Amazing bid by Thiemann to absorb string theory into LQG

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Thomas Thiemann's paper presents a novel approach to string theory by integrating Loop Quantum Gravity (LQG) quantization techniques with Algebraic Quantum Field Theory (AQFT) and integrable systems. The work offers a new solution to the representation problem for closed bosonic quantum strings in flat target space, suggesting that stable representations may exist without the need for extra dimensions or supersymmetry. This could potentially resolve significant issues in string theory, such as the cosmological constant problem, while demonstrating that critical dimensions and supersymmetry are not inherent predictions of the theory. Thiemann emphasizes the importance of exploring the algebraic perspective of string theory to uncover richer representation theories. The findings encourage further investigation into simpler, phenomenologically sensible models of string theory.
  • #61
Originally posted by selfAdjoint
I remember this categorization. It was overtaken by events...

You may be mistaken about the "Northern/Southern" distinction going away. I looked at Eric's link to his conversation with Baez and Baez defined southern as Gambini and Pullin

Gambini and Pullin's work continues as a distinct offshoot from the rest. I am uncertain about just why, but it is quite different. They seem to consider a different set of problems---having resolved to their satisfaction some problems that bother the others.

Baez classified Rovelli, Smolin, Ashtekar, Thiemann as "northern". I would rather just call them "majority".

IIRC there were several papers in 2002-2003 from the Gambini/Pullin folks. I guess it is nicer to call them "southern" because of living in Argentina and Louisiana, than to call them "splinter group". Baez always tactful.

Sorry not to be more informative about what makes Gambini and Pullin's work different, maybe looking at their recent papers would
make it clear to anyone curious about it.
 
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  • #62


Originally posted by marcus
congratulations on the diamond complexes you two, will have a look.

puzzled about something: lots of separable Hilbert spaces in Rovelli's book. have a look. download it from his website.
I think it has to do with choosing the diffeomorphism gauge group to include maps that can have a finite number of points where they aren't smooth. Maybe I am not remembering correctly so I'll see if I can find a page reference

Yeah. Section 6.4.2 Page 173
"The Hilbert Space K-Diff is Separable"

But look above on page 164

He says his kinematical Hilbert space \mathcal K (parallel to Thiemann's \mathcal H_{kin}) is inseparable, but don't worry, that's just guage.

The point is that you build a huge Hilbert space of "raw states" and then fillet out of it your separable space of "physical states".
 
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  • #63
The Hilbert space is separable, what's all this non-separable?

Urs, Eric, what am I missing? why do you say Loop hilbert space is non-separable?

Rovelli's book section 6.4.2 page 173

"The 'excessive size' of the kinematical hilbert space K reflected in its nonseparability turns out to be just a gauge artifact."

To make the theory diffeo-invariant you have to mod out the diffeomorphism group so K is not the final version
In rovelli's notation the diffeo group is Diff
and Diff* is the same thing but including maps that have a finite number of points where they arent smooth.
Modding out by Diff*
replaces spin network states by their equivalence classes (knots).
an equivalence class of networks is a knot
the space of knot states
K_{Diff*}

For Rovelli the real kinematical hilbert space is this separable one. When he is done saying this he says
"This concludes the construction of the kinematical quantum state space of LQG...Now it is time to define the operators."

?
 
  • #64


Originally posted by lumidek
Is this categorization correlated with the difference between the spinfoam path integral approach vs. the canonical operator approach? Or do you have a better description? Thanks.

Hi Lubos,

The best categorization is probably found in that URL I gave. I probably couldn't do any better so I'll just quote it.

In early work on loop quantum gravity, folks assumed this derivative existed for the kinematical states of interest. But then Ashtekar and Lewandowski constructed a very nice Hilbert space of kinematical states, clearly "right" in many ways, but with the unfortunate property that the derivative does NOT exist.

Now loop quantum gravity is split into two broad schools, which one could call the "northern" and "southern" schools. The northern school uses the Ashtekar-Lewandowski Hilbert space of kinematical states, and gives up on using loop derivatives. The southern school attempts to make sense of loop derivatives, and works with a space of kinematical states with no clear Hilbert space structure. The main exponents o the southern school are Rodolfo Gambini, Jorge Pullin and their collaborators, mostly from South America. The northern school includes Abhay Ashtekar, Carlo Rovelli and Lee Smolin.
 
  • #65


Originally posted by marcus
why do you say Loop hilbert space is non-separable?

Because Urs says so? :)

I don't really know if the Hilbert space is separable or not. So far the only thing I've said that was based on my own knowledge is that I don't like the inner product of the northern approach (it's easy to throw mud). I haven't even attempted to read Thiemann's paper yet and I'm sure it would go way over my head if I did try.

I became infatuated with the loop derivative (as you can see if you follow that link I posted), so I've never been a fan of the northern approach. I'm hoping that the southern approach can find a good inner product and I'm halfway hoping that Urs' and my paper may help in that direction, but the chances are admittedly slim.

If Thiemann's paper ends up casting doubt on their approach to LQG, I'd point the finger at the inner product. It's not clear to me that this is directly related to the issue of "seperable"ness though.


Eric
 
  • #66
separability redux

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

Let us see why this might be so. Everything here concerns the circle S^1 and we have the following theorem (Oresme-Tcehbyschev:

Let k be a coordinate on the circle that is incommensurable with 2\pi, then the integer multiples of k are dense in the circle.

Thus Urs' nondenumberable covering of S^1 by closed intervals has a countable subcovering by intervals beginning and ending on different multiples of k, leading to a separable space of functional states.
 
  • #67
I don't care so much about the extended hilbert space, there is quite a bit of literature in the 50s where analysts play around with that idea (and moving to Banach spaces etc etc). Sometimes they even managed to make working (but equivalent) theorems.

What I don't get in Thiemanns paper, is precisely why he expects the Poisson bracket to carry over into quantum mechanics operators. It strikes me that he would have to do incredible violence to the geometry to pull that off.
 
  • #68
Poisson brackets & operators

That's just normal canonical quantization. Define the operator algebra and turn the Poisson brackets, mutatis mutandis, into commutators.
 
  • #69
The poisson algebra is classical, to properly quantize it you typically only end up with some subalgebra at best, something new in general.

The rules for quantizing depends on the scheme you use, in canonical quantization you *replace* the Poisson bracket with commutator brackets (curly brackets to square brackets rule). Presto Classical mechanics into Operator mechanics.

Here its not so clear, this almost looks like a *semi* classical derivation of the bosonic sector. Its not so surprising no anomalies show up.

Unless I'm missing something deep in the mathematics of LQG that allows this sort of quantization.
 
  • #70
Originally posted by jeff




Describe this "new 'space-time' perception" and explain why it's "great!"

For JEFF:

Exhibit 'A':http://uk.arxiv.org/PS_cache/hep-th/pdf/0311/0311011.pdf

These may have relavance to others:


This paper is for those scanning the posts who would like a summary of interest:http://uk.arxiv.org/ftp/physics/papers/0401/0401128.pdf

Recent papers by Moffat:http://uk.arxiv.org/abs/gr-qc/0401117

and one by Gambini-Porto-Pullin:http://uk.arxiv.org/abs/gr-qc/0401117

Martinetti and Rovelli:http://uk.arxiv.org/PS_cache/gr-qc/pdf/0212/0212074.pdf really great!
 
  • #71
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.
 
  • #72


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).
 
  • #73
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.
 
  • #75


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:smile:

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:wink:
 
  • #76
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.
 
  • #77
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
 
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  • #78
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...).
 
  • #79
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?
 
  • #80
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).
 
  • #81
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?
 
  • #82
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
 
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  • #83


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
 
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  • #84


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
 
  • #85
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.
 
  • #86
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.
 
  • #87
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
 
  • #88
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.
 
  • #89


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/
 
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  • #90
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
 
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