometric Transportation Laws, by going from 'whole' to 'bits'.
Now as far as I see it change the Dimension and you automatically change the effects one imposes on Space, in simplistic terms The Laws Of Physics will differ from frame to frame. You cannot be sure of a number of important factors, one major factor is that measure and what we are used to in a 3-D frame becomes something completly different. Distance and Geometric paramiters observed from a 3-D background are not Equivilent emmbedded into a 2-D background, a 'distance' in 3-Dimensions is not the same as a 'distance' in 2-Dimensions.
One can argue that 'quanta' and its effects are different from dimensional frame to frame. This I believe was taken up by Smolin and Magueijo and also Stepanov amongst others. In Geometric framing, a 3-D 'somthing' has a defined boundery, thats what shape/structure in Geometry is! In measuring structure, the devise you use MUST be in contact with what is being measured, for instance a tape-measure has increments that enable you to measure a three dimensional piece of wood form end to end, it is a precise 3-D object itself, as precise as one can be within 3-D.
The tape measure we are used to in our ordered 3-D world suits us well, the dimensionality of the tape measure itself does not come into question, it has length-width-depth in sufficient quantity as to be an ideal tool, but it has a dimensional limit that it constrains to.
If one was to use the piece of wood to measure the tape-measure? how would this effect measure? (try wrapping a piece of wood around a single increment of a tape measure!
The limits of measure from a 3-D frame to a 2-D frame relies upon what is certain ( 3-D structures) and what is not (2-D background). The HUP principle adheres to measure, for HUP to be effective then everything that is being measured must be connected, in our 3-D world this enables us to see matter in small enough quantities to pick-up things and move them around in 3-D Space. A speck of dust between your fingers is a sure thing if one can observe it and move it from A to B.
All well and good if A and B are dimensionally equivalent, but we know that a 3-D frame is not all there is. A good example is energy that commutes from a lower 2-D dimensional frame , up into our 3-D world that is Particle Creation!, 2-d >> 3-D. The converse of this is the reduction of Particles back down into a reduced dimensional field 3-D >> 2-d.
It is by no cincedence that HUP has to have continuous motion as a 'fixer' for Uncertainty, the more you constrain P the less position X exists for you to measure. The whole of HUP is inter dimensional transactions. As you measure something in a 3-D frame, you push part of it away from you and into another dimension (hidden variable). The continuation of measure ends from the perspective of the object in 3-D, it has a limit, this limit is in fact a Dimensional Geometry Bound.
Geometry = observation Measure within Three Dimensions, Quantum Measure is unobservable from this domain. In order to measure something in the Quantum Realm, you need an infinite amount of precision, you will always be taking the initial measure from a 3-D world, thus you cannot quantify a measure going from a 3-D world into the 2-d Quantum world, PRECISION IS FIXED INTO 3-Ds at all Times.
Let me state this again, the Precision for measure favours our 3-D world, because thats where we exist and that is where we always make our measurments from. If you cannot measure something precisely, then it cannot be quantizised, or discontinued. Put another way something that is in continuation cannot be measured in 'bits'.
There are inroads to the re-formulation of Einsteins SR and GR, the Laws of Relativity can be seen as the Laws of Dimensional Space(Geometry) while the SR can be seen as having a Varying Speed of Light,(because there are varying Dimensions of Space-Times) where there are domains of no Observers, less than 3-D space's.
Taken further, when we look out into the Cosmos from our 3-D constant frame, we observe Light as unchanging by default, it will always have a precision as long as we are within a 3-D frame(Galaxy), and will always be seen as 'coming' to us along a 3-D observation, but via a Background field that is 2-D, so it comes along a route from another Galaxy that is 3-D (other galaxy) >>2-D (EM-Vacuum-CMB-) >> (Our Galaxy) 3-D) Where it is Relative to our Dimensional Frame.
When we look inwards from our 3-D world down into the Quantum Realm 2-D, we expect a continuation of the same Laws, but this is not so purely by the fact that we are measuring and observing from 3-D to 2-D, and not as stated earlier with Cosmic Relative observations, there is a missing factor, the Cosmic Fact that there is other Galaxies and other 3-Dimensional worlds ,(us) 3-D >> 2-D >> 1-D (SPACE-SINGULARITY)2-D>> 3-D (them-other galaxies), while observing from inside our Galaxy to the Quantum Realm there is only (us) 3-D >>2-D>>1-D.
We do not observe other Galaxies from wihtin our 3-D world going in the direction of Space Reduction Quantumly, one just gets to a 2-D field of Particle Production, which technically replace's the Galaxies as you reduce down to microscopic Black Hole Singularity.
The problem I see it is String Theorists neglect the fact that their world-lines are continuous, and therby have no discreteness about them, plus the major fact they always expect their initial backgrounds to be Universal! they can not move along 3-dimensional spaces and down into the Quantum Realm, without an infinite amount of dimensional Explinations. There are no more than 3-Dimensional Space's in existence in Our Universe, this is bourne out by the fact that Structure(geometry) does not need excess 'Dimensions' for energy to transport from one frame to another, its 'where' you measure from that is fundemental.
Originally posted by Urs
By the way:
First-class constraints are those whose Poisson bracket closes on the set of constraints, i.e. is a linear combination of any of the constraints of the theory. Second class constraints are those whose Poisson brackte does not give another constraint.
hmm... my knowledge of classical mechanics is a little insufficient here. the definition of "First class constraint" that i mentioned above is that it is a constraint which holds even if the equaiton of motion is not satisfied.
so what do you mean that the Poisson bracket closes on the set of constraints? i calculated this:
[p^2-m^2,f]_\mathrm{P.B.}=-2p^\mu\partial_\mu f
where f is any function on phase space. is this supposed to vanish? or give some linear combination of the constraint p^2-m^2?
Originally posted by Urs
As you said, the operator version of the single constraint of the free relativistic particle is obtained by the usual correspondence rule
p^\mu \to \hat p^\mu = -i \hbar \frac{\partial}{\partial x^\mu} and yields nothing but the Klein-Gordon equation
\partial^\mu \partial_\mu \phi = -m^2 \phi
(up to factors of c,\hbar).
OK, so you just impose the operator version of the constraint.
for some reason i thought there was some issue about imposing only the expectation value of the constraint, as in Gupta-Bleuler. but i guess not.
incidentally, i have always been a little confused about in what sense i can call this quantization "the usual correspondence rule". in nonrelativistic quantum mechanics, we can perform the substitution x\rightarrow\hat{X} and p\rightarrow\hat{P}, subject to the canonical commutation relation [\hat{X},\hat{P}]=i[x,p]_\mathrm{P.B.}=i
it's easy enough to show that this is equivalent to \hat{X}\psi(x)=x\psi(x) and \hat{P}\psi(x)=-i\partial\phi/\partial x when working in the coordinate basis.
the canonical commutation relations give the basis independent quantization procedure. but in the relativistic theory, we don't have these canonical commutation relations, since there is no time operator, and it seems like we are forced to work in the position basis.
can we impose a more basis independent quantization procedure here?
Now let's see how LQG tells us to quantize the free relativistic particle:
There we are told not to consider the constraint \varphi itself but the group which is generated by it by means of Poisson brackets. I.e. we are supposed to look at the group elements
U(\tau) = \exp\left([\phi,\cdot]_\mathrm{PB}\right)
where [\cdot,\cdot]_\mathrm{PB} is the Poisson bracket and this guy is supposed to act on classical observables, i.e. functions on phase space.
hmm... why is this thing a group now? i guess the Poisson algebra of observables is a Lie algebra, so we might expect that exponentiating it would yield a group, but i believe that infinite dimensional Lie algebras do not always exponentiate to Lie groups, only with finite dimensional Lie algebras do we have this guarantee.
the fact that you are exponentiating [\phi,\cdot]_\mathrm{P.B.} instead pf just \phi means that you are using the adjoint representation of this Lie algebra? this way we have a group of operators on the classical algebra of observables?
But we could also choose something very different. This is the great ambiguity that I was referring to. For instance, if we followed the tretament by Ashtekar, Fairhurst and Willis of the LQG-like quantization of the 1d nonrelativistic particle, than we'd want to use a nonseparable Hilbert space on which the momentum operator \hat p is not representable. In this case, which is the precise analog of what Thomas Thiemann does in the 'LQG-string' the above choice for \hat U is not an option.
OK, so they are considering representations of the group generated by the constraint, instead of the constraint itself.
this is the same way the Stone-von Neumann theorem goes, right? it says that there is only one theory that satisfies the exponetiation of the canonical commutation relations (the Weyl relation, i think this is called?). but i have read on s.p.r that there can be inequivalent (and perhaps even physically relevant) representations of the commutation relations themselves
so it seems like the choice to only look at the group version loses you generality?
so this choice is what allows Thiemann to get rid of the anomoly?
for some reason i thought there was some issue about imposing only the expectation value of the constraint, as in Gupta-Bleuler. but i guess not.
This is an additional subtlety but not the issue wrt LQG/standard quantization. Since the single constraint of the KG particle is self-adjoint it should make no difference. So if you want consider the Gupt-Bleueler quantization method. It doesn't alter the point about the LQG-quantization at all.
but in the relativistic theory, we don't have these canonical commutation relations, since there is no time operator,
I am not sure why you think so. We have
[\hat x^\mu , \hat p^\nu] \sim \eta^{\mu\nu}
which translates to
[\partial_\mu , x^\nu] = \delta_\mu^\nu
\,.
There is however a subtlety with defining the Hilbert space of physical states, since these do not live in L^2(M^4), obviously (since they don't decay in the time direction). There are many ways to handle this, the most elegant and advanced being gauige fixing by means of BRST methods. But for our discussion all this does not really matter.
hmm... why is this thing a group now? i guess the Poisson algebra of observables is a Lie algebra, so we might expect that exponentiating it would yield a group, but i believe that infinite dimensional Lie algebras do not always exponentiate to Lie groups, only with finite dimensional Lie algebras do we have this guarantee.
I am not aware of the problems that you are hinting at, do you have a reference? Note that in the case of the Virasoro algebra, which is infinite dimensional of course, the classical group does exist all right. In any case, this would not affect the KG particle, which clearly has a finite constraint algebra.
OK, so they are considering representations of the group generated by the constraint, instead of the constraint itself.
Yes! That's the point. But note that 'representing the constraints themselves' is usually accompanied by much more structure. We are not just looking for any set of operators which has the same algebra as the constraints. We want these operators to be built from the canonical data of the classical system, i.e. canonical coordinates and momenta, by some sort of 'correspondence rule'. All this information about the physical system is lost in the 'represent the group without the rest'-approach.
this is the same way the Stone-von Neumann theorem goes, right? it says that there is only one theory that satisfies the exponetiation of the canonical commutation relations (the Weyl relation, i think this is called?). but i have read on s.p.r that there can be inequivalent (and perhaps even physically relevant) representations of the commutation relations themselves
Stone-von Neumann says that iff the Weyl algebra is represented weakly continuously, then the canonical coordinates and momenta \hat x,\hat p do exist as operators, too, otherwise they do not. And if they exist the Weyl algebra elements are the exponetiations of the Heisenberg algebra elements. See http://citeseer.nj.nec.com/355097.html
so it seems like the choice to only look at the group version loses you generality?
No, it gives you too much generality. Using these strange reps it is possible to built strange theories.
so this choice is what allows Thiemann to get rid of the anomoly?
Yes. There is no technical subtlety hidden in this 'getting rid of the anomaly'. There is the classical conformal group and Thomas Thiemann points out that one can built a Hilbert space on which operators exist which represent this classical group. That's nothing deep. On large enough Hilbert spaces there exist operators which represent almost everything.
so what do you mean that the Poisson bracket closes on the set of constraints?
That the Poisson-bracket of two constraints is again a linear combination of constraints. In your example f is not a constraint, so the bracket you compute need not be a constraint, either. Since there is only a single constraint for the KG particle the only bracket to test is the Poisson-bracket of the single constraint with itself, which vanishes.
But compare the Virasoro generators. The Poisson bracket of two Virasoro generators is again a Virasoro generator, up to a factor. Hence their algebra closes and they are 1st class.
Originally posted by Urs
I am not aware of the problems that you are hinting at, do you have a reference? Note that in the case of the Virasoro algebra, which is infinite dimensional of course, the classical group does exist all right. In any case, this would not affect the KG particle, which clearly has a finite constraint algebra.
i will check for a reference shortly, i am just talking out of my memory from class, so take it with a grain of salt, but i believe that the virasoro algebra does not generate a group. it generates conformal transformations in 2D, right? and something about there being local conformal transformations that do not have an inverse globally, and hence to not form a group.
hey, by the way, when is the voting for sci.physics.strings going to begin?
Hey lethe,
Originally posted by lethe
can we impose a more basis independent quantization procedure here?
I thought maybe you'd be interested in knowing that we can avoid having to choose specific canonical position and conjugate momentum variables entirely by formulating the classical theory in terms of a symplectic form Ωμν, the non-degenerate closed 2-form) on phase space that serves as the fundamental structure needed to define hamiltonian dynamics. One then quantizes by replacing poisson brackets for Ωμν with commutators etc.
lethe -
right, exponentiating a Lie algebra always only gives you the group locally.
Regarding s.p.s.: We are currently waiting for one of the 'Volunteer Votetakers' to volunteer taking votes. We are being told that this should happen in the days/weeks. I am hoping it will happen soon, but currently we cannot do anything to speed up the process.
The Call For Votes for the proposed USENET newsgroup sci.physics.strings, supposed to be concerned with discussion of string theory, has now been published at
http://groups.google.de/groups?selm=1077593588.15146%40isc.org .
Everybody may vote. Detailed instructions for how to vote are given at the above link.
Originally posted by Urs
I am not sure why you think so. We have
[\hat x^\mu , \hat p^\nu] \sim \eta^{\mu\nu}
which translates to
[\partial_\mu , x^\nu] = \delta_\mu^\nu
i infer from this expression that you have an operator on your Hilbert space \hat x^0 that acts on states like multiplication by t. this is the straightforward application of the nonrelativistic quantization to the relativistic particle.
but something that i have learned from reading s.p.r is that there is no such operator. since learning that fact, it has been a big question mark in my mind as to whether there actually exists a theory that could really be called relativistic quantum mechanics of a particle.
i have wanted to understand what is going on with that for a while. since we were doing quantization of the relativistic particle, i thought i would toss in a question about that for you, but i can certainly appreciate that it is a bit off topic for the current discussion
There is however a subtlety with defining the Hilbert space of physical states, since these do not live in L^2(M^4), obviously (since they don't decay in the time direction). There are many ways to handle this, the most elegant and advanced being gauige fixing by means of BRST methods. But for our discussion all this does not really matter.
perhaps this issue about decays is the reason for this thing that i have read about the nonexistence of the time operator, and perhaps this BRST process is the way to resolve it?
Originally posted by Urs
Note that in the case of the Virasoro algebra, which is infinite dimensional of course, the classical group does exist all right. In any case, this would not affect the KG particle, which clearly has a finite constraint algebra.
ahh... this is related to my other confusion. i was going to exponentiate the Poisson algebra of classical observables (which is infinite dimensional), not the subalgebra of constraints (which is closed under the Poisson bracket since the contraints are first-class). i suppose this is the reason i also screwed up and tried to take the Poisson bracket before with some generic classical observable instead of another constraint.
um... i guess i need to learn a bit more about this notion of the Poisson bracket closing on the constraints. and here, i thought i already knew all the classical mechanics i would ever need to know.
Originally posted by Urs
lethe -
right, exponentiating a Lie algebra always only gives you the group locally.
OK, i guess i am not so familiar with spaces which are only locally a group. but this sounds reasonable.
Originally posted by Urs
Yes! That's the point. But note that 'representing the constraints themselves' is usually accompanied by much more structure. We are not just looking for any set of operators which has the same algebra as the constraints. We want these operators to be built from the canonical data of the classical system, i.e. canonical coordinates and momenta, by some sort of 'correspondence rule'. All this information about the physical system is lost in the 'represent the group without the rest'-approach.
Stone-von Neumann says that iff the Weyl algebra is represented weakly continuously, then the canonical coordinates and momenta \hat x,\hat p do exist as operators, too, otherwise they do not. And if they exist the Weyl algebra elements are the exponetiations of the Heisenberg algebra elements. See http://citeseer.nj.nec.com/355097.html
yes, actually, i have read the paper you reference here, and i think that paper is exactly what i had in mind with my above comments.
in Theorem 1 of that paper, they state that any pair of family of operators that satisfies the Weyl form of the CCRs, is unitarily equivalent to the Schrödinger representation (Weyl exponentiated form)
in theorem 2, they give some conditions that imply that any pair of operators P and Q that satisfy the canonical commutation relation (Heisenberg form [q,p]=i, not Weyl form) are equivalent to the Schrödinger representation (Heisenberg/non-exponentiated form).
but then in the text, the authors makes reference to physically relevant systems for which those conditions are not met, and therefore may have inequivalent represntations to the Schrödinger.
so it seems to me like finding a rep of P and Q is more general than finding a representation of their exponentiations. there are many inequivalent representations for the former, and only one representation for the latter.
this is what i took from the paper you referenced, so what am i missing?
No, it gives you too much generality. Using these strange reps it is possible to built strange theories.
see my above complaint. in short, there is only one rep of the exponentiated unitary form, and uncountably many reps of the self-adjoint non-exponentiated form of the CCR. so i conclude that the latter is more general, it allows for more systems. should i not conlcude that there is some ambiguity in choosing such a rep?
Haelfix
Feb26-04, 07:44 PM
'OK, i guess i am not so familiar with spaces which are only locally a group. but this sounds reasonable.'
Exponentiating an algebra gives you a group, but it need not be the unique group, even in the finite dimensional case. There could be global topological features that have to be checked for. In fact, its a big pain, in practise you have to go through many tables to check for consistency.
In the infinite dimensional case, its even worse. For instance, there exists examples, where you can take an infinite dimensional group element arbitrarily close to the identity, but they are not exponentials of the lie algebra.
Lethe -
I bet that when you have heard that there is no time operator this was referring to an operator conjugate to a Hamiltonian. This is something different that we had been discussing.
Take a non-relativistic QM system with Hamiltonian H. Can there be an operator T such that [H,T] \sim 1?
As far as I remember the argument is that there cannot, because two operators satisfying a CCR as will act like multiplication/differentiation with respect to each other's eigenvalues and hence be unbounded from below and from above. But the spectrum of a decent Hamiltonian is supposed to be bounded from below (have a ground state), so it cannot satisfy any CCR.
But this argument doe not apply to systems which do not have an ordinary Hamiltonian. For instance the KG particle that we were discussing is governed by a constraint, not a Hamiltonian evolution. Here time is on par with the spatial dimenions.
If you wish, you can regard the constraint of the KG particle as the Hamiltonian with respect to parameter evolution, where the parameter is an auxiliary variable along the worldline of the particle. This plays formally the role of time in non-relativistic QM and the above argument would show that there is not operator associated with the worldline parameter which has the CCR with the constraint.
For more details on the quantization of the KG particle and its relations to non-relativistic QM you might want to have a look at http://www-stud.uni-essen.de/~sb0264/TimeInQM.html .
Regarding your summary of the Stone-vonNeumann theorem I do not quite agree. I think the message is that there are many reps of the Weyl algebra and that if and only if these reps are weakly continuous does the Heisenberg algebra exist and then the Weyl rep is the exponentiation of the Heisenberg algebra.
LQG like approaches play with the possibility that even if the Heisenberg algebra does not have a rep still a rep of the Weyl algebra exists.
Originally posted by Urs
For more details on the quantization of the KG particle and its relations to non-relativistic QM you might want to have a look at http://www-stud.uni-essen.de/~sb0264/TimeInQM.html
Nice! :)
I just read this and highly recommend it.
Eric
PS: Urs, this constraint business is something I haven't thought much about. It gives me a new interpretation of the subspace of paths for which \partial^2 = 0 . This is like the "physical" space.
Originally posted by lethe
i infer from this expression that you have an operator on your Hilbert space \hat x^0 that acts on states like multiplication by t. this is the straightforward application of the nonrelativistic quantization to the relativistic particle.
but something that i have learned from reading s.p.r is that there is no such operator. since learning that fact, it has been a big question mark in my mind as to whether there actually exists a theory that could really be called relativistic quantum mechanics of a particle.
i have wanted to understand what is going on with that for a while. since we were doing quantization of the relativistic particle, i thought i would toss in a question about that for you, but i can certainly appreciate that it is a bit off topic for the current discussion.
When one moves from a Three Dimensional Relativistic network, down into the Quantum Mechanical 'Hidden-Variable-Network', the equations themselves have to be able to transform one set of data (with observations) back-and-forth. This is where the problem lay, and you clearly have touched upon a deeper meaning in your post quoted above?
What 'single' type of formula's can trancend both QM and GR?..will there be a single formula that can both describe an event in GR and QM without changing the Mathematical formula?..the answer is no, both theories are by their very nature uncompatable, you cannot Unify events in 3+1 dimensions with events in 1+1 dimensional reduced fields.
The transformation has to occur with the Mathematical interpretations, for instance a 3+1 Network has collisions in space, dimensionally 'whole'?.. Particles move around and collide in the 3+1 network that allows them this freedom. In reduced QM Dimensional Networks, this cannot happen as there are no 3 Dimensional 'Whole-Particles' that can exist similtainiously in 3-D and 2-D!
If one uses a formula that traces a Particle wherever it goes (which is what Einstien formulated in GR!), then there comes a point where not only does the formula cease to exist, the Particle itself has been removed from the 3+1 network, as the point in Spacetime(3+1)is reduced to a point in just Space..no TIME = no Observation = Hidden Vaiables = no Collisions = just space/fields = dimensional backgrounds = different (Special) formula's!
Now the bigger picture can be viewed by many 'theorists' into whatever formula's takes their fancy, for instance a simplyfied String Theorist would create 'extra' formulas to exist in 'extra' dimensions, all of which are technically sound, as they have no need of verifacation, and cannot-be verified by observations, the removal of 'SpaceTime' is a natural consequence of the removal of observations. The extra dimensions that some Mathematicians 'create' just simply do not exist.
From a dimensional perspective within GR and SR, one can go from 3+1 (4-D), to 2+1 (3-D) to 1+1 (2-D) TO A SINGULARITY NETWORK that is 0+1...1+0 . Of course the energies that are replacing Particles in 'Identity' terms as one reduces the Particles into Fields, also end up as Creationary Energies when one reaches the simplistic 1+1 AREA NETWORK! around a Blackhole, which happens to reside at the Core of every single Galaxy. Some would offer an explination that Science needs a Dimensional perspective alteration to the Existence of our place within a Spacetime Galaxy (3-D+t), surrounded by Fields of QM Networks that is Electro-Magnetic-Vacuum Space (2+1) with no further need of Mathematical Extensions to Reality.
Originally posted by ranyart
When one moves from a Three Dimensional Relativistic network, down into the Quantum Mechanical 'Hidden-Variable-Network',
hidden variable network??? wtf?
hidden variable network??? wtf?
Maybe, maybe, maybe he is thinking of Smolin's latest attempt at merging Nelson's stochastic QM with quantum gravity
http://xxx.uni-augsburg.de/abs/gr-qc/0311059
where spin networks are indeed used as 'hidden variables' to produce QM dynamics from classical statistics.
Last time Lee Smolin tried the same with BFSS Matrix Theory
http://xxx.uni-augsburg.de/abs/hep-th/0201031 .
I used to consider this interesting,
http://groups.google.de/groups?selm=ahe52s%241a2q%241%40rs04.hrz.uni-essen.de
though I am not so sure anymore.
Anyway, this is what ranyart's avant-garde poetry reminded me of. As with every piece of modern art, you have to search the answer within yourself. ;-)
Originally posted by Urs
As far as I remember the argument is that there cannot, because two operators satisfying a CCR as will act like multiplication/differentiation with respect to each other's eigenvalues and hence be unbounded from below and from above. But the spectrum of a decent Hamiltonian is supposed to be bounded from below (have a ground state), so it cannot satisfy any CCR.
right, this is what i had in mind.
But this argument doe not apply to systems which do not have an ordinary Hamiltonian. For instance the KG particle that we were discussing is governed by a constraint, not a Hamiltonian evolution. Here time is on par with the spatial dimenions.
ok, this is interesting
If you wish, you can regard the constraint of the KG particle as the Hamiltonian with respect to parameter evolution, where the parameter is an auxiliary variable along the worldline of the particle. This plays formally the role of time in non-relativistic QM and the above argument would show that there is not operator associated with the worldline parameter which has the CCR with the constraint.
i think i can see that now. thank you, that was very helpful for me.
Regarding your summary of the Stone-vonNeumann theorem I do not quite agree. I think the message is that there are many reps of the Weyl algebra and that if and only if these reps are weakly continuous does the Heisenberg algebra exist and then the Weyl rep is the exponentiation of the Heisenberg algebra.
LQG like approaches play with the possibility that even if the Heisenberg algebra does not have a rep still a rep of the Weyl algebra exists.
before i think about your point here, I am confused as to what you are refering to when you say "Weyl algebra". i am thinking it should be the set of operators you get after exponentiation, but do these things form an algebra? i expect them to form a group, but i wouldn't expect the sum of two of these guys to be another one of these guys.
in short, the operators in the Weyl relation are the Lie group corresponding to the Lie algebra spanned by the operators in the canonical commutation relations (the Heisenberg algebra)
Hi lethe -
in short, the operators in the Weyl relation are the Lie group corresponding to the Lie algebra spanned by the operators in the canonical commutation relations (the Heisenberg algebra)
Wait, we have to get out nomenclature in sync.
What I am calling a Weyl algebra are operators U(a),V(a) which satisfy
U(a)V(b) = \exp(i 2\pi ab/\hbar)V(b)U(a). These
need not come from exponentiating elements of a Heisenberg algebra. But the Stone-vonNeumann theorem tells us that iff U and V are weakly-continuous, then they do come from an exponentiated Heisenberg algebra. Otherwise they don't. If they are weakly continuous, then you are right that U and V give the Lie group of the Heisenberg algebra, namley the Heisenberg group.
LQG is based on throwing away the Heisenberg algebra and concentrating on reps of the Weyl algebra U and V which are not weakly continuous.
Originally posted by Urs
Maybe, maybe, maybe he is thinking of Smolin's latest attempt at merging Nelson's stochastic QM with quantum gravity
http://xxx.uni-augsburg.de/abs/gr-qc/0311059
where spin networks are indeed used as 'hidden variables' to produce QM dynamics from classical statistics.
Last time Lee Smolin tried the same with BFSS Matrix Theory
http://xxx.uni-augsburg.de/abs/hep-th/0201031 .
I used to consider this interesting,
http://groups.google.de/groups?selm=ahe52s%241a2q%241%40rs04.hrz.uni-essen.de
though I am not so sure anymore.
Anyway, this is what ranyart's avant-garde poetry reminded me of. As with every piece of modern art, you have to search the answer within yourself. ;-)
Ah!..with a little detective work I see what you mean (which is not a literal reference to me seeing into your mind!)
QM from QG
Right, that's what the Smolin paper is called. Unfortunately the statement implied by the title is either a tautology or circular.
No, if the idea is that QG is a new theory above QM, so that any classical field or particle defined inside the QG theory will magically be a quantum field or particle. Still, the paper does not go as fas as his title, because Nelson stochasticity is imposed, not deduced.
This paper has some relevence to this post.
http://arxiv.org/PS_cache/hep-th/pdf/0403/0403108.pdf
arivero
Mar10-04, 07:50 AM
Originally posted by ranyart
This paper has some relevence to this post.
http://arxiv.org/PS_cache/hep-th/pdf/0403/0403108.pdf
But quantisation of the strings is a very different matter, isn't it? To begin with, the string is already by itself a many-particle entity, so first quantisation should be enough.
pelastration
Mar10-04, 08:16 AM
Originally posted by arivero
... the string is already by itself a many-particle entity
many-particle ... can you explain more?
selfAdjoint
Mar10-04, 10:01 AM
Thomas Larson has today posted a possible way forward for LQG of sci.physics research, here (http://groups.google.com/groups?dq=&hl=en&lr=&ie=UTF-8&group=sci.physics.research&selm=24a23f36.0403090210.41ccbff0%40posting.google .com) . Recall that Urs had said LQG required a factor in the commutator that he called V to obtain the Virasoro algebra, and noted that LQG theorists set V = 1.
Now Larson points us to a paper on the math-ph arxiv (http://arxiv.org/abs/math-ph?0002016) which discusses a great many (all?) the possible candidates for V.
More about Thiemann's LQG-string
(the project to realize string theory within the context of LQG)
Robert C. Helling, Giuseppe Policastro
String quantization: Fock vs. LQG Representations
19 pages
http://arxiv.org/abs/hep-th/0409182
---abstract---
We set up a unified framework to compare the quantization of the bosonic string in two approaches: One proposed by Thiemann, based on methods of loop quantum gravity, and the other using the usual Fock space quantization. Both yield a diffeomorphism invariant quantum theory. We discuss why there is no central charge in Thiemann's approach but a discontinuity characteristic for the loop approach to diffeomorphism invariant theories. Then we show the (un)physical consequences of this discontinuity in the example of the harmonic oscillators such as an unbounded energy spectrum. On the other hand, in the continuous Fock representation, the unitary operators for the diffeomorphisms have to be constructed using the method of Gupta and Bleuler representing the diffeomorphism group up to a phase given by the usual central charge.
---end quote---
Robert Helling's papers go back to 1998, he has co-authored with
Hermann Nicolai, has been much of the time at Albert Einstein Institute,
Potsdam MPI, has specialized in M-theory (from the looks of it)
It looks like Hermann Nicolai, a director at AEI Potsdam, who organized last year's String meet Loop conference (with Abhay Ashtekar), has perhaps encouraged Thiemann to try this merger of theories in the first place
this was what the StringMeetLoop conference last October was supposedly to lay the groundwork for. Nicolai does String/M and particle theory and he co-organized it with Ashtekar who does Loop and is a relativist.
At the conference they wanted to get the HEP people---the particle physicists---talking to the relativists---the General Relativity people. Both being concerned with quantizing gravity in their respective fashions.
But after, when Thiemann took the first step the reception was not so hopeful or encouraging, as I thought. More like the bluejays in the front yard when the neighbor cat comes to visit. All kinds of reasons offered why it could not possibly be right.
This Robert Helling article has a different tone of voice
here is the original Thiemann article, in case you have not already seen it:
The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
Thomas Thiemann
46 pages
http://arxiv.org/hep-th/0401172
selfAdjoint
Sep19-04, 10:08 PM
"...at least in the case of the quantum oscillator the polymer state is unphysical." That seems to be the death-knell fot the Thiemann approach. I wasn't aware of the fierce properties of the polymer state (no momenta), which as the authors say, is much used in LQG derivations. Back to the drawing board.
---quote from Robert Helling SPS post---
Thanks for noting our paper. Unfortunately, I am about to leave
Cambridge (my next postdoc is at IU Bremen, back in Germany) and all
my papers and notes are stored away in boxes and unaccesible to me at
the moment. So I cannot answer Urs' qustions about signs and
(anti)-commutativity. He might well be right and we screwed those up
but those would be just typos and wouldn't change anything substantial
in the conclusions. Furthermore, I don't have my laptop's network
connection currently running, thus I have to use google groups rather
than my regular news reader.
> Their description of the harmonic oscillator looks particularly strange
> because if we did not agree what is the physics of the quantum harmonic
> oscillator, we could probably agree about nothing in the world.
Mybe you missed that point but our philosophy was to say "this is what
you get when you apply LQG methods to the harmonic oscillator". Be
careful with them im general because the _physical_ consequences (esp.
the spectrum) are not what you meassure in this easy example. It was
important to us not just to say that functions that jump have no place
in physics because you could never observe them. That would be too
easy (and in fact plain wrong: If you describe a D-brane (whose
physical existence I understand Lubos does not doubt) by a skyscraper
sheef then this is done exactly by a function (of the transverse
coordinates) that is zero everywhere except at one point where it has
a finite value).
[Moderator's note:....LM]
So the point of our discussion of the harmonic oscillator is that there
are measurable consequences of doingit 'the wrong way'.
[Moderator's note:... LM]
> I am sure that many of us tried to deal with divergent sums and divergent
> integrals, and various other singular objects in various ways that can be
> proved "wrong" on physical grounds.
Actually, this is a major reason for why this formalism is more
involved than the usual one: In the algebraic language (and this is
just mathemtically more careful language, no physical difference to
the usual approach) great care is taken to avoid divergent (and
similar) sums etc so no ambiguities (or ways to do it wrong) appear
from that. This is why careful people deal with the Weyl operators
e^{ix} and e^{ip} instead of the usual x and p: The Weyl operators are
bounded and thus problems with domains of definition etc do not arise.
For example, in the position representation p is the deriviative. But
not all functions in L^2(R) are differentiable. Only a dense subset is. But all are translateble. Thus one saves some complications (if
ones intend is to be careful) if one uses the better defined Weyl
operators instead. I am not saying that it cannot be done with x and
p, it's just you either close your eyes to mathematical subtleties
(which is what we physicists do most of the time and it works fine
most of the time) or you have to deal with limits and that stuff.
[Moderator's note: ... LM]
> All Hilbert spaces obtained from these wrong assumptions are
> non-separable, unphysical, and the only way how the non-separability can
> be cured is if the resulting theory is completely topological and all
> these values of "x" are eventually unphysical, perhaps except for their
> ordering.
The separability is not the issue.
[Moderator's note:... LM]
In fact, there are (accepted) physical systems with a non-separable
Hilbert space: One (as we remark in a footnote) are Bloch electrons.
That is electrons in a periodic potential.
[Moderator's note:...LM]
Then you know that the wave function is periodic as well. Ahem no, not
quite, only the physics is periodic. So the wave function is periodic up
to a phase. And by doing the intergral over the whole infinite crystal,
you find that two wave functions with different phases are orthogonal. So
for each point in the interval [0,2pi) of phases there is an orthogonal
sector in the Hilbert space. Thus the total Hilbert space is kind of the
L^2 of the unit cell to times the number of points in that interval, clearly a non-separable space.
[Moderator's note:... LM]
You could say that this happens only in the infinite crystal size
limit.
[Moderator's note:... LM]
But this idealization people usually are happy to make. Otherwise (with an
IR cut-off) there would for example be no phase transitions. But that is a
different matter.
> The states in this model represent unphysical mixtures of a
> hugely infinite number of superselection sectors - it is another
> description of Helling et al. comments about "discontinuity" of
> Thiemann's representation, I think. Each of these sectors is made of a
> single state. In physics, it is legitimate to study each single
> superselection sector separately - and if these superselection sectors are
> made of a single state, the theory is physically vacuous.
At least in the mathematical sense (and that is supposed to coincide
with the physical sense), a super-selection sector is a representation
of the quantum algebra. To states are in different sectors if they
are in inequivalent representations.
> > Everybody knows that first quantization is a mystery.
>
> "Why the world is quantum?" may be a mystery and the most counterintuitive
> insight about the world, but the mathematical operation behind the first
> quantization does not seem mysterious in any way, and it also does not
> seem ambiguous. Moreover, I don't know why you chose the "first
> quantization" because even it is a mystery, it is a smaller mystery than
> the "second quantization". ;-)
Was this just a joke? If not, here is why people say this (and usualy
this continues with "second quantization is a functor"): Of course if
your classical system has R^n as configuration space and its cotangent
bundle as the symplectic space then every child knows how to quntize:
Take L^2(R^n) as your Hilbert space and replace all x's by
multiplication operators and all p's by derivatives. Oh, and when
there's an ordering ambiguity, follow one or the other prescription
(but do that consistently).
However, what do you do, if I just give you some symplectic space and
don't tell you which are the simple prefered position and momentum
coordinates (and that's what x and p are, just coordinates). And as
the real world does not come with coordinates written on everything
one should have some recepy how to deal with this more general
situation. And then check that this reduces to the usual story (or an
equivalent one) in the simple situation. And this is what we have done
in the paper.
It is known, that there is no unique way to do this map from a
symplectic space to a Hilbert space with operators in general. There
are further choices involved.
[Moderator's note:... LM]
> I did not quite like their comments describing the algebras with different
> central charges as different "representations" (with an exclamation mark).
When we say algebra, we mean the C*-algebra of the observables. And
those are indeed the same (and only the represenations differ).
[Moderator's note:... LM]
These algebras are the algebras of the X's (or the W(f) after some
massaging). Then this Weyl algebra has representations. And on those
representations there is a symmetry (Lie-)algebra acting by unitary
operators. And this symmetry algebra is some Virasoro alegbra in both
cases. But these symmetry algebras have different central charges in
the two representations of the Weyl algebra. So: There are two kinds
of algebras, don't confuse them.
Furthermore, even if we didn't talk about it, in the Virasoro algebra
the central charge is just an abstract element usally called c. It
commutes with everything, so in an irreducible representation it is
represented by a number. And again, this number depends on the
representation. This number, together with h, the eigenvalue of L_0 in that representation, label a highest weight representation of the
algebra.
[Moderator's note:... LM]
> The only way how can one understand this sentence is that they claim that
> the Virasoro algebras with different central charges are isomorphic to
> each other.
Nobody claimed that. As I just said: In the algbra, c is an abstract
element, it becomes a number only in a representation. And nobody
claims that representations with different c are equivalent.
[Moderator's note:... LM]
> There is a common theme in Thiemann's papers which, I'm afraid, may
> unfortunately be shared by Helling and Policastro - which is that they
> often look at the "classical limit" of an algebra, and treat all the
> modifications implied by quantum mechanics as unimportant - and perhaps
> annoying? - details whose relevance for any of their conclusions goes to
> zero.
Could you be more specific with this claim?
[Moderator's note:... LM]
> By the way, this purely quantum viewpoint will be even more important if
> we want to get more insight into the (2,0) theory or even M-theory at the
> generic point of the moduli space - because these theories (at least in
> some backgrounds) clearly indicate that they cannot be fully obtained from
> a classical theory by quantization - and they almost certainly cannot be
> obtained from a *unique* classical theory.
Quantization is a game that always involves a classical system.
However nobody claimed that every qunatum system arises from the
quantization of a classical theory.
[Moderator's note:... LM]
-----end quote---
there was so much mod comment inserted into Helling's text that it was hard to see the overall intent of his post, I have elided the mod comment,
as in the preceding, to get a sense of the original.
---quote from Helling's SPS post---
> There are other examples in which Thiemann et al. try to make this sort of
> "quantization without quantization". They want the commutators to be
> always equal to the Poisson brackets;
I hope you don't include us in "et al". We impose the Poisson goes to
commutator rule only for linear combinations of what would be x and p,
not for higher powers. And I doubt Thomas would do commit that crime
either.
[Moderator's note:... LM]
> One may be trying to obtain a completely different framework of
> "quantization" - but there are several but's.
We tried hard to spell out the general framework of the quantization
procedure used in the two approaches. We say, that you can include the
polymer quantization if you do not impose the at first rather
technical condition of weak continuity. But then this has huge
observable consequences. So don't confuse framework and consequences.
[Moderator's note:... LM]
Could you spell out the rules for your framework that clearly rules
out the LQG one? It should be a machine that turns a symplectic space
with its observables into a Hilbertspace with its operators.
[Moderator's note:... LM]
> First of all, this procedure
> is not really quantization because it tries to preserve those properties
> of the classical theory that *cannot* hold in what is normally called a
> "quantum theory" - such as the exact equality between the commutators and
> the Poisson brackets.
Nobody imposes that. We only ask for a unitary representation of the
diffeomorphism symmetry. And those might obey the group law of the
diffeo group or not (because of an anomaly).
[Moderator's note:...LM]
> Second of all, it is not physics because no one has
> certainly seen a Thiemannian harmonic oscillator
Right. That we meant by "(un)physical" in the abstract.
- and no one ever will,
> simply because non-separable Hilbert spaces cannot be "seen".
See above.
> I find it mildly entertaining that the normal procedures of quantization -
> including quantization of the harmonic oscillator - are themselves
> pictured as an alternative approach.
Where?
[Moderator's note:... LM]
We describe both quantizations in a single framework. There is
one choice to be made. And that has physical consequences. In the
mechanics example, those consequnces are unphysical, so the choice was
wrong. Everybody is free to deduce something about the choice in the
string case.
> Well, of course that we do not need
> nonsensical non-separable spaces to describe the harmonic oscillator. Not
> only that: non-separable spaces do *not* describe the harmonic oscillator
> and they never did. Moreover, the standard procedure has been known since
> the mid 1920s, and it is the only one that can give physical predictions
> that reduce to the classical oscillator in the appropriate limit. It's
> great to rediscover this cool method of quantization in 2004, but it
> should not be viewed as something new.
We didn't say there is anything wrong with the standard harmonic
oscillator. Rather we used it as a test bed for the quantization
procedure. This was to counter arguments along the lines of 'nobody
has yet seen a string in nature".
> That's nice to hear because Robert Helling was the person who patiently
> required (in "Re: Background Independence", 2004-09-14 04:30:48 PST) that
>
> > RH: 4) The ground state of (quantum) GR should be diffeomorphism invariant
> > as otherwise diffeomorphisms would be spontaneously broken.
Let's face it: In that thread you didn't get it that I was playing the
devil's advocate.
[Moderator's note: Well, sometimes I am confused who is your devil and who
is your God, or whoever is your devil's "alternative". ;-)
> > it is discussed that the singular GNS state can be interpreted as a thermal
> > state of infinite temperature!
One should be carefule as this is world sheet temperature and not
target space.
> > I didn't know this before and like that insight, because it points at a way
> > to understand a larger framework in which various "different" quantizations
> > (of the string for instance) appear as different aspects of the same thing.
>
> I am not getting the purpose of these attempts. Is the goal to be nice and
> to prove that no one can ever be completely silly?
If you like to express it that way...
Sorry, right now, I do not have more time to reply to the more polemic
parts of your post.
Robert
---end quote---
"...at least in the case of the quantum oscillator the polymer state is unphysical." That seems to be the death-knell fot the Thiemann approach. I wasn't aware of the fierce properties of the polymer state (no momenta), which as the authors say, is much used in LQG derivations. Back to the drawing board.
Your take on this was shared by Thomas Larsson today
http://physicsforums.com/showthread.php?t=44495
He says the way (according to Helling/Policastro) that LQG handles
the harmonic oscillator is fatal for LQG.
In effect, he tells the Loop Gravitists to go "back to the drawing board".
the first thing I notice is I am pleased with our reflexes
I posted notice of H/P on 19 September
then it appeared (I think the next day) on SPS
and now discussion has started (24 September) on SPR
You were sadly shaking your head 5 days before Thomas Larsson.
By now I am accustomed to surprises so I am waiting to see how this turns out and cannot really give a reaction.
I recall that Rovelli (and Daniele Colosi) had a paper last year about the Harmonic Oscillator. Their paper was called:
A simple background-independent hamiltonian quantum model
"...Our main tool is the kernel of the projector on the solutions of Wheeler-de Witt equation, which we analyze in detail..."
http://arxiv.org/abs/gr-qc/0306059
this is not to say that their paper has any bearing on the fatal harmonic oscillator disease discovered by Helling/Policastro (wouldnt seem to but I suppose it might)
selfAdjoint
Sep24-04, 04:37 PM
Well, the Colosi-Rovelli paper doesn't contain the word polymer at least!
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