Amazing bid by Thiemann to absorb string theory into LQG

[Urs]

separability of Hilbert spaces

I have just received email by Thiemann where he confirms that his kinematical Hilbert space in hep-th/0401172 (the 'LQG'-string) is indeed non-separable.

selfAdjoint wrote regarding this question:

Thus Urs' nondenumberable covering of by closed intervals has a countable subcovering by intervals beginning and ending on different multiples of k, leading to a separable space of functional states.

It may have a countable subcovering, but it remains true that there are more states in the Hilbert space than associated with this countable sub-covering. Just imagine: Every subset of S^1 which is the union of a finite number of closed intervals defines a state in H_kin which is orthogonal to any state associated with any other such subset! This are clearly uncountably many mutually orthogonal states.

Thiemann seems to regard his as separable. See his discussion leading into 5.7: "We now use the well-known fact that H_kin, if separable, can be represented as a direct integral of Hilbert spaces...".

Right. But this seems to be just a review of the 'Direct Integral Method' which is not used any further in the paper.

Let me note that the physical Hilbert space in Thiemann's paper is indeed separable. But that's no surprise, the physical Hilbert space is by construction much 'smaller'.

In order to see clearly how this should be compared to the usual approach, consider this:

The ordinary Hilbert space of the OCQ (old covariant quantization) or BRST quantization of the (super-)string is a kinematical Hilbert space because it contains physical and non-physical states. The physical Hilbert space is only the subspace which is generated by acting with DDF operators on physical massless (or tachyonic states). Since there are no constraints, i.e. no equations of motion to be imposed on the physical Hilbert space it is really quite inessential whether it is separable or not (it might for instance be an uncountable product of separable superselection sectors). But I believe that it is the non-separability of the kinetic Hilbert space on which all the action with operators and constraints happens, which allows Thiemann's non-standard quantization. See this entry for more details.

I did a little seraching for literature on Hilbert spaces in LQG in general. The situation is pretty confusing for non-specialists, since there seem to be lots of different Hilbert space that were studied. But I think a general pattern is that the kinematical Hilbert spaces are non-separable, while the physical ones often have separable superselection sectors.

I don't see the point in arguing that the non-separability is 'only a gauge artefact'. Yes, sure it is, since the systems we are talking about have no dynamics except for those imposed by constraints.

The question that I consider crucial is whether the space on which the constraints are imposed as operator equations is separable or not. If it is not, apparently very unusual things can happen, as in Thiemann's paper.

 

[marcus]

Originally posted by Urs
I have just received email by Thiemann where he confirms that his kinematical Hilbert space in hep-th/0401172 (the 'LQG'-string) is indeed non-separable...

There seems to be a difference in what the two authors call the kinematical Hilbert space. I quoted Rovelli (page 173
secton 6.4.2) saying the "kinematical Hilbert space of LQG", the one he uses that is, is separable.

I wonder how far this difference between Rovelli and Thiemann's terminology extends, or if it could be only in the paper
you mentioned (the "LQG-string" one).

 

[Urs]

kinematical, almost physical, physical

Hi Marcus,

I don't have Rovelli's book (anything available online?) but from what you wrote before it seems pretty clear what is going on:

The full kinematical Hilbert space of LQG is non-seperable.

After solving the spatial reparametrization constraints (but not yet the Hamiltonian constraint) we are left with something like an 'almost physical' Hilbert space. This is apparently what is called the 'kinematical Space' by Rovelli. It is separable - since lots of constraints have been solved.

But in Thiemann's paper of the 'LQG-string' there are only 2 constraints (at a given point of the string, of course) and he solves them both at the same time. Actually, because the theory splits into the left- and right-moving sectors, there is in a certain sense only one constraint (at a given point). So here it makes little sense to first solve some of the constraints and then the rest, getting an 'almost physical separable kinematical Hilbert space' as an intermediate object.

The full kinematical Hilbert space of the 'LQG-string' on which all of the original constraints are imposed is non-separable, and I guess that this is true for all LQG quantizations.

The point is that we want to compare the LQG quantization with ordinary quantizations. In the standard OCQ/BRST quantization of the string the Hilbert space if also 'fully kinematical' in the sense that all constraints have to be solved inside this space. No constrains have been dealt with before constructing this space. This means that the fact that the analogous space in Thiemann's construction is non-separable is a real difference to the standard approach, not just a gauge artifact or something like that.

 

[marcus]

Originally posted by Urs
Hi Marcus,

I don't have Rovelli's book (anything available online?)...

Yes,

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

His homepage is

http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

and if you scroll down there's the link to the book PDF file
a fairly recent draft
its now at the press but I just have the draft

 

[marcus]

Originally posted by Urs


...but from what you wrote before it seems pretty clear what is going on:

... not just a gauge artifact or something like that.

thanks for the explanation

I'm still not completely clear about how Rovelli's approach differs but that can wait.

Rovelli gets a separable space before he imposes physical constraint
and he says specifically that non-separability is "a gauge artifact".
If you happen to look at that page in the draft of "Quantum Gravity" let me know.

It is curious. something like what you describe must be going on but I am not completely sure.

He gets a non-sep Hilbert space. Then he takes equivalence classes under the action of a certain group
(diffeo gauge group)
so in effect he mods out

identifies kinematic states that are equivalent modulo diffeomorphisms

then the Hilbert space is separable
----------------------

maybe modding out is morally equivalent to imposing a constraint

 

[selfAdjoint]

Still struggling with non separability

Urs, your original development of the non separability argulment was this:
He starts with the classical algebra of phase space functions

W(I) = exp(int_I Y),

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

We have the well defined product relation

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

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

W(I) Omega

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

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

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

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


Now I claim that every exp(Int_I Y), where I is a Borel interval is in the closure of a dense countable set of exp(Int_K Y), where K is an Oresme-Tschebyschev interval. Is this false? And if true, is this not separability?

The orthogonal side of it doesn't phase me so much because this is explicitly not a Hilbert space of physics states. So instead of "arrows at right angles" or whatever we just have a zero inner product.

 

[marcus]

Hi sA,
perhaps the separability is not the key issue
but rather *Algebra reps and GNS are more central
to Thiemann's actual paper

but even tho it may be a side issue, separability of
LQG kinematic state space is interesting to discuss.
I appreciate your going back to the General Topology
definition so to speak---the countable dense subset.

In the Rovelli context things tend to be simpler
(and is this necessarily a deceptive simplicity?)

For instance, as I understand it a separable H space is
just one with a countable basis.

So it is no big deal for rovelli to show that his kinematic
state space K_diff is separable.
It just comes out of the spin-network basis.
The spin-networks embedded in the manifold M span the
(non-sep)hilbertspace K

But the theory has to be diff-invariant so we were always
planning to take diff-equivalence classes of states.
This was in the cards. Diffeomorphisms are gauge.
Two elements of K which differ merely by a diffeo are
the same state

But when you look at equivalence classes of spin-networks
there are only a countable number of them
they are abstract knots---only distinguished by their topology, so to speak,
and abstract knots are something you can count combinatorially

So the hilbertspace of equivalence classes has a countable basis.

It is pretty simple, all there on page 173 and whatever that discussion refers to. Only technicality is that he uses
an extended set of diffeos----they can be unsmooth at a finite set of points so like they are "almost-everywhere" diffeomorphisms

this approach----seeing there is a countable basis to the vectorspace---seems more intuitive than the General Topology approach, tho
perhaps less fundamental. does it seem ok to you?

oh, rovelli seems to find the separable kinematic state space convenient for calculating. must be a consideration

 

[Urs]

Hi selfAdjoint,

in my original post about non-separability here on PF I mistakenly focused on disjoint Borel sets. But that's totally irrelevant because in Thiemann's Hilbert space the states associated with exp(Int_I Y) and exp(Int_J Y) are orthogonal iff I != J. Even if I and J overlap but are not identical the respective states are orthogonal. Therefore it if I is, for instance, the union of J1 and J2, then the states associated with I, J1, J2 are all different and mutually orthogonal.

Now I claim that every exp(Int_I Y), where I is a Borel interval is in the closure of a dense countable set of exp(Int_K Y), where K is an Oresme-Tschebyschev interval. Is this false? And if true, is this not separability?

In which sense do you want to take the closure of these states?

As far as I understand you are arguing that there is a countable set OT of Borel subsets of S^1 such that any arbitrary <Borel subset of S^1> can be written as a (possibly infinite but countable) union of elements in OT.

Even if this is true I don't see how it shows that there are countably many states associated with these sets. That's because a different state is associated with every different <Borel subset of S^1>. Even if the Borel subset I is the union of J1 and J2 the states associated with I,J1, and J2 are all different and mutually orthogonal. So you are right that there are countably many states associated with the Borel subsets in OT, but these are not all the states in the Hilbert space, nor can all the other states be written as linear combinations of the states in OT. That's because any <Borel subset of S^1> that is not an element of OT (even though it may be the union of sets in OT) defines a state which is orthogonal to all the states associated with elements in OT.

Phew! :-)

BTW did you see that over at the Coffee Table Jacques Distler is claiming that Thiemann (and I, for that matter :-( ) makes elementary technical mistakes in his paper? I don't think that his criticism is legitimate, but I guess such a discussion is worthwhile (even though I would rather not be the one disagreeing with Jacques...).

 

[selfAdjoint]

Okay, then on separability

Urs, thanks for the clarification. I do see if the states being orthogonal if the intervals differ at all sidetracks my idea.

No I didn't see that attack on the technical competence of the paper. I firmly agee with your approach, to take the paper seriously and see where it leads to different physics. BTW how much of the different physics could be due to the fact that this is a radically non perturbative theory?

 

[Urs]

Hi -

I don't think that non-pertubativity is any issue at all. The worldsheet CFT usually used to describe the string is also non-perturbatively defined - after all it can be solved exactly (for flat target space, as also in Themann's paper, at least)!

One must well distinguish between the worldsheet theory of the single non-interacting string and the spacetime theory. The latter is defined only perturbatively by summing over an infinite number of CFTs on various Riemann surfaces. It is this summing which makes string theory pertuabtive. Every single contribution to this sum is well defined and exactly defined (nonperturbative on the worldsheet).

 

[selfAdjoint]

Distler

I think the answer to Distler is Rovelli, or else Ashtekar. We're not talking something that was made up yesterday by amateurs. The Ashtekar group, including Thiemann, has publications in this area going back to the 80's. And just now, Distler discovers elementary errors?

 

[eforgy]

Inner Products

I still think the issue is a crappy inner product :)

They have this inner product that gives completely orthonormal basis elements.

[psi_I,psi_J] = delta_{IJ}

even if psi_I and psi_J overlap. This is analogous to what we found (sorry to keep bringing up our work, but I think its relevent). We corrected this problem by introducing a "g-modified" inner product

[A,B]_g = [A,gB]

for some self-adjoint operator (with respect to the original inner product) of grade zero that "mixes" up the components. I wonder what would happen if they introduced a similar g-modified inner product?

Eric

 

[marcus]

Originally posted by eforgy

...They have this inner product that gives completely orthonormal basis elements.

[psi_I,psi_J] = delta_{IJ}

even if psi_I and psi_J overlap.

...
...
...for some self-adjoint operator (with respect to the original inner product) of grade zero that "mixes" up the components. I wonder what would happen if they introduced a similar g-modified inner product?

Eric

You have got me curious. do you want to be more specific in describing the LQG inner product?

by psi_I and psi_J do you mean spin network states?
how about hinting how the innerproduct of two such things is defined?
perhaps someone should, just for intelligibility

 

[eforgy]

Originally posted by marcus
You have got me curious. do you want to be more specific in describing the LQG inner product?

Hi Marcus,

I'm no expert for sure, but check out page 142 of


http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

5.3.6 Lattice scalar products, intertwiners, and spin network state.

Equation (5.160) is the culprit (in my opinion).

Eric

 

[Urs]

selfAdjoint wrote:

I think the answer to Distler is Rovelli, or else Ashtekar. We're not talking something that was made up yesterday by amateurs. The Ashtekar group, including Thiemann, has publications in this area going back to the 80's. And just now, Distler discovers elementary errors?

Yes, I think this is an indication of the fact that there has been very little serious interaction. I hope that string theorist who are convinced that LQG is flawed take the opportunity of the 'toy-example' that Thiemann has provided to precisely pinpoint which steps they do not accept. People have pointed out many physical oddities of LQGs which to many makes it look unacceptable as a theory of quantum gravity (e.g. the lack of semiclassical states so far, or the results of the BH entropy calculation - many string string theorists say that these results are not good at all). But I would very much like to understand the technical problems, if there are any. If it should turn out that there are consistent quantizations of the string worldsheet, for instance, which are inequivalent to the standard one, then I will want to understand this. Maybe my hope is just to understand why I can reject the non-standard quantizations. But to do so I first have to understand the details. That's my goal here.

I am trying hard to answer Distler's charges, see here. He seems to be getting impatient with me. If anyone wants to chime in I'd appreciate it!

In my last reply to Distler I discovered that I don't fully understand the following point of Thiemann's paper:

Are the operators exp(i L_n), where L_n are the Virasoro generators, even represented on his Hilbert space?

I am asking because a priori only polynomials in the W have explicit representations. For the Pohlmeyer charges, which instead need the Y ~ ln W there are already lots of subtleties, discussed by Thiemann in section 6.5. Can something similar be done for the exp(i L_n)? Are even the subtleties for the Pohlmeyer charges fully resolved?

And BTW: Why don't we just use the classical DDf invariants instead of the Pohlmeyer invariants? They have a much nicer algeba, nicely describe the string's spectrum and have a generalization to the superstring.

 

[selfAdjoint]

DDF Operators

I think I know why Thiemann did not use the DDF operstors. In GSW they are derived in the light cone gauge, and in Polchinski by CFT methods. Thiemann's effort is to build a nonperturbative quantized string without either of those approaches. So if he wanted to use DDF operators he would have to construct them anew himself. Whereas the Pohlmeyer operators were ready made and available. But this is certainly an effort that could pay off in the future, remembering that they have to be defined within the Thiemann quantization, not by mixing it with pertubrative methods.

 

[Urs]

Hi -

there is absolutely no problem in copying the idea of the CFT DDF states and turn them into classical invariants. Just replace integrals over dz z^n by integrals over d sigma e^{in sigma}, replace partial X by something proportional to pi + X' (just as in the Thiemann paper, too) and so on. I have done that once. I'll provide the details tomorrow. I have to catch some sleep now (night over germany...).


Urs

 

[selfAdjoint]

Request to Urs

Urs, when you read this tomorrow, could you do this? Write up your Coffee Table desription of DDR states here, in LaTex? You just use your ordinary LaTex syntax, and put it between boxes [ tex ] and [ /tex ] without the blanks. If you need a style sheet there is both a quickie one and a more complete one available by link at the "Introducing LaTeX Math Typesetting" which is the second thread in the General Physics board up above here.

I have been trying to follow your Coffee Table description but my browser is IE and it comes out hard to understand.

Thanks.

 

[jeff]

Originally posted by selfAdjoint
I have been trying to follow your Coffee Table description but my browser is IE and it comes out hard to understand.

Download a copy of mozilla

http://www.mozilla.org/products/mozilla1.x/download/

 

[Urs]

classical DDF invariants

Hi selfAdjoint -

as jeff says, since the Coffee Table uses MathML to display its math you need a browser which understands this standard. For Wintel that's currently only Mozilla, which is freely available. You might furthermore need to install a font, which is also available on the net for free. For more details see here.

But thanks for explaining me how to write pretty-printed math here on PF. However, instead of reproducing what I have written at the Coffee Table already I have opted for taking the time to write all this stuff down cleanly in a pdf file:

Urs Schreiber, http://www-stud.uni-essen.de/~sb0264/p5.pdf .

It is just a set of private notes. Let me know what you think! :-)

$$\exp(i\pi) + 1 = 0$$

 

[selfAdjoint]

Thanks so much! I have printed it off and I am going to start studying it NOW! What a lot of work went into it.

I am really leery of downloading Mozilla, to fill up my hard drive with unknown software that has unknown interactions with my operating system. Does anyone else have Mozilla with Windows NT? Any war stories?

 

[Urs]

Mozilla and NT

Hi selfAdjoint -

I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.

 

[marcus]

Originally posted by Urs
Hi selfAdjoint -

I am using Mozilla on NT in parallel with IE 6.0 and there are no problems as far as I can see.

Urs, I see you are online at the moment
has anyone found anyone equation in TT's paper that
you can put your finger on and say is wrong?

could you point me to some "equation number such and such on page
soandso" and say what is wrong with it

or is the controversy(if there is still some discussion) only about the verbal interpretations
and what conclusions to draw?

 

[Urs]

problems with the LQG string

Hi Marcus -

1) Jacques Distler say that this step is not allowed. That would invalidate the second and third paragraph on p.20.

2) If the constraints themselves are not represented on H_kin then (5.3) to (5.5) would not make sense. I don't know if they are, see here.

 

[jeff]

Originally posted by marcus
could you point me to some equation...and say what is wrong with it...or is the controversy only about the verbal interpretations and what conclusions to draw?

It was clear right away that the LQG-string is unphysical, and whether it's mathematically sound isn't a matter of interpretation.

 

[marcus]

Originally posted by Urs
Hi Marcus -

1) Jacques Distler say that this step http://golem.ph.utexas.edu/string/archives/000299.html#c000530
is not allowed. That would invalidate the second and third paragraph on p.20.

Urs, thanks for the reply! As I indicated I am looking for some
specific step in the argument or some equation, that is believed mathematically unsound.

I have a better idea now (Jacques says second and third paragraph
on page 20 are invalid)
but I still do not know what false step is supposed to make them invalid because
your link is to coffeetable and covers a lot of ground.
If not too much trouble could someone please point to which particular step
on some page in the TT paper
that Jacques says is wrong.

In the meanwhile I will have a look at those paragraphs on page 20.

 

[marcus]

Are there actually any math problems with the LQG string

does anyone else know what "step" Jacques thought
could be mathematically unsound?

I mean what page and what line on that page
where he says some definite thing that is invalid.

It isn't easy to read the coffeetable and
I don't see any specific reference there to
some equation number or some sentence that Urs
link points to.

what is the wrong "step" that makes second and third
paragraphs of page 20 invalid?

 

[marcus]

Are there actually any math problems with TT's paper

Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.

Here is one of those two paragraphs. What is wrong with it, can anybody say?

"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."

 

[selfAdjoint]

For the benefit of others, here are those two paragraphs.
1. Second paragraph on p. 20_________________________
Next we construct from $$\mathfrak{P}$$ bounded functions on $$\mathcal{M}$$ which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on $$\mathfrak{P}$$ be promoted to commutators divided by $$i\hbar$$ and the adjoint respectively. Denote the resulting *-algebra by $$\mathfrak{A}$$.
____________________________
Notes on 1. $$\mathfrak{P}$$ is a classical Poisson subalgebra constructed from the Pohlmayer charges. $$\mathcal{M}$$ is the target manifold (background space).

2. Third paragraph on page 20._____
The automorphism groups $$\alpha^{\pm}_{\varphi}, \varphi \in Diff(S^1)$$ generated by the Virasoro constraints s well as the Poincare automorphism group $$\alpha_{a,L}$$ extend naturally from $$\mathfrak{P}$$ to $$\mathfrak{A}$$ simply by $$\alpha (W(Y_{\pm})) = W (\alpha,(Y_{\pm}))$$. A general representation of $$\mathfrak{A}$$ should now be such that the automorphism groups $$\alpha$$ are represented by inner automorphisms, that is, by conjugation by unitary operators representing the corresponding group elements. Physically the representation property amounts to an anomaly-free implementation of both the local gauge group and the global symmetry group while unitarity implies that expectation values of gauge invariant or Poincare invariant observables does not depend on the gauge or frame of the measuring state. Finally, the representation should be irreducible or at least cyclic.
_______________________________

 

[selfAdjoint]

Originally posted by marcus
Let's take it step by step. Jacques purportedly claims that two paragraphs on page 20 are mathematically unsound.

Here is one of those two paragraphs. What is wrong with it, can anybody say?

"Next we construct from P bounded functions on M which still separate the points and promote them to operators by asking that Poisson brackets and complex conjugation on P be promoted to commutators divided by ihbar and the adjoint respectively. Denote the resulting star-algebra by A."

Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.

 

[marcus]

Originally posted by selfAdjoint
Now that I have Mozilla up, I read the dialog between Distier and Urs. I don't see that Distier ever explicitly addressed Thiemann's text. He made a general sniffy comment that the symbols were not defined, but never specified what he meant by this. Urs defended by beginning a derivation a la GSW (which to my mind, as Urs implied later in the dialog, was irrelevant; Thiemann is not doing anything within perturbative string theory, and cannot be successfully attacked from within perturbative string theory). Distier then criticized the derivation and the rest of the dialog was about that.

selfAdjoint many thanks for this report. I am glad that you have
Mozilla running and can read coffeetable.

I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.

It is certainly possible that TT's paper is flawless mathematically, I should say, and that no one will be able to point to any specific line in it where TT makes a false move.

But as you know it is not uncommon either for math papers to need corrections when they are first circulated in draft and it would be helpful to TT if anyone can find some error or unclear point, which he could be told about so he could have a chance to fix it.

The overall conclusions certainly are interesting, are they not?

 

[selfAdjoint]

The conclusions are strong in my opinion. BTW I have been trying to fit Urs' DDF operators into Thiemann's scheme, so far without success (I just don't heve this stuff sufficiently at the tip of my mind).

Here is Distler's first comment on THE LQG String:
Why don’t I just close my eyes, click my heels and wish away all anomalies?

What are the rules here?

It is well known that it is impossible to preserve all of the relations of the classical Poisson-bracket algebra as operator relations in the quantum theory.

What principle allows Thiemann to decide which relations will be carried over into the quantum theory?

Where does he discuss which relations fail to carry over?

To which the answer is, see the last five years of LQG theory, especially by the Ashtekar school.

And here is his second comment.
No, I don’t believe there’s any quantization scheme that takes the full Poisson-bracket algebra of the classical theory and carries it over — unaltered — into the operator algebra of the quantum theory.

Depending on the quantization scheme, you may be able to carry over some subalgebra (the prototypical example, being the CCRs).

And his third comment (getting down to some detail).
OK, so you (he) claim(s) that there is a quantization in which the commutation relations of X '(ó) , Ð (ó) , T + + (ó) and T - - (ó) are carried over from the classical Poisson-bracket algebra, unaltered (i.e., the commutators of the T ’s do not pick up a central term)?

Certainly, that’s not true if the T ’s lie in the universal enveloping algebra generated by X '(ó) , Ð (ó) — as is conventionally the case.

Without being sure, I suspect he's still working from inside string theory here.

And finally, the nuv of his argument.
You mean aside from the fact that none of the symbols are well-defined?

Look, this is elementary stuff.

We can expand everything in Fourier modes. If T + + is in the universal enveloping algebra of the Fourier modes of X ' and Ð , then its Fourier modes (conventionally called L n ) are some expressions quadratic in those modes.

Since the Fourier modes of X ' and Ð (the “oscillators”) don’t commute, you need to specify an ordering. I don’t care what ordering you choose, but I insist that you choose one.

Now compute the commutator of two L n ’s. Again, you will obtain something which is at most quadratic in oscillators (there will, in general, also be a piece 0 th -order in oscillators). And it must be re-ordered to agree with your original definition of the L n s.

Carrying out this computation, you obtain the central term in the Virasoro algebra, and I believe that it is a theorem that the result is independent of what ordering you chose for the L n s.

Note that I never mentioned what Hilbert space I hope to represent these operators on. So I don’t see where its separability (or lack thereof) enters into the considerations.

It's pretty clear here that the X' and $$\Pi$$ he is talking about come out of the echt string context. "Oscillators"!

 

[jeff]

Originally posted by selfAdjoint
The conclusions are strong in my opinion.

Respectfully, what conclusions?

 

[selfAdjoint]

Jeff, I meant the conclusions of the Thiemann paper, The LQG String.

Notice that I hold that criticism of it based on the techniques and constraints of string physics are by the point, or at least that they have to be explicitly shown to bear on what Thiemann is doing. Almost all of his paper, including the Hilbert space and operator algebra parts, is common to the developments of LQG over the last few years, and much of it is common to the work of mathematical physicists over the past several decades - the GNS construction, for example is truly classic.

I believe the weakest point of the paper is the Pohlmeyer charges, which Thiemann seems to have used as he found them, but I also think that Urs has provided the beginning of a fix for that in his generalized (un-string-ized) DDF charges. The problem with these as far as I understand is that he has provided a classical pre-quantum development of them, and Thiemann is set upon introducing his charges post-quantization. That difficulty is only temporary, I am sure.

BTW, thank you for the link to the Mozilla page. As I said above, I have installed it (browser only) and it seems to be working fine.

 

[lethe]

Originally posted by marcus

I have always been afraid to try to install Mozilla because of not knowing how it would cohabit in the same house with Internet Explorer. I gather you felt a similar trepidation but steeled yourself and took the plunge.

mozilla will not threaten your existing Internet Explorer installation. it is perfectly safe. the mozilla suite includes a web browser, email client, and html editor, all rolled into one. if you only need a web browser, you can get just the web browser component alone as an application called http://www.mozilla.org/products/firebird/ .

marcus, i think you should give firebird a try. not only is it the only browser around that will let you read MathML, but pop up advertisements will become a thing of the past, and tabbed browsing is very useful.

furthermore, firebird doesn t "install" in your computer at all. you just download the application, unzip it, and double click. don t like it? got tired of it? just delete it, you don t even have to uninstall. it is completely safe and completely free, and there is a chance that you will like it so well, that you will never know why you stuck with IE.