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My reply to Lubos can be found at the this entry of the.String Coffee Table.
Amazing bid by Thiemann to absorb string theory into LQG
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Singularities in quantum field theory
Dear Urs,
concerning your (more precisely, Thiemann's) comments that you can get rid of all ordering constants by exponentiating something, I hope that you don’t really believe it because this would count as a rudimentary misunderstanding of the singularities (and anomalies) in quantum field theory. The exponentials of something always store the same information as “something”, and if one of them has some ordering constant contribution, you see it in the other as well.
For example, X(z) X(0) have logarithmic OPEs - it behaves as ln(z). This implies that exp(i.K.X(z)) has a power law OPE with exp(-i.K.X(z)). It’s totally nonsensical at quantum level to imagine that exp(-i.K.X(z)) is an inverse operator to exp(i.K.X(z)). Do you understand why? This is a very important point. The singularities are not invented by people who want to make their lives more difficult. They can be derived from the rules of quantum field theory. In quantum field theory, everything (fields) fluctuate, and the total fluctuation summed over all the modes simply implies that the expectation value of X(z)X(z') diverges as z approaches z'. All the operators in QFT are potentially infinite, and one must be very careful in regulating their products. It is not possible to deal with the operators as with ordinary numbers, and most of them can't be inverted.
While for the Virasoro group without the central charge you would be able to write the explicit “exponentiated” elements of the reparameterization group and - because they have a clear geometric interpretation - you could invert them without anomalies, it is simply not true for the Virasoro operators generating the reparameterization of X’s. Because of the term c/z^4 in the OPE of two stress energy tensors, you must know very well that exp(-V) can’t be treated as the inverse of exp(+V). You can only imagine that exp(V) is an honest element of a group if the OPEs of V with itself - and all other “V“‘s that you want to use - only have the 1/z term, corresponding to the commutator. This is the whole point of "anomalies" that quantum mechanics simply prevents us from imagining that the classical, naive symmetries survive in quantum mechanics. A quantum mechanical theory that would respect the naive classical symmetries simply does not exist, if one can calculate the anomalies. The conformal anomaly prevents us from defining the Virasoro "group" where objects can be inverted in a naive way. Recall that
O1(z) O2(0) ~ [O1,O2] (z) / z
the coefficient of 1/z is schematically the commutator of the two operators. If you integrate a stress energy tensor etc., it is also OK to have the 1/z^2 term in the OPEs of the stress energy tensor because it reflects the worldsheet dimension of the stress energy tensor and tells you how should you integrate it to get scalars etc.
But the OPE of the stress energy tensor (of the X^mu CFT) with itself contains an extra 1/z^4 term. This is just a fact that you can calculate in many ways, and this simply means that exp(V) where V is a Virasoro generator, or some integrated combination of the stress energy tensor, does not behave as an honest element of some group, and exp(-V) is not in any naive sense inverse to exp(V) because these two *operators* have singularities.
Note that his naive operation, involving the (wrong) application of the formula e.g.
exp(C.D.C^{-1}) = C exp(D) C^{-1}
which is OK for matrices, is incorrect in our “usual” representation of CFT, because of singularities between C and C itself. Thiemann's sloppy methods would certainly allow you to derive a lot of incorrect "results" in ordinary CFT, too. You can’t imagine that C^{-1} is inverse to C - there are just no meaningful operators on the Hilbert space that would look like C=exp(V) and were inverse to one another. Because C^{-1}.C is not really one, you can’t derive the formula you derived either, unless c=0 where the classical intuition is OK. Note that it even requires you, for C=exp(V), to consider exp(exp(V)…). These are heavily singular operators, and all these confusions simply come from his/their wrong intuition that you can work with the operators in CFT as with ordinary classical numbers. They don’t understand where the normal ordering terms come from, they don’t understand singularities of operators in quantum field theories, they don’t understand the difference between classical and quantum field theory - and perhaps between classical and quantum physics in general.
Even if Thiemann did a better job and counted the quantum contributions properly, the framework of his paper would be terribly far from a construction of a meaningful theory. In a meaningful theory, he would have to consider torus diagrams, for example, and so on. The modular invariance would be definitely broken for his "new representations" if he had no CFT backing him. The modular invariance is derived from the very special functions associated with CFT, Poisson resummation, modular functions, and so on. Without doing the torus diagram, he does not really need to discover the critical dimension (D=26 was first found from unitarity at the one-loop torus level), but if he fails to understand why the critical dimension is necessary, it is very far from having evidence that it does not have to be necessary. All these things are very sensitive and they must be done very exactly, if the theory is supposed to be consistent. His treatment is a naive application of classical reasoning, involving brutal eliminating of the terms that are absolutely essential for consistency of the theory.
It’s just sad. Ignorance about the basics of quantum field theory should not be sold as a "new, revolutionary proposal in physics", and every student in theoretical physics should be able to identify the errors in papers similar to Thiemann's paper.
All the best
Lubos
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
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Originally posted by Urs
My reply to Lubos can be found at the this entry of the.String Coffee Table.
I'm having browser trouble getting the "string coffee table" and
the font is too small on my screen as well, so unless you object I will copy your reply here so I can read it:
-----quote from Urs------
Re: Thiemann’s quantization of the Nambu-Goto action
Hi Luboš,
thanks for your answer!
I see your general point, but would like to look at some of the issues you raised in more detail.
You say that the Nambu-Goto action is ‘obsolete’. But of course the NG action is classically equivalent to the Polyakov action and I think that in the critical number of dimensions the equivalence extends to the quantum theory. Furthermore, the Nambu-Goto action for the string is essentially the Dirac-Born-Infeld action (up to the worldsheet gauge field) of the D-string.
As far as I can see the constraints that Thiemann arrives at in equation (2.4) of his paper follow from standard canonical reasoning. One finds that the canonical momenta π μ of the Nambu-Goto action as well as of the DBI action classically satisfy two identities which can be identified as constraints. At the classical level these constraints are precisely the (classical) Virasoro constraints that one also obtains by varying the worldsheet metric in the Polyakov action. Since the two actions are classically equivalent this is no surprise.
My point is that there should be a priori nothing wrong with looking at the Nambu-Goto action when studying the string. Indeed this is frequently done for instance when F-strings and D-strings are considered at the same time, as for instance in
Y. Igarashi, K. Itoh, K. Kamimura, R. Kuriki, Canonical equivalence between super D-string and type IIB superstring.
In equations (2.3) and (2.4) of this paper the authors in particular give the same two bosonic constraints of the Nambu-Goto action that Thiemann arrives at. Their action also involves superfields and the worldsheet gauge field, but this does not affect the general result that the Virasoro constraints follow from a canonical analysis of the Nambu-Goto action. I have spelled out the derivation (for the bosonic DBI action) in a recent entry. (By setting the worldsheet gauge field and the C fields to zero this derivation directly restricts to that for the ordinary Nambu-Goto action).
My point is that it is maybe not fair to say that Thiemann artificially or freely chooses the constraints - at least not at the classical level. The constraints that he uses are, classically, the Virasoro constraints of the closed bosonic string.
My suspicion is rather that Thiemann devitates from standard reasoning when he defines what he wants to understand under quantizing the Virasoro constraints. Would you agree with this?
Let’s ignore the way on which we arrived at the classical Virasoro constraints (by starting from one of various classically equivalent actions) and concentrate on the question what it means to quantize them.
The standard procedure is to make Gupta-Bleuler quantization and use either creation/annihilation operator normal ordering or CFT techniques to make sense of the quantum representation of the classical Virasoro generators. This leads in the usual way to the anomaly, the shift a in (L_0 - a) and so on.
Thiemann claims (based on a large literature on quantization of constrained systems that is also the basis for loop quantum gravity) that there is an at least superficially different technique that can also be addressed as quantization of the Virasoro constraints. In the simple case at hand this is imposing the constraint the way mentioned right below equation (5.4), which essentially says that <ψ|exp( constraints )|ψ ′>=<ψ|ψ ′>, where the Hilbert space and the representation of the operators is not necessarily the usual Fock representation.
This is not equivalent to and not even implied by saying that <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 don't come out but at least we get some idea of Urs reply
i've tried to edit back in some of the symbols
there is another reply further down
all seems pretty interesting
Last edited:
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So much easier to read in the larger PF font----instead of the coffee table small font on green background!
------quote of Urs next reply-----
Re: Thiemann’s quantization of the Nambu-Goto action
Hi again, Luboš!
Yes, I understand everything that you say here. I know that : exp(-V): is not the inverse to : exp(V): in CFT and I do understand where the 1 /z 4 terms come from. When you go back to my original entry you’ll see that I address precisely this phenomenon by mentioning that things like : exp(k⋅X): have conformal dimension depending on k in CFT, which is another aspect of this phenomenon.
But, yes, I was taking for granted that Thiemann is using a rep of his operators that allows him to ignore all normal ordering issues and work with them as with matrices and hence not as in CFT. He is referring to lot’s of mathematical theorems, using the GNS construction etc. (that I obviously haven’t checked myself and I am trusting that he applies them correctly) and even though he does not say so explicitly I deduced from his paper, in particular from the the third paragraph on p. 20, that he does use exp (C⋅D⋅C - 1)=Cexp(D)C - 1. I do understand that this does not make sense in CFT (or even any other quantum field theory in the usual sense) but I also believe that a large number of mathematically versed people in the LQG camp do think that this can be given good meaning by using all these mathematical constructions that Thiemann alludes to. Unfortunately I am not an expert on this stuff.
I think the key ingredient is the GNS construction, which tells you that a unital *-algebra can be represented faithfully i.e. without normal ordering issues just like matrices on some Hilbert space. That’s the content of the relation in the 9th line from below on p.15: [ a]=[ab]. On the right hand side is the classical multiplication of the algebra, on the left hand side we have operator multiplication. Whenever this is true we do have ( exp( π ω(a) ) ) - 1 =exp(-π ω(a)).
There is some fine print to this construction which I am maybe not fully aware of. In particular things need to be bounded for this to make sense. That’s why Thiemann uses the operators W^ =exp(iY^) instead of the Y^ themselves, because these would be unbounded.
----------end of quote, sorry symbols not coming out-----
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recalling the Lubos/Urs discussion
The recent Lubos/Urs discussion was carried on in three places on the web, partly here at PF and the "coffee table" and also at SPR.
To try boil it down and get an idea of the general drift of this interesting exchange, I will exerpt Lubos recent PF posts (look back for detailed assertions) to recall the general tenor. And quote an Urs reply just posted at SPR which seems to sum up his response in the most concise way. First here are exerpts from two Lubos posts:
Here now is Urs SPR post which I just saw a few minutes ago:
----------quote from SPR----
Lubos and I had a little exchange about my original question over at
http://golem.ph.utexas.edu/string/archives/000299.html#c000504 .
It turns out that a crucial point for talking about the methods used in Thiemann's paper (hep-th/0401172) is that the ordinary lore of quantum field theory does not (or is not supposed to) apply in that framework.
For instance, in the ordinary version of quantum field theory the operator
:exp(-V):
is not the inverse of
:exp(V):,
due to the normal ordering, which is indicated by the colons. But in
Thiemann's paper (and, as far as I understand, in similar LQG papers) it is used that there is a representation of QFT operators, obtained by means of the GNS construction, which satisfy
pi(a) pi(b) = pi(ab),
where a and b are classical observables and pi(a) is the operator
representation of the observable a. This would imply that
pi( exp(-V) )
is indeed the inverse of
pi( exp(V) )
and I believe that this is a relation which is used heavily in Thiemann's paper. For instance this seems to be the basis for the claim in the 3rd paragraph of p. 20 that
alpha(W(Y_+-)) = W(alpha(Y_+-)) ,
where Y^mu_+- = p^mu +- X'^mu are essentially what is usually written as
partial X and bar partial X,
i.e. the left- and right-moving bosonic fields on the worldsheet,
W(Y) is the exponentiation of Y
and
alpha is the action of the exponetiated Virasoro constraints.
It is pretty obvious that if this is true then no anomaly does appear, since the elements generated by exponentiating the operator constraints behave exactly as those generated by exponentiating the classical constraints.
Is it hence true that we can alternatively have QFTs that have neither
normal ordering issues, nor anomalies, nor non-trivial OPEs, etc? If not, is there something wrong about Thiemann's assumptions? What is going on here?
-------------end quote--------------
Urs post contains a link to the "coffee table" discussion, which
will be useful to those whose browsers accommodate it gracefully.
The recent Lubos/Urs discussion was carried on in three places on the web, partly here at PF and the "coffee table" and also at SPR.
To try boil it down and get an idea of the general drift of this interesting exchange, I will exerpt Lubos recent PF posts (look back for detailed assertions) to recall the general tenor. And quote an Urs reply just posted at SPR which seems to sum up his response in the most concise way. First here are exerpts from two Lubos posts:
Originally posted by lumidek
Let me summarize a small part of his fundamental errors again. He believes many very incorrect ideas, for example that...
...Once again, all these things are wrong, much like nearly all of his
conclusions (and completely all "new" conclusions).
...Every time one can calculate something that gives them an
interesting but inconvenient result, they claim that in fact we don't need to calculate it, and it might be ambiguous, and so on. No, this is not what we can call science...
...There are hundreds of people who understand the quantization of a free string very well, and they can judge whether Thiemann's paper is
reasonable or not and whether funding of this "new kind of science"
should continue.
[from the next post]
...It’s just sad. Ignorance about the basics of quantum field theory should not be sold as a "new, revolutionary proposal in physics", and every student in theoretical physics should be able to identify the errors in papers similar to Thiemann's paper.
Here now is Urs SPR post which I just saw a few minutes ago:
----------quote from SPR----
Lubos and I had a little exchange about my original question over at
http://golem.ph.utexas.edu/string/archives/000299.html#c000504 .
It turns out that a crucial point for talking about the methods used in Thiemann's paper (hep-th/0401172) is that the ordinary lore of quantum field theory does not (or is not supposed to) apply in that framework.
For instance, in the ordinary version of quantum field theory the operator
:exp(-V):
is not the inverse of
:exp(V):,
due to the normal ordering, which is indicated by the colons. But in
Thiemann's paper (and, as far as I understand, in similar LQG papers) it is used that there is a representation of QFT operators, obtained by means of the GNS construction, which satisfy
pi(a) pi(b) = pi(ab),
where a and b are classical observables and pi(a) is the operator
representation of the observable a. This would imply that
pi( exp(-V) )
is indeed the inverse of
pi( exp(V) )
and I believe that this is a relation which is used heavily in Thiemann's paper. For instance this seems to be the basis for the claim in the 3rd paragraph of p. 20 that
alpha(W(Y_+-)) = W(alpha(Y_+-)) ,
where Y^mu_+- = p^mu +- X'^mu are essentially what is usually written as
partial X and bar partial X,
i.e. the left- and right-moving bosonic fields on the worldsheet,
W(Y) is the exponentiation of Y
and
alpha is the action of the exponetiated Virasoro constraints.
It is pretty obvious that if this is true then no anomaly does appear, since the elements generated by exponentiating the operator constraints behave exactly as those generated by exponentiating the classical constraints.
Is it hence true that we can alternatively have QFTs that have neither
normal ordering issues, nor anomalies, nor non-trivial OPEs, etc? If not, is there something wrong about Thiemann's assumptions? What is going on here?
-------------end quote--------------
Urs post contains a link to the "coffee table" discussion, which
will be useful to those whose browsers accommodate it gracefully.
Last edited:
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Hi Marcus -
sorry for the problems caused by the Coffee Table layout that you encountered. I don't know why that occurred. For me it is just the other way round: I have a large and nicely readable font when reading the String Coffee Table, with very nice pretty-printed formulas! :-) I also like that I can use the usual LaTeX commands to write math at the Coffee Table - and - well, that when composing messages I am not disturbed by a pack of smileys sitting next to the editor pane staring at me and doing weird things! ;-) But of course I don't want to pull away any discussion from the PF! So if you'll copy the Coffee Table comments any I could just as well post them here myself. :-)
Anyway, Lubos tells me that he will look more closely at the GNS construction and related things after lunch. I am looking forward to hearing his comments.
Urs
sorry for the problems caused by the Coffee Table layout that you encountered. I don't know why that occurred. For me it is just the other way round: I have a large and nicely readable font when reading the String Coffee Table, with very nice pretty-printed formulas! :-) I also like that I can use the usual LaTeX commands to write math at the Coffee Table - and - well, that when composing messages I am not disturbed by a pack of smileys sitting next to the editor pane staring at me and doing weird things! ;-) But of course I don't want to pull away any discussion from the PF! So if you'll copy the Coffee Table comments any I could just as well post them here myself. :-)
Anyway, Lubos tells me that he will look more closely at the GNS construction and related things after lunch. I am looking forward to hearing his comments.
Urs
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Originally posted by Urs
Hi Marcus -
... when composing messages I am not disturbed by a pack of smileys sitting next to the editor pane staring at me and doing weird things! ;-)
I know what you mean, especially this one
although this one is, in its own way, equally terrible[zz)]
Actually I appreciated your linking from here to the "coffee table"
and keeping us apprised of the discussion by links. Personally I don't really have to copy everything---I just put my nose closer to the monitor and I can read the "coffee table".
So don't feel you have to copy everything over from there (unless you just want to!)
I hope that some clarification and discussion occurs at SPR since
at least for the time being that is where the widest collection of knowledgeable people seems to be. It is always good to hear from a variety of viewpoints. Maybe your recent post at SPR will get some replies.
[imagine friendly smiley here]
lumidek
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Group averaging & Pohlmeyer charges
Yes, the GNS theorem is alright as a piece of abstract math, and for a unital *-algebra that satisfies certain properties, it returns the Hilbert space on which the algebra acts, and so on. There are many subtleties that would not work in this case, because of the noncompact character of the gauge group, and so on. But there is something more serious going on here. The GNS construction is a formal mathematical theorem, but it is creating less optimism if one sees what Thiemann wants to insert into this theorem, and what can he get.
On page 18, equation (5.5), he's getting the "Dirac observables" by averaging over the whole group. Of course, the Virasoro/reparameterization group is noncompact and the integral does not converge, so he defines the integral as the limit of the average over a finite piece of the (infinite-dimensional) group manifold. The infinite-dimensional integrals are usually ill-defined, but all these things are irrelevant technicalities and it is much easier to see why his construction is not physical.
It is easy to see that the only polynomial operators in the finite oscillators of x,p that can survive - that can average to anything nonzero - are the zero modes of the momentum, i.e. the total momentum of the string. There are just no other operators that would be invariant under the whole Virasoro group. In Gupta-Bleuler quantization, there are *states* that are invariant under one half of the generators (the positive frequency ones), and this fact follows from the special properties of the ground state that is annihilated by all positive frequency oscillators. In modern covariant BRST quantizations, we can find states that are BRST-invariant, which is nearly equivalent to saying that their matter part is invariant under the whole Virasoro group.
But there are no nontrivial "finite energy" Virasoro-invariant operators constructable out of the standard oscillators. So the intersection of the algebra of "Dirac observables", that he inserts into his Hilbert-space-generating theorems, with the regular stringy operators is really the algebra of "D" components of the momentum zero modes. The resulting representation that he can derive implies that particles can have various momenta. What a big deal! It has nothing to do with the internal dynamics of the usual physical string, derived from the oscillator modes.
The operators that Pohlmeyer and Thiemann want to consider are the Wilson lines of a gauge field. The vector potential of this GL(N,C) gauge field is taken to be the derivative of X^mu, contracted with some constant complex N times N matrices T_mu. Of course, a Wilson line is independent of the parameterization. The Wilson line is a path-ordered exponential, and we can Taylor-expand this exponential to get the individual terms, the Pohlmeyer charges (multiplied by a trace of a product of the complex constant matrices T_mu).
The first Pohlmeyer charges - linear in T_mu - are just the string's overall momentum and winding. The second Pohlmeyer charge is still well-defined in the correct spectrum of string theory, but it gives no new information. Look at eqn. (3.29), page 11 of Thiemann's paper. Z is the charge - it is gotten as a sum over cyclical rearrangements of a similar charge called R. For N=2 - the 2nd term in the expansion - it is easy to see that we integrate over all pairs of positions on the string because two points on a circle are always cyclically arranged in the right way - there is one way only - and therefore this charge is simply the first Pohlmeyer charge squared. Again, the overall momentum and possible the overall winding of the string, nothing more.
OK, so now we must go to the first nontrivial case N=3, the third Pohlmeyer charge. Now we only integrate over some intervals. In terms of the usual stringy oscillators, we get some combination of many terms cubic in the oscillators, with coefficients that don't go to zero too quickly as the mode number goes to infinity. Obviously, we would avoid the name "charge" for such an operator because no finite energy state of the string - and now I am considering the Hilbert space of the *real* string theory, not the LQG/AQFT genetically modified string - is an eigenstate of such a "charge".
At any rate, the Pohlmeyer charges form a very small (and sort of non-finite-energy) subalgebra of the polynomials constructed from the real oscillators in string theory, if you fix the usual gauge. You might think that those people would admit that the real string theory is at least one of their solutions. Of course, it is not, because they define wrong commutation relations for these charges. If you look at page 17, paragraph I (Kinematical algebra...), you will see that they simply define the commutators to be equal to the classical Poisson bracket - of the functions on the phase space. That's not what we have in usual string theory: commutators are more than just the Poisson brackets. They are equal in the "semiclassical" physics, but not in the full quantum physics.
This obscure definition of the commutators was inserted in, and not surprisingly, it comes out. Garbage in, garbage out. Everyone knows that no working quantum theory can be based on this obscure redefinition of the commutators. More importantly, it seems to me that the author confuses the *-algebra with a Lie algebra. While he might define a strange Lie algebra where the commutators are replaced by i.hbar times the Poisson brackets, this is not enough for a *-algebra. For a *-algebra, he would have to define the actual products. But of course, it is impossible. Classical physics gives us its simplified picture of commutators - namely the Poisson brackets - but it can't give us any simplified generalization of the multiplication itself. The multiplication in quantum mechanics is approximately equal to the multiplication of the corresponding classical objects, but not quite - for example, it depends what the order of the factors is.
OK, so they define some classical Poisson bracket algebra that just pretends to be a quantum algebra, but nevertheless they claim that there is a representation of it according to the theorems. As I noted, this conclusion is based on the confusion between the Lie algebra and the *-algebra. They can't define a product itself of the Pohlmeyer operators such that AB-BA will be the Poisson bracket. This is just impossible. OK, let's forget for a while that they don't have the required *-algebra because they only defined the commutators, not the products. Imagine that they would fix this bug in some way. What would be the relation of the resulting representation and the (standard) string theory?
By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets. Even though the task is different because they artificially define not-completely-quantum commutators that they decide to pick, we can see that the constraint for the Hilbert space to form a representation of *this* algebra is much weaker than the constraint that the Hilbert space can be obtained from a regular quantization of the oscillators.
All the best
Luboš
Yes, the GNS theorem is alright as a piece of abstract math, and for a unital *-algebra that satisfies certain properties, it returns the Hilbert space on which the algebra acts, and so on. There are many subtleties that would not work in this case, because of the noncompact character of the gauge group, and so on. But there is something more serious going on here. The GNS construction is a formal mathematical theorem, but it is creating less optimism if one sees what Thiemann wants to insert into this theorem, and what can he get.
On page 18, equation (5.5), he's getting the "Dirac observables" by averaging over the whole group. Of course, the Virasoro/reparameterization group is noncompact and the integral does not converge, so he defines the integral as the limit of the average over a finite piece of the (infinite-dimensional) group manifold. The infinite-dimensional integrals are usually ill-defined, but all these things are irrelevant technicalities and it is much easier to see why his construction is not physical.
It is easy to see that the only polynomial operators in the finite oscillators of x,p that can survive - that can average to anything nonzero - are the zero modes of the momentum, i.e. the total momentum of the string. There are just no other operators that would be invariant under the whole Virasoro group. In Gupta-Bleuler quantization, there are *states* that are invariant under one half of the generators (the positive frequency ones), and this fact follows from the special properties of the ground state that is annihilated by all positive frequency oscillators. In modern covariant BRST quantizations, we can find states that are BRST-invariant, which is nearly equivalent to saying that their matter part is invariant under the whole Virasoro group.
But there are no nontrivial "finite energy" Virasoro-invariant operators constructable out of the standard oscillators. So the intersection of the algebra of "Dirac observables", that he inserts into his Hilbert-space-generating theorems, with the regular stringy operators is really the algebra of "D" components of the momentum zero modes. The resulting representation that he can derive implies that particles can have various momenta. What a big deal! It has nothing to do with the internal dynamics of the usual physical string, derived from the oscillator modes.
The operators that Pohlmeyer and Thiemann want to consider are the Wilson lines of a gauge field. The vector potential of this GL(N,C) gauge field is taken to be the derivative of X^mu, contracted with some constant complex N times N matrices T_mu. Of course, a Wilson line is independent of the parameterization. The Wilson line is a path-ordered exponential, and we can Taylor-expand this exponential to get the individual terms, the Pohlmeyer charges (multiplied by a trace of a product of the complex constant matrices T_mu).
The first Pohlmeyer charges - linear in T_mu - are just the string's overall momentum and winding. The second Pohlmeyer charge is still well-defined in the correct spectrum of string theory, but it gives no new information. Look at eqn. (3.29), page 11 of Thiemann's paper. Z is the charge - it is gotten as a sum over cyclical rearrangements of a similar charge called R. For N=2 - the 2nd term in the expansion - it is easy to see that we integrate over all pairs of positions on the string because two points on a circle are always cyclically arranged in the right way - there is one way only - and therefore this charge is simply the first Pohlmeyer charge squared. Again, the overall momentum and possible the overall winding of the string, nothing more.
OK, so now we must go to the first nontrivial case N=3, the third Pohlmeyer charge. Now we only integrate over some intervals. In terms of the usual stringy oscillators, we get some combination of many terms cubic in the oscillators, with coefficients that don't go to zero too quickly as the mode number goes to infinity. Obviously, we would avoid the name "charge" for such an operator because no finite energy state of the string - and now I am considering the Hilbert space of the *real* string theory, not the LQG/AQFT genetically modified string - is an eigenstate of such a "charge".
At any rate, the Pohlmeyer charges form a very small (and sort of non-finite-energy) subalgebra of the polynomials constructed from the real oscillators in string theory, if you fix the usual gauge. You might think that those people would admit that the real string theory is at least one of their solutions. Of course, it is not, because they define wrong commutation relations for these charges. If you look at page 17, paragraph I (Kinematical algebra...), you will see that they simply define the commutators to be equal to the classical Poisson bracket - of the functions on the phase space. That's not what we have in usual string theory: commutators are more than just the Poisson brackets. They are equal in the "semiclassical" physics, but not in the full quantum physics.
This obscure definition of the commutators was inserted in, and not surprisingly, it comes out. Garbage in, garbage out. Everyone knows that no working quantum theory can be based on this obscure redefinition of the commutators. More importantly, it seems to me that the author confuses the *-algebra with a Lie algebra. While he might define a strange Lie algebra where the commutators are replaced by i.hbar times the Poisson brackets, this is not enough for a *-algebra. For a *-algebra, he would have to define the actual products. But of course, it is impossible. Classical physics gives us its simplified picture of commutators - namely the Poisson brackets - but it can't give us any simplified generalization of the multiplication itself. The multiplication in quantum mechanics is approximately equal to the multiplication of the corresponding classical objects, but not quite - for example, it depends what the order of the factors is.
OK, so they define some classical Poisson bracket algebra that just pretends to be a quantum algebra, but nevertheless they claim that there is a representation of it according to the theorems. As I noted, this conclusion is based on the confusion between the Lie algebra and the *-algebra. They can't define a product itself of the Pohlmeyer operators such that AB-BA will be the Poisson bracket. This is just impossible. OK, let's forget for a while that they don't have the required *-algebra because they only defined the commutators, not the products. Imagine that they would fix this bug in some way. What would be the relation of the resulting representation and the (standard) string theory?
By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets. Even though the task is different because they artificially define not-completely-quantum commutators that they decide to pick, we can see that the constraint for the Hilbert space to form a representation of *this* algebra is much weaker than the constraint that the Hilbert space can be obtained from a regular quantization of the oscillators.
All the best
Luboš
lumidek
- 92
- 0
Lie algebra and *-algebra
Concerning Thiemann's bombastic claims that you can define quantum theory with a Hilbert space such that the ordering issues (and double Wick contractions) disappear, let me summarize it by saying what is the main technical error that leads his to this ludicrous conclusion.
He defines some reparameterization-invariant operators, based on the Pohlmeyer charges, as operators formally associated with the functions on the phase space. And on page 17, he defines their commutators as the Poisson brackets, which is OK to define a Lie algebra. But it is not enough to define the *-algebra because in the *-algebra, you must actually know the product of each pair of operators. Of course, there is no product AB such that AB-BA would be always equal to the Poisson bracket (exactly), and therefore he has no *-algebra, and he cannot apply the formal theorems from axiomatic or algebraic quantum field theory.
Concerning Thiemann's bombastic claims that you can define quantum theory with a Hilbert space such that the ordering issues (and double Wick contractions) disappear, let me summarize it by saying what is the main technical error that leads his to this ludicrous conclusion.
He defines some reparameterization-invariant operators, based on the Pohlmeyer charges, as operators formally associated with the functions on the phase space. And on page 17, he defines their commutators as the Poisson brackets, which is OK to define a Lie algebra. But it is not enough to define the *-algebra because in the *-algebra, you must actually know the product of each pair of operators. Of course, there is no product AB such that AB-BA would be always equal to the Poisson bracket (exactly), and therefore he has no *-algebra, and he cannot apply the formal theorems from axiomatic or algebraic quantum field theory.
eforgy
- 30
- 0
Naive question about Nambu mechanics (don't shoot me)
Hello,
I know very little about string theory besides that it involves strings. Woohoo! :)
Anyway...
Urs' apparent enthusiasm about Thiemann's paper has got me following along with interest. One of Lubos' comments
"By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets."
particularly caught me attention. For some unrelated reason, I recently read the paper
http://www.arxiv.org/abs/hep-th/9501141
Dynamical Symmetries and Nambu Mechanics
My first question: is this the same Nambu that you guys are talking about?!
Lubos' comment reminded me of this paper even before I put 2+2 together. Anyway, Nambu mechanics looks really interesting to me and I had never seen it before. In a way, it is a generalization of Hamiltonian system, where instead of a symplectic structure W, you have an antisymmetric "Nambu tensor" N and a Nambu bracket defined as
{f1,...,fn} = N(df1,...,df2).
compared to the Poisson bracket
{f1,f2} = W(df1,df2).
Let me quote the conclusion from the paper:
"We have demonstrated that several Hamiltonian systems possessing dynamical symmetries can be realized in the Nambu formalism of generalized mechanics. For all but one of these systems, an extra freedom was found in the choice of the generalized Hamiltonians needed for their Nambu construction. Finally, one may speculate that since the harmonic oscillator is a very important example in quantum mechanics, its Nambu formulation may lead to a better understanding of the yet unsolved problem of the quantization of Nambu mechanics."
This made me think about what a "Nambu commutator" might be, which may give some insight into the apparent unsolved problem of quantizing Nambu mechanics. My first guess would be the Schouten-Nijenhuis bracket, but that is a wild (sort of) guess.
If I hazard a speculative dive bomb, I wonder if there could be made any sense out of the diagram
point particles -> Poisson bracket
strings++ -> Nambu bracket.
Kind of like p-form electromagnetism.
I'm wondering if the "unresolved" problem of quantizing Nambu mechanics has actually been completed already, but it is called "string theory."
Is there a chance I am making any sense at all? I doubt it, but I thought I'd bring it up.
Cheers,
Eric
Hello,
I know very little about string theory besides that it involves strings. Woohoo! :)
Anyway...
Urs' apparent enthusiasm about Thiemann's paper has got me following along with interest. One of Lubos' comments
"By definition, it is different from string theory because string theory's commutators are *not* generally equal to the Poisson brackets."
particularly caught me attention. For some unrelated reason, I recently read the paper
http://www.arxiv.org/abs/hep-th/9501141
Dynamical Symmetries and Nambu Mechanics
My first question: is this the same Nambu that you guys are talking about?!
Lubos' comment reminded me of this paper even before I put 2+2 together. Anyway, Nambu mechanics looks really interesting to me and I had never seen it before. In a way, it is a generalization of Hamiltonian system, where instead of a symplectic structure W, you have an antisymmetric "Nambu tensor" N and a Nambu bracket defined as
{f1,...,fn} = N(df1,...,df2).
compared to the Poisson bracket
{f1,f2} = W(df1,df2).
Let me quote the conclusion from the paper:
"We have demonstrated that several Hamiltonian systems possessing dynamical symmetries can be realized in the Nambu formalism of generalized mechanics. For all but one of these systems, an extra freedom was found in the choice of the generalized Hamiltonians needed for their Nambu construction. Finally, one may speculate that since the harmonic oscillator is a very important example in quantum mechanics, its Nambu formulation may lead to a better understanding of the yet unsolved problem of the quantization of Nambu mechanics."
This made me think about what a "Nambu commutator" might be, which may give some insight into the apparent unsolved problem of quantizing Nambu mechanics. My first guess would be the Schouten-Nijenhuis bracket, but that is a wild (sort of) guess.
If I hazard a speculative dive bomb, I wonder if there could be made any sense out of the diagram
point particles -> Poisson bracket
strings++ -> Nambu bracket.
Kind of like p-form electromagnetism.
I'm wondering if the "unresolved" problem of quantizing Nambu mechanics has actually been completed already, but it is called "string theory."
Is there a chance I am making any sense at all? I doubt it, but I thought I'd bring it up.
Cheers,
Eric
eforgy
- 30
- 0
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
- 30
- 0
Originally posted by jeff
eforgy,
Generally I don't complain about off topic posts, but I'm making an exception in this case.
Hi Jeff,
I thought the issue of Poisson brackets and commutators was sort of on topic and was one of the cruxes of Lubos' post.
I can't help but think that Nambu mechanics and its quantization is somehow relevant here. I could be wrong. It wouldn't be the first time.
I apologize for the "beer" comment, but I was somewhat excited to see that my instinct wasn't too far off the mark.
There is certainly a lot of interesting things going on here and I'm looking forward to following along (more quietly).
Eric Forgy, Ph.D.
MIT Lincoln Laboratory
PS: If any of the experts have any words to say about this Nambu perspective, I'd very much appreciate it.
ranyart
- 368
- 0
Originally posted by eforgy
Hi Jeff,
I thought the issue of Poisson brackets and commutators was sort of on topic and was one of the cruxes of Lubos' post.
I can't help but think that Nambu mechanics and its quantization is somehow relevant here. I could be wrong. It wouldn't be the first time.
I apologize for the "beer" comment, but I was somewhat excited to see that my instinct wasn't too far off the mark.
There is certainly a lot of interesting things going on here and I'm looking forward to following along (more quietly).
Eric Forgy, Ph.D.
MIT Lincoln Laboratory
PS: If any of the experts have any words to say about this Nambu perspective, I'd very much appreciate it.
Being no expert I have an interest in this thread, Having been busy the last couple of weeks doing some heavy reading, I can point you to this paper, it has relevence, and Marcus's original thread will not be comprimised:http://uk.arxiv.org/PS_cache/gr-qc/pdf/0401/0401114.pdf
Abay Ashtekar has a number of recent papers, with a good insight to the development of Hamiltonian methods, this recent paper will be interesting coupled to Thiemann's paper, when one take the crux of it into consideration, a new 'space-time' perception is emerging and it great!
Science Advisor
- 648
- 1
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
Staff Emeritus
Gold Member
Dearly Missed
- 6,845
- 11
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.
Science Advisor
- 124
- 0
Hi -
let me first answer Eric's question: Nambu brackets are related to the quantization of strings and branes in general because they give you a nice way to write the determinant of the pull-back of the target space metric to the worldvolume, and that's what enters the Nambu-Goto action of the p-brane (and the string), which is , slightly generalized (with an additional gauge field on the wolrdvolume), rather known as the Dirac-Born-Infeld action. Of course this detreminant simply (or rather its square root) simply measures the proper volume of the worldvolume.
Here is why this is the way it is:
On a p-dimensional worldvolume the Nambu bracket
{X^m1,X^m2,...,X^mp}
of the embedding coordinates X^mi(sigma^a) (a = 1,2,..p) that map every point sigma of the worldvolume to a point X(sigma) of the target space (our spacetime) is defined as
{X^m1,X^m2,...,X^mp}
= epsilon^{a1 a2 ... ap} (partial_a1 X^m1)(partial_a2 X^m2)...((partial_a1 X^m1)) .
Using this it is pretty easy to convince yourself that the expression
{X^m1,X^m2,...,X^mp}G_m1n1 Gm2n2 ...Gmpnp {X^n1,X^n2,...,X^np}
is, up to a combinatoric prefactor, equal to det(G_ab), where G_ab is the pull-back of G_mn (the target space metric) to the worldvolume.
You can find the details of this construction reviewed in my diploma thesis (analog of master thesis, roughly)
http://www-stud.uni-essen.de/~sb0264/sqm.html
See pp. 175 for the canonical analysis of the general Nambu action using Nambu brackets and page 179 for definition and discussion of the Nambu bracket themselves.
You might in this context also want to look at the Coffee Table Entry
Canonical analysis of D-string action
where I use the Nambu-bracket for the string while canonically analyzing the DBI action of the bosonic D-string.
2d Nambu brackets are particularly nice, because they can be approximated by commutators of large matrices (just like Poisson brackets can be approximated by operator commutators).
For the string, the 2d Nambu brackets appear in the
Lagrangian (in this context one usually speaks of the Green-Schwarz string in Schild gauge). When you replace these Nambu brackets for the string by commutators of large matrices you get the IKKT or Type IIB matrix model.
For the membrane, the Lagrangian usus 3d Nambu brackets, and people are still trying to get a better handle on these, because that would facilitate the covariant quantization of membranes.
However, the 2d Nambu brackets appear in the Hamiltonian of the membrane, obviously. If you replace these Nambu brackets in the membrane Hamiltonian with matrxi commutators you arrive at the BFSS matrix model of M-theory.
Ok, enough for now. Lubos can surely say more about this.
let me first answer Eric's question: Nambu brackets are related to the quantization of strings and branes in general because they give you a nice way to write the determinant of the pull-back of the target space metric to the worldvolume, and that's what enters the Nambu-Goto action of the p-brane (and the string), which is , slightly generalized (with an additional gauge field on the wolrdvolume), rather known as the Dirac-Born-Infeld action. Of course this detreminant simply (or rather its square root) simply measures the proper volume of the worldvolume.
Here is why this is the way it is:
On a p-dimensional worldvolume the Nambu bracket
{X^m1,X^m2,...,X^mp}
of the embedding coordinates X^mi(sigma^a) (a = 1,2,..p) that map every point sigma of the worldvolume to a point X(sigma) of the target space (our spacetime) is defined as
{X^m1,X^m2,...,X^mp}
= epsilon^{a1 a2 ... ap} (partial_a1 X^m1)(partial_a2 X^m2)...((partial_a1 X^m1)) .
Using this it is pretty easy to convince yourself that the expression
{X^m1,X^m2,...,X^mp}G_m1n1 Gm2n2 ...Gmpnp {X^n1,X^n2,...,X^np}
is, up to a combinatoric prefactor, equal to det(G_ab), where G_ab is the pull-back of G_mn (the target space metric) to the worldvolume.
You can find the details of this construction reviewed in my diploma thesis (analog of master thesis, roughly)
http://www-stud.uni-essen.de/~sb0264/sqm.html
See pp. 175 for the canonical analysis of the general Nambu action using Nambu brackets and page 179 for definition and discussion of the Nambu bracket themselves.
You might in this context also want to look at the Coffee Table Entry
Canonical analysis of D-string action
where I use the Nambu-bracket for the string while canonically analyzing the DBI action of the bosonic D-string.
2d Nambu brackets are particularly nice, because they can be approximated by commutators of large matrices (just like Poisson brackets can be approximated by operator commutators).
For the string, the 2d Nambu brackets appear in the
Lagrangian (in this context one usually speaks of the Green-Schwarz string in Schild gauge). When you replace these Nambu brackets for the string by commutators of large matrices you get the IKKT or Type IIB matrix model.
For the membrane, the Lagrangian usus 3d Nambu brackets, and people are still trying to get a better handle on these, because that would facilitate the covariant quantization of membranes.
However, the 2d Nambu brackets appear in the Hamiltonian of the membrane, obviously. If you replace these Nambu brackets in the membrane Hamiltonian with matrxi commutators you arrive at the BFSS matrix model of M-theory.
Ok, enough for now. Lubos can surely say more about this.
Last edited by a moderator:
Science Advisor
- 124
- 0
Thiemann's Hilbert space is not seperable
Now let me reply to Lubos' message:
Thiemann indeed does not show that the group averaging in (5.5) is mathematically well defined and that it does indeed project onto the space of Pohlmeyer charges. But I am willing to believe that the proof could be given. In any case, (5.5) does not seem to be essential to his construction. What is essential is that the classical Pohlmeyer charges are classical invariants of the string in the sense that they Poisson-commute with the classical Virasoro constraits. What Thiemann is looking for are quantum versions of these invariants, namely operators that commute with all the operator Virasoro constraints.
Now the big question is: How are these operators defined?
Let me say that I do think that his construction of a Hilbert space and of the quantum operators is well defined and that indeed the commutators that he considers reproduce just the classical Poisson brackets. I have tried to indicate why this can be true and why there are no higher order Wick contractions in Thiemann's framework in this comment.
We should not use CFT reasoning when thinking about Thiemann's quantization. There are no OPEs in his framework, no Wick contraction, no normal ordering, etc. His commutators are essentially the same as Poisson brackets. We have to find out what exactly it is that allows him to do away with all this - and if it is viable. Under the shower this morning I had an insight: I believe that the crucial thing is that Thiemann's Hilbert space is non-seperable, i.e. has no countable basis. This implies that it is much larger than ordinary Hilbert spaces and that operators with small worldsheet distance may sit next to each other without feeling each other's presence in form of singular terms of OPEs.
Let me try to recapitulate Thiemann's construction of the Hilbert space:
He starts with the classical algebra of phase space functions
W(I) = exp(int_I Y),
where I is a Borel subset of the circle, i.e. a union of closed intervals.
We have the well defined product relation
W(I)W(J) = phase factor times W(I + J) .
Now the absolutely crucial and non-standard step is to built a Hilbert space where every single one of the W(I) for I in a set of pairwise disjoint closed subsets of S^1 defines a linearly independent state. That's because states are of the form
W(I) Omega
(where Omega is some sort of "GNS-vacuum state"). And states for disjoint Borel sets are orthogonal
< W(I) | W(J) >_Omega = 0 if I disjoint J .
This follows directly from the algebra of the W and the definition of the scalar product <|> by (6.20).
But since there are non-countable many sets of pairwise disjoint closed subsets of the circle (simply because there are uncountably many points) this means that a basis for Thiemann's Hilbert space also is not countable and hence the space is not seperable. This is a mathematically consistent but physically highly pathological Hilbert space. It's non-seperability explains why there are no OPEs and the like, i.e. why the W(I) are not sensitive to 'neighbouring' W(J): The Hilbert space is by construction so large that W({x}) and W({x+epsilon}) can sit right next to each other without noticing each other. They just commute. This is so by construction. It is not a mathematical inconsistency, I think. But it is apparently physically pathological.
Maybe the use of non-seperable Hilbert spaces is the crux of loop quantum gravity. Because in full LQG, too, the Hilbert space of spin networks is taken to be non-seperable. This is carried over by Bojowald to "loop quantum cosmology" and leads to the very non-standard quantization of the Wheeler-DeWit equation.
It would be tempting to conclude that all these LQG constructions are nothing but a proof by counterexample that nature does not want to be quantized on non-seperable Hilbert spaces. :-)
Now let me reply to Lubos' message:
Thiemann indeed does not show that the group averaging in (5.5) is mathematically well defined and that it does indeed project onto the space of Pohlmeyer charges. But I am willing to believe that the proof could be given. In any case, (5.5) does not seem to be essential to his construction. What is essential is that the classical Pohlmeyer charges are classical invariants of the string in the sense that they Poisson-commute with the classical Virasoro constraits. What Thiemann is looking for are quantum versions of these invariants, namely operators that commute with all the operator Virasoro constraints.
Now the big question is: How are these operators defined?
Let me say that I do think that his construction of a Hilbert space and of the quantum operators is well defined and that indeed the commutators that he considers reproduce just the classical Poisson brackets. I have tried to indicate why this can be true and why there are no higher order Wick contractions in Thiemann's framework in this comment.
We should not use CFT reasoning when thinking about Thiemann's quantization. There are no OPEs in his framework, no Wick contraction, no normal ordering, etc. His commutators are essentially the same as Poisson brackets. We have to find out what exactly it is that allows him to do away with all this - and if it is viable. Under the shower this morning I had an insight: I believe that the crucial thing is that Thiemann's Hilbert space is non-seperable, i.e. has no countable basis. This implies that it is much larger than ordinary Hilbert spaces and that operators with small worldsheet distance may sit next to each other without feeling each other's presence in form of singular terms of OPEs.
Let me try to recapitulate Thiemann's construction of the Hilbert space:
He starts with the classical algebra of phase space functions
W(I) = exp(int_I Y),
where I is a Borel subset of the circle, i.e. a union of closed intervals.
We have the well defined product relation
W(I)W(J) = phase factor times W(I + J) .
Now the absolutely crucial and non-standard step is to built a Hilbert space where every single one of the W(I) for I in a set of pairwise disjoint closed subsets of S^1 defines a linearly independent state. That's because states are of the form
W(I) Omega
(where Omega is some sort of "GNS-vacuum state"). And states for disjoint Borel sets are orthogonal
< W(I) | W(J) >_Omega = 0 if I disjoint J .
This follows directly from the algebra of the W and the definition of the scalar product <|> by (6.20).
But since there are non-countable many sets of pairwise disjoint closed subsets of the circle (simply because there are uncountably many points) this means that a basis for Thiemann's Hilbert space also is not countable and hence the space is not seperable. This is a mathematically consistent but physically highly pathological Hilbert space. It's non-seperability explains why there are no OPEs and the like, i.e. why the W(I) are not sensitive to 'neighbouring' W(J): The Hilbert space is by construction so large that W({x}) and W({x+epsilon}) can sit right next to each other without noticing each other. They just commute. This is so by construction. It is not a mathematical inconsistency, I think. But it is apparently physically pathological.
Maybe the use of non-seperable Hilbert spaces is the crux of loop quantum gravity. Because in full LQG, too, the Hilbert space of spin networks is taken to be non-seperable. This is carried over by Bojowald to "loop quantum cosmology" and leads to the very non-standard quantization of the Wheeler-DeWit equation.
It would be tempting to conclude that all these LQG constructions are nothing but a proof by counterexample that nature does not want to be quantized on non-seperable Hilbert spaces. :-)
eforgy
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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 consensus 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.
Eric
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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.
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
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Inseparable Hilbert Space
Urs, I believe you are correct about the inseparability of Thiemann's initial Hilbert space [tex]H_{kin}[/tex]. 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.
Urs, I believe you are correct about the inseparability of Thiemann's initial Hilbert space [tex]H_{kin}[/tex]. 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.
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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 [tex]H_{aux}[/tex].
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.
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 [tex]H_{aux}[/tex].
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.
eforgy
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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.
Eric
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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.
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.
eforgy
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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=...06232045.77d132a7%40posting.google.com&rnum=1
lumidek
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South vs. North
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.
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.
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Rovelli's Hilbert space is separable, isn't Thiemann's too?
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"
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.
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"
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selfAdjoint
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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.
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