Amazing bid by Thiemann to absorb string theory into LQG

[selfAdjoint]

Jeff, I still say you are misinterpreting Thiemann's repeated use of "by definition" on page 3. I read him as meaning "By the definitions of our systems (which are valid in and of themselves) these results fall out". What he is certainly not saying is "We are just going to write in whatever we want".

 

[jeff]

Originally posted by selfAdjoint
Jeff, I still say you are misinterpreting Thiemann's repeated use of "by definition" on page 3. I read him as meaning "By the definitions of our systems (which are valid in and of themselves) these results fall out". What he is certainly not saying is "We are just going to write in whatever we want".

We disagree not about the interpretation, but rather about it's significance: thiemann is making these "definitions" without the further justification that is required by the unavoidability of operator ordering anomalies as jacques distler pointed out and as verified by urs'.

 

[selfAdjoint]

Jeff, bear with me. The following is not intended at all as an insult to string physicists, a group of people I admire greatly.

Think of a cartoon where a group of mountaineers are laborously and bravely ascending a difficult peak. At last they reach the top and are stunned to see a bunch of people in festive clothes, lounging around with refreshments. How can this be? they shout. Easy, says the group, there's an escalator up the other side of the mountain!

Now imagine what use are the following arguments.

- Everybody knows you can't reach the top without crossing the ice field. They didn't cross the ice field so they couldn't have come up. They're a mirage!

- Everybody knows that you can't ascend this mountain without using pitons. If you don't place pitons, you will fall. They didn't place pitons, so they will shortly fall!

IF (big if!) Thiemann has discovered a way to come at string theory from another direction, using different but well tested methods, then repeating over and over that they aren't using YOUR methods or obeying YOUR constraints is just fruitless. And saying that the work is only mathematical not physical only repeats a canard; if they can predict things and be right, that's physics. Everything else is just different schools of math.

The way to falsify The LQG String is to search into it's methods, learn their basis, and show they are false, or that the string cannot be described from that direction. But remember: the string he claims to describe is the nonperturbative string, so telling us things about a perturbative string is not going to work. You may be an expert on snarks, but Thiemann is talking about a Boojum.

 

[Urs]

so what are the implications?

So it seems the basic issue has been resolved. We had long arguments about how Thomas Thiemann can avoid the anomaly and related issues. (I, for one, did learn something in the process. Thanks to everybody who participated in the discussion!) But in the end it turned out that all these arguments about technical issues of representing the string constraint algebra had no relevance to Thomas Thiemann's paper. That's because he does not represent the Virasoro constraints and does not deal with their quantum algebra. What he does is this:

He says that there exist operators U_\pm(\varphi) on his Hilbert space which act on the string oscillators just as classical Poisson brackets would. He declares that the quantum theory is solved when states are found which are invariant under these U_\pm(\varphi).

There is no doubt that these U_\pm(\varphi) do exist. They are defined by their desired action on the states.

The intensive discussion about how these U_\pm(\varphi) might be constructed in terms of the canonical variables \pi^\mu and X^{\prime\mu} was vain: They are not constructed or indeed constructible from these canonical operators.

As Jacques Distler says: It seems that we haved achieved nothing by pointing out that the U_\pm(\varphi) exist.

I believe that this point can be made quite clear by the following simple argument:

I can by straight analogy define similar operators U_\pm(\varphi) on the standard string Hilbert space, i.e. a 2-parameter family of operators that has the algebra of the classical conformal algebra in 2d. (Simply define these operators by their action on every single state.) If I were to declare that the quantum string is given by states invariant under these operators on the standard string Hilbert space I would see no anomaly here, either!

Thomas Thiemann appeals to Dirac's quantization procedure. But this procedure says that the classical first class constraints should be implemented as operator constraint equations in the quantum theory. However, the U_\pm(\varphi) are not the exponentiations of the quantized classical first order constraints. They are not expressible in terms of these quantized constraints even in principle.

The reason why no quantum effects are seen in Thomas Thiemann's paper now is seen to be the result of the fact that the quantum constraints are not imposed, but an auxiliary set of constraints which is modeled after the classical theory but does not follow from any standard quantization procedure.

As an example, consider what it means that L_0 is not represented on Thomas Thiemann's Hilbert space: It means that he does not get the Klein-Gordon equation, which is nothing but L_0|\psi\rangle = a|\psi\rangle for string states of a given mass. This is not even representable in his approach. So not only relatively esoteric issues like the anomaly are lost, but even the well known behaviour of point particles of a given mass is not describeable.

I am afraid that I have to second Jeff's point.

(I also want to repeat that it is not true that Thomas Thiemann's approach is distinguished from the standard quantization by being non-perturbative. It is, but the standard quantization is, too. One must make sure not to confuse what the non-perturbativity of string theory is referring to, namely to the physics of target space, not that on the worldsheet. The standard quantization of the Nambu-Goto or the Polyakov action can be completely and exactly solved, the space of solutions being spanned by the DDF states. If a theory is completely solved it is non-perturbatively understood, obviously. The perturbation expansion in string theory appears instead when we compute target space amplitudes by summing CFT correlators for worldsheets of arbitrary genus. Every single such correlator (an expectation value for a CFT on a surface of given topology) is nonperturbatively and even rigorously defined. What is not really defined is the infinite sum of all these correlators which is supposed to give the target space amplitude.)


With this insight into the LQG quantization of a 1+1 dimensional rep invariant field theory it would now be highly interesting to have a second look at the quantization procedures used in LQG for the quantization of 1+3d gravity.

 

[marcus]

Originally posted by Urs

With this insight into the LQG quantization of a 1+1 dimensional rep invariant field theory it would now be highly interesting to have a second look at the quantization procedures used in LQG for the quantization of 1+3d gravity.

Urs, it would be great if you and others would examine the approach to quantizing 1+3-dimensional gravity taken in TT's book, which is expected to come out this year.

Also if you wish, compare the approach to LQG in Rovelli's book, which is quite different:
it does not use such abstract algebraic and functional-analytic machinery as the GNS construction nor does it force certain conditions to be met by algebraic "fiat".

Rovelli's construction of LQG is far more "down-to-earth". But I don't want to suggest that one version is more valid than the other---although the contrast is stark (as Rovelli points out in his Preface).

(As I noted earlier, Rovelli's kinematical state space is separable but AFAIK that difference may be merely superficial and arise from the order in which things are defined.)

Originally posted by selfAdjoint
... You may be an expert on snarks, but Thiemann is talking about a Boojum.

selfAdjoint, does TT's strategy remind you of how some professors construct the complex numbers as
R[x] modulo an ideal generated by the polynomial x²+1.
They make a polynomial ring and factor it down by the ideal denoted by (x²+1).
So the complex numbers are written
R[x]/(x²+1)

And thus they force the equation x²= -1
to be true by fiat.

and then there remain things to be proved about that object
(you do not save work you just change the order in which the work
must be done, but it has a kind of "mod" elegance)

the Boojum method

 

[Urs]

Hi Marcus -

it will be hard to find the spare time to look into the LQG constructions for 1+3d gravity in full detail. But I have a rough idea of what they are doing (from review papers, talks, and discussions with LQGists).

But maybe, now that I/we know what we have to be looking for we can simply make the experts tell us. Over at the Coffee Table Thomas has joined the discussion and I am beginning to ask him about 1+3d gravity.

He already told me that there the spatial diffeo constraints are imposed and solved by the same method as in the LQG-string, while the Hamiltonian constraint is imposed the ordinary way. (Does that resonate with what you see in Rovelli's book?)

That's, now that we have understood the LQG-string, a very valuable and interesting information. Maybe the whole diagreement about the viability of LQG can be condensed to the question whether it is really allowed to call constraints of the form used in the 'LQG-string' a quantization.

I think that's precisely where the problem lies.

 

[eforgy]

Originally posted by Urs
I think that's precisely where the problem lies.

Hi Urs,

Is this as potentially deadly as it sounds? It almost seems like LQG is one proof away from going up in smoke.

Eric

 

[jeff]

Originally posted by selfAdjoint
Jeff, bear with me. The following is not intended at all as an insult to string physicists...

However they initially strike me, my assumption that your posts aren't meant as insults is automatic so you really don't need to worry about that.

Originally posted by selfAdjoint
You may be an expert on snarks, but Thiemann is talking about a Boojum.

I looked up "snark" and "boojum" in my dictionary (shorter oxford). Are you saying I'm expert on elusive truths while thiemann is expert on dangerous imaginary theories?

Anyway, you already understand the main ideas of LQG well enough. Maybe it was the breadth of urs' perspective that allowed him to have so much fun with this.

 

[selfAdjoint]

I did really enjoy Urs' development of the classical DDF states. That was educational too. And who knows, ther may be life in the old LQG yet, I don't think we have penetrated to the bottom of what Thiemann is about yet.

By Snark and Boojum I only meant that the mysterious and deadly Boojum was an as yet undescribed variety of the commonplace and innocuous Snark. Undescribed in Carrol's poem because it killed everyone who ever saw it. But it was just two varieties of a single thing with very different characterisics that I brought it up.

 

[Urs]

Snarks and Boojums

I hope the gamekeeper is around! :-) I have read only the last Harry Potter in the original English version, so I guessed Snarks and Boojums might have appeared in Hagrid's class in the years before... ;-)

Anyway, Eric writes:

Is this as potentially deadly as it sounds? It almost seems like LQG is one proof away from going up in smoke.

Let me focus on technical issues which have a definite answer:

I know understand something quite important which was not apparent to me before:

Not all of the constraints used in LQG are represented as operators on some Hilbert space.

This has just been confirmed by Thomas at the Coffee Table. There he writes

I understand that the question whether constraint quantization can be done with the group or the algebra is controversial. The uneasy feeling may come from your experience with Fock spaces of which perturbative path integral quantization is just another version. In those representations one usually deals with the algebra, however, notice that one can work as well with the group. So you question my procedure by using an example where both approaches work. I would say
that there is no evidence for concern. For instance in LQG we have a similar phenomenon with respect to the spatial diffeomorphism group. We can only quantize the group, not its algebra. Yet the solution space consists of states which are supported on generalized knot classes which sounds completely right. There are other examples where the group treatment, also known as group averaging or refined algebraic quantization produces precisely the correct answer.
See for instance [23] and references therein.

So is the question: "Group or algebra?"?. I am not sure. To me the problem rather seems to be that the group constraints that Thomas uses are built by hand, modeled after the classical group action. But I think we would rather want the constraints drop out of the quantization process by a quantization mechanism (compatibility to the path integral woudn't hurt). This mechanism gives us the quantized first class constraints, i.e. the quantized constraint algebra. Is it ok to simply ignore it and construct different operators and using them as constraints? I'd say the LQG-string shows that this is not ok.

But at this point we are bitten by the fact that we are physicists, not mathematicians. One can always claim that the new, modified, quantization is what really describes nature. Maybe it would not even help if we could see the LQG quantization of some system that can actually be tested experimentally. If the approach failed to comply with experiment one could still claim that this ordinary system is not described by LQG quantization, but that quantum gravity is!

On the other hand, quantum gravity can be tested, right now. The 0th order approximation is classical gravity. LQG could still be tested by showing that it can, or cannot, reproduce smooth, locally flat space, gravitons, etc. As you know, so far this has not been done.

But, if there are gravitons to be found in the theory, it must of course have some description in terms of a path integral, at least in the appropriate limit. It looks quite problematic then that the fundamental theory cannot be described by a path integral.

Ok, I am rambling. To answer your question a little bit clearer: Personally I am more sceptic about LQG after having seen the LQG-string then I was before.

 

[eforgy]

Originally posted by Urs

Ok, I am rambling. To answer your question a little bit clearer: Personally I am more sceptic about LQG after having seen the LQG-string then I was before.

Hi Urs,

This is interesting. I guess a more constructive question would be to ask if something can be done to fix the situation? You seemed to be suggesting an alternative in the String blog. Is there anything to that line of thought? I hope all this ends up bringing about more discussion among the two camps.

Of course every theory of today is sick is some way or another and could use improvement.

Eric

 

[Haelfix]

Why should it necessarily have a description in terms of path integrals? Functional integrals as we all know, are not well defined.

In fact its not even that they are not defined, its the fact that they can be WRONG! Its a well known fact that canonical quantization and path integration need not produce the same results, past the classic solutions; often a clever mathematician has to tinker with them to get something that looks realistic.

Physicists that naively expect it to produce 100% accuracy, in every context are 1 step shy of being delusional.

The path integral should in correct a future lorentz invariant quantum field heory, yield something new, more general and more mathematically sound 'quoth A. Zee'

However, the last part is of course valid. LQG must produce flat space as a limit at some point for it to be considered more than a mathematical excursion.

 

[selfAdjoint]

Where did you get all this about functional integrals not being well defined? It's bushwa.

 

[lethe]

Originally posted by selfAdjoint
Where did you get all this about functional integrals not being well defined? It's bushwa.

i m not exactly sure what he is talking about, but it is probably the complaint heard from mathematicians all the time that there is no invariant measure with which to do path integrals.

 

[Haelfix]

Functional analysis is the branch of mathematics that should incorporate path integrals, unfortunately so far its never been able too. There is no precise definition analogous to say a Lebesgue measure, or an epsilon delta proof for analysis.

In short, its not well defined, much like say the Dirac delta wasn't defined until distribution theory came along.

Its somewhat painful for physicists to accept this, and is curiously not common knowledge the exact extent of the discrepancy. Feynmann, was very well aware of the problems, in fact some of his first papers include appendixes describing them. Many physicists of the time, felt that it was simply a calculational scheme, and that canonical quantization was the real heart of *real* QFT.

For instance, in most books on Quantum field theory, they'll show a proof of the duality between canonical quantization and path integrals (pionered by Dyson). Unfortunately, its wrong, valid only in certain circumstances (say when the action is nice and positive quadratic) or when it approaches regular quantum mechanies (the Wiener measure)! Mathematicians, will point out that in general there are serious operator ordering issues, and considerable abuse of complex structure. Wicks theorem is completely abused for instance, all throughout field theory.

In short, if you go through the details, you'll find that in general path integration must be treated on a case by case basis with considerable care and often adhoc assumptions to fix the problems. Branches of field theory have tried to make all this rigorous (one approach is the algebraic/axiomatic QFT people) but have been so far unsuccessful in making everything work and mantaining the nice experimental success that functional integrals enjoy.

So I mantain, that its premature and silly from a logical and mathematical point of view, to blindly excpect that all this works in quantum gravity. It might, but then there is reason to think that it might not!

 

[Urs]

Path integrals

It is true that path integrals in general, or the conclusions that are drawn by 'using' them, are not rigorously defined - as is true for QFT in general. In his introduction to the Yang-Mills problem in the Clay Millenium Prize questions Witten emphasizes that mathematicians should try to come to terms with QFT in the future. He knows that many beautiful results are hidden there.

Still, many results in QFT are obtained by using path integrals in a semi-heuristic way and somehow it works. Most of modern quantum field theory is in fact defined in terms of perturbative path integral calculations.

Anyway, for special cases path integrals are rigorously defineable. This is in particular true for Gaussian ones describing free field theories. The wordlsheet theory of the string (in trivial flat background) happens to be free and the path integral is well defined. The Wick rotation is also unproblematic (as discussed in Polchinski) and one can work on the Euclidean worldsheet. The path integral quantization of the string gives of course the 'standard' result, including the anomaly etc.

If we ignore the fact that we are dealing with a string which is supposed to be embedded in some target space and simply regard the theory as an example of a free quantum field theory in 1+1 dimensions, sort of as a toy example for interacting field theories in 3+1 dimensions, then it is sort of disconcerting that the 'LQG-string' does not reproduce the path integral quantization result.

But it is no surprise that it does not: The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using.

 

[arivero]

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

Sorry the pedantic mode, but "we" in a scientific text means "the author and all the people who is following the reasonment"

 

[marcus]

Originally posted by arivero
Sorry the pedantic mode, but "we" in a scientific text means "the author and all the people who is following the reasonment"

It seems likely that this is what Thiemann meant when he said "we" in his abstract. (My post consisted of a quote of the abstract, essentially without comment, to start a thread of discussion.)

BTW Alejandro, am I right that you have discussed DSR in past threads? I seem to recall your mentioning Amelino-Camelia, maybe also Magueijo. Might you contribute your current opinions if we began a thread on DSR?

 

[selfAdjoint]

Urs, you wrote

If we ignore the fact that we are dealing with a string which is supposed to be embedded in some target space and simply regard the theory as an example of a free quantum field theory in 1+1 dimensions, sort of as a toy example for interacting field theories in 3+1 dimensions, then it is sort of disconcerting that the 'LQG-string' does not reproduce the path integral quantization result.

But it is no surprise that it does not: The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using.

I have a couple of questions. How do you go about quantizing the 1+1 toy model? And where does the "by hand" come into Thiemann's derivation? Is it imported with group averaging? I guess what I am asking here is whether the fault you see goes back to what we might call the Potsdam school of algebraic quantization or is specific to Thiemann's use of that material.

One other thing (actually this is to anyone reading this) on the Coffee Table site, Distler recommended a paper:

Alvarez-Gaumé and Nelson, “Hamiltonian Interpretation Of Anomalies,” Commun. Math. Phys. 99 (1985) 103.

The online issues of that journal don't go back to 1985, and are behind a Springer pay wall anyway. I live in a small Wisconsin town hundreds of miles from any university that might carry such a journal. Do you know of any other source that I might find the same information in? A morning of scrounging around in the cites of Thiemann's paper and of the group averaging papers he does cite haven't brought me any enlightenment on the anomaly question in the general modern algebraic quantization of Hamiltonian systems with constraints. Neither did arxiv search on hamiltonian quantization anomaly. Couldn't find anything on it in the math phys books I have, Haag and Baez-Siegel-Zhou, either.

 

[Urs]

Hi selfAdjoint -

well what do you mean by asking how to quantize the toy model? Begin by deriving the constraints from either the Nambu-Goto action or the Polyakov action and impose them as usual to get the usual spectrum of the string=1+1d toy model.

The "by hand" in Thomas Thiemann's approach comes in at the point where he ignores the quantized constraints (the quantum Virasoro generators) and chooses to call physical states those which are invariant under his operators U_\pm(\varphi). These do not follow from the quantization procedure but are constructed by hand so that they have the desired group algebra. This is where his approach differs from the standard approach.

I don't really quite know what the definition of 'algebraic quantization' is and if that's related to the way Thiemann chooses the constraints. If you do, please let me know. I guess that algebraic quantization refers to taking a classical Poisson algebra and trying to deform it into a quantum commutator algebra. Thiemann is doing that, too, in his paper, but that's a different issue.

I understand this better in terms of the classical DDF invariants. When you impose the constraints the way Thiemann does ('by hand') then the classical DDF invariants are literally the same when quantized. But when the proper quantization is used then these logarithmic correction terms have to be included that I discuss in my draft. These logarithmic correction terms are related to the anomaly and stuff, which is missing in Thiemann's approach. I haven't checked yet, but the logarithmic terms should be essential for making the 'longitudinal' DDF states null, as it should be to avoid negative norm states.

I have ordered a pdf copy of the article that Distler mentioned. As soon as I obtain it (might take a couple of days) I'll send you a copy.


Urs

 

[arivero]

Originally posted by marcus
BTW Alejandro, am I right that you have discussed DSR in past threads? I seem to recall your mentioning Amelino-Camelia, maybe also Magueijo. Might you contribute your current opinions if we began a thread on DSR?

Yes I also remember to take a look to the papers on doubly special relativity a year ago but I do not remember which my opinion was. So I guess I was not very excited. Still it could be interesting to open a thread on it. I will keep an eye on the forum.

 

[Urs]

spectral action

Hi Alejandro -

you are an expert on NCG: Do you have any experience with using the spectral action principle for getting gravity and gauge theory form a spectral triple?

I am asking because Eric Forgy and myself are planning to insert our http://www-stud.uni-essen.de/~sb0264/p4a.pdf into the spectral action principle in order to get a theory of discrete gravity. Any help by experts is appreciated! :-)

In general, I would be interested in hearing your comments on the paper at the above link. The introduction has a brief overview over the central ideas.

All the best,
Urs

 

[jeff]

Originally posted by selfAdjoint
How do you go about quantizing the 1+1 toy model?

I'm not sure what's puzzling you here.

Originally posted by selfAdjoint...where does the "by hand" [i.e., urs: "The classical gauge group is imposed by hand in the quantum theory in the approach Thomas Thiemann is using"] come into Thiemann's derivation?

I don't want to put words in urs's mouth, but I think he meant that thiemann defines his theory so that the structure of the classical lie algebra of the gauge group is preserved in the quantum theory. See the paragraph of equation (5.2) for the general idea.

 

[Urs]

Hi Jeff -

yes, exactly, that's what I meant. The point is that what Thiemann does is find some operators which represent the classical gauge group on his Hilbert space. These operators, the U_\pm(\varphi) don't come from quantiizing the classical first class constraints. They come from using analogy with the classical theory.

Note that on a large enough Hilbert space it is possible to find operators that represent all kinds of groups. (Essentially because one can think of these operators as being large matrices and almost everything can be expressed in terms of large matrices.)

 

[selfAdjoint]

OK, now I am baffled. Group averaging is a method of quantizing a hamiltonian system with constraints.

Thiemann says:
Alternatively in rare cases it is possible to quantize the finite gauge transformations generated by the classical constraints by the classical constraints provided they exponentiate to a group.

It seems to be this exponentiation that you regard as put in by hand. I have been trying to track down the source of Thiemann's statement above but so far have been unable to find it.

Once he starts on group averaging he has available the theorem in the Giulini and Marolf paper he cites as [23], if you can meet the hypotheses of their group averaging construction, then the quantization is essentially unique, at least there is a unique "rigging function" from the kinematical Hilbert Space to the physical one. So if there is a quantization at all, and G.A. applies, then the G.A. quantization is that existing quantization.

So it doesn't look like G.A. takes us away from the familiar quantization with the anomalies; it must be the exponentiation that does it.

So is it your thesis that the exponentialion of the constraint operators is arbitrary? Thiemann seems to be saying he has support for doing it, and it's not just an arbitrary procedure.

 

[Urs]

so what's really the problem?

Hi selfAdjoint -

good that you insist on clarifications about this point. It is THE most crucial point, apparently.

Yes, the group averaging method as such is most probably completely uncontroversial as a mathematical technique.

But note that Thiemann writes "provided the CLASSICAL constraints exponentiate to a group". So he checks if the classical constraints exponentiate to a group and then he constructs an operator representation U(t) of that group on a Hilbert space. But these U(t) are not the exponentiated QUANTUM constraints (if there is an anomaly)! They are just some operators that represent the classical symmetry group of the system.

So this way the classical symmetry is put in 'by hand'. Distler says, an I think that he is right, that this is not how QFT works. We should not just try to represent the classical symmetry on a Hilbert space.

Even though mathematically this can be done, it is not related (or so the claim of the critics of this approach is) to the physical process called quantization. Quantization rather requires that you represent the classical constraints C as operators \pi(C) on a Hilber space and demand that
\langle \psi|\pi(C)|\psi\rangle = 0. In Thiemann's approach the classical constraints themselves are not even representable on his Hilbert space.

Maybe this should be compared to the 'quantization' of a simple mechanical system where the Hamiltonian itself is not representable on the Hilbert space but where the claim is that the time evolution operator U(t) is. Surely this would be very different from the ordinary notion of 'quantization'.

Of course Thiemann's procedure is no really 'arbitrary'. After all he models his quantum symmetry after the classical sysmmetry. The point is, however, that this is a choice of procedure different from the ordinary rules.

 

[jeff]

Originally posted by selfAdjoint
...is it your [urs's] thesis that the exponentialion of the constraint operators is arbitrary?

Adding a bit to what urs said, let's look at the paragraph of definition (5.2) I mentioned. Thiemann, under the assumptions mentioned above (5.2), takes (5.2) to be valid. He then infers from this definition that the classical lie algebra carries over to the quantum one appearing just below (5.2). The problem is that this inference is mistaken since gauge symmetries of the classical theory will pick up anomalies upon quantization so that the quantum algebra as given below (5.2) is simply wrong and thus the definition (5.2) makes no sense.

 

[Urs]

True, this isn't even self-consistently formulated, because in this section he is explicitly dealing with quantum Cs. Later on (as Thomas has also emphasized again in the Coffee Table discussion) however he defines the action of the U as in equation (6.25)
<br /> U_\omega(g)\pi_\omega(b)\Omega_\omega<br /> =<br /> \pi_\omega(\alpha_g(b))\Omega_\omega<br />.
This is what I am referring to all along as implementing the classical symmetry by hand on the Hilbert space. But you are right, this does not follow from (5.2) and this is what apparently confused me for quite a while, because all this discussion about non-separability of the Hilbert space etc. was essentially an attempt to understand how both (5.2) and (6.25) can be true. But (5.2) doesn't make sense at all in his quantization of the string, because
\pi(C_I) does not even exist! As Thiemann has emphasized in the Coffee Table discussion, the Virasoro constraints $C_I$ are not representable on his Hilbert space!

Hm...

 

[selfAdjoint]

So it all turns on that "Alternatively.." phrase up above. That's the cop-out from the requirement that the \pi(C_{\mathcal{I}}) be densely defined "on a suitable domain of \mathcal H_{Kin}". Wish I knew where he got that, and just what its hypotheses are.

 

[Urs]

BTW, Distler just indirectly said at the Coffee Table that for 1+3d gravity the \pi(C_I) cannot exist even in principle (anomaly or not) because their commutators cannot make sense.