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."
From Thiemann's conclusions paragraph "...Let us conclude by stressing once more that the claim of this paper is certainly not to have found a full solution of string theory. Rather, we wanted to point out two things: First of all, that canonical and algebraic methods can be fruitfully combined in order to analyze the string. Secondly, that the specific Fock representation that one always uses in string theory is by far not the end of the story: The invariant representation theory of the quantum string, as we have defined it here, is presumably very rich and we encourage string theorists to study the string from the algebraic perspective and to systematically analyze all its representations. This might lead to a natural resolution of major current puzzles in string theory, such as the cosmological constant puzzle [38] (120 orders of magnitude too large), the tachyon condensation puzzle [39] (unstable bosonic string vacua), the vacuum degeneracy puzzle [40] (huge moduli space of vacua upon compactification), the phenomeology puzzle [41] (so far the standard model has not been found among all possible string vacua of the five superstring theories currently defined, even when including D – branes) and finally the puzzle of proving perturbative finiteness beyond two loops [42]. See the beautiful review [43] for a status report on these issues. Namely, it might be that there are much simpler representations of the string, especially in lower dimensions and possibly without supersymmetry, which avoid or simplify all or some these problems. While this would be attractive, the existence of new, phenomenologically sensible representations would demonstrate that D = 10, 11, 26 dimensions, supersymmetry and the matter content of the world are tied to a specific representation of string theory and hence would not be a prediction in this sense...." ---------end of exerpt-------- He seems to be getting rid of the rolled-up dimensions. taking string back down to "phenomenologically sensible" FOUR spacetime dimensions. Ye gods. He says that all those extra dimensions and supersymmetry are not, after all, needed for string theory or necessary predictions of the theory, because he sees a chance to make the theory work without all that extra baggage, in a background independent context using Quantum Gravity tools. BTW in his references he gives 2004 as the publication date for Rovelli's book. He says "Quantum Gravity, Cambridge University Press 2004." I suspected it would be out this year since Rovelli has sent in the MS and how long can it take? Lots happening.
Hi everybody - Marcus wrote: Please let us know who "we" is. Thanks! :-) 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. 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. 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? The tachyon seems to disappear - but with it disappear many other features of the ordinary string quantization, such as the rest of the mass spectrum. I would like to point out that the tachyon is not an inconsistency of the bosonic string, but merely an indication of an instability of its background. For instance the tachyon of the open bosonic string is very well understood as to be due to the instability of the space-filling uncharged D25 brane. The depth of the tachyon potential is precisely the energy density of this unstable brane and while the tachyon rolls down its potential well the brane decays. This way open bosonic string theory decays to closed bosonic string theory and the fact that this process can be understood in terms of single strings, branes, and string field theory shows that it gives a consistent physical picture. This is analogous to all the tachyons that appear when superstrings stretch between brane-antibrane pairs. They are not a problem in the theory but a physical feature: The brane and antibrane pairs annihilate as the tachyons roll down their potential. In fact, there is an interesting flavor of string cosmology, where the tachyon serves as the inflaton field and the high initial value of the inflaton is nicely explained by the collision of a brane-antibrane pair. (The initially parabolic potential of the tachyon evolves into a Mexican-Hat type potential as the branes approach.) My point is that the disappearance of the tachyon is not a advatage per se. Even the closed bosonic string (in the usual quantization) might not be inconistent at all, but might decay into a conistent 2d bosonic string or 10d superstring. See this discussion at the String Coffee Table for more details. Now let me ask some random questions: Can you recover the usual Fock representation within the GNS construction framework? (I guess not, since no anomaly will ever show up in this framework, right?) If yes, is there anything (apart from the obvious differences) that distinguishes the Fock representation from other representations obtainable by doing the GNS construction ? What happens when Thiemann's approach is applied to the Polyakov action? A priori this looks like the case more closely related to LQG, since it is the Polyakov action which describes 1+1 dimensional gravity on the worldsheet. There are several independent ways to arrive at the usual quantization of the F-string. There is the old covariant approach, the light-cone quantization, the BRST quantization, the path-integral quantization. These quantization schemes have superficially many differences, and yet they all give the same result - which disagrees with Thiemann's result. How could that be understood? What is going on here? This is maybe the most interesting question. H. Nicolai initiated the recent research into LQG quantization of 1+1 dim gravity coupled to scalar matter (at the "Strings meet Loops" symposium in Potsdam last year) by asking if LQG methods can reproduce the very well established results concerning the quantization of the Polyakov action. The idea is that 1+1 dimensions may be an accesible laboratory for understanding how the LQG approach is different from other quantization approaches. And finally: When Thiemann's approach is generalized to the fermionic string, might it be of any help to know that the constraints of the superstring are deformed exterior (co)derivates on the form bundle over the configuration space of the string and that all massless bosonic backgrounds manifest themselves as deformations of these exterior operators as described in this paper? :-) Best, Urs
"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!
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.
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! :-) Yup, that's me. :-) 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.
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
I found the Nicolai slide-talk comparing you-know-whats http://www.aei-potsdam.mpg.de/events/StringmLoops/Nicolai.pdf It is a seven-page slide-talk comparing Loop Gravity and String. This was his introductory talk, opening the symposium which you attended. It 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.
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.
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 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.
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
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: 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!
that sounds like a safe bet! 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.