## October 17, 2006

### Lessons from the LQG string

#### Posted by Robert H.

It’s now two years, that Giuseppe and I have put out out our paper comparing the usual quantisation of the bosonic string to Thiemann’s loop inspired version. A bit to my surprise, that paper was of interest to a number of people and the months afterwards I was lucky to tour half of Europe to give seminars about it (in that respect it was my most successful paper ever; the only talk I have given more often is my popular science talk “Phaser, Wurmloch, Warpantriebe” about physics with a Star Trek spin prepared for the Max Planck society public outreach).

That paper had quite a resonance in the blogosphere as well, but its results have not always been presented in a way we intended them. This might also be because the paper was in large parts quite technical and some of the main messages were burried in mathematical arguments.

So I thought it might be a good idea to put out a “mainly prose” version of the argument which leaves out the technicalities to bring home the main messages. This I did and you should be able to find it on hep-th as you read this.

Remember the philosophy of this investigation: The loopy people always insist that diffeomorphism invariance is so central to gravity that it is important to build it into a theory of quantum gravity right from the beginning and all the problems one has with perturbatively quantising GR are due to ignoring this important symmetry or at least not building it into the formalism but expanding around some background.

As GR is a complicated interacting theory it is easy to get lost in the technical difficulties and one should consider simpler examples to test such claims.

The world sheet theory of the bosonic string is such an example as it is extremely simple being a free theory but still has an infinite dimensional symmetry of diffeomorphisms of the lightcone coordinates. It is thus the ideal testbed for approaches to diffeomorphism invariant theories where one can compute everything and check if it makes sense.

The first part of today’s paper explains all this and shows that the difference in the treatments can be summarised by saying that the usual Fock space quantisation of the string uses a Hilbert space built upon a covariant state whereas the loopy approach insists on invariance of that state which is a much stronger requirement.

My point is that covariance is the property which is physically required (and in fact states in the classical field theory are covariant but not invariant) and thus statements like the LOST theorem have too strict assumtions.

If you insists on invariance you end up with a Hilbert space representation which is not continuous as this is what LOST like theorems tell you. The question now is if this discontinuity makes your theory useless as a quantum theory. Well, everybody is free to set up the rules of the game they call “quantisation” and in the end only theories which do not disagree with experiments are good theories. But as we are all well aware, there are not too many experiments performed today which study properties of quantum gravity or bosonic string and thus this test is not available for the time being.

A weaker test would be to apply your rules of quantisation to other systems which are available for experimentation and see what they give there. Thus the second part (as in the original paper with Giuseppe) deals with a loop inspired quantisation of the harmonic oscillator. The old paper was criticised for providing a solid argument that it is observationally possible to distinguish the loopy oscillator from the Fock oscillator.

The second part of the new paper I think provides such an argument: It couples the oscillator to an electromagnetic radiation field and computes the absorption spectrum. Remember that usually the absorption for a transition between states $\mid m⟩$ and $\mid m\prime ⟩$ goes like

$\frac{1}{\left(\Omega -{\omega }_{m}+{\omega }_{m\prime }{\right)}^{2}}\phantom{\rule{thinmathspace}{0ex}}.$

Here, $\Omega$ is the frequency of the radiation. Now, the loopy result is proportional to

$\frac{1}{\mathrm{sin}\left(\left(\Omega -m+m\prime \right)/N{\right)}^{2}}$

where $N$ is a large natural number characterising the states. Thus if $\Omega \ll N$ the two expressions agree but for large $\Omega$ they don’t (don’t worry about an overall constant).

Thus if I am only allowed to measure within a finite frequency band for $\Omega$ the states can be made similar by choosing $N$ large enough. But once that $N$ is chosen the experimenter can reveal the difference by studying the behaviour at large frequencies.

So are they the same or not? Well, that’s a long story for which you have to read the paper.

After you’ve done that, you can come back here and comment.

Posted at October 17, 2006 3:15 PM UTC

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### Re: Lessons from the LQG string

What you write in the beginning of section 3, that the anomaly of Fock quantization is fatal, is factually incorrect. According to the no-ghost theorem, clearly stated in section 2 of GSW, the free bosonic string can be quantized with a ghost-free spectrum for all D <= 26. The anomaly turns the classical conformal gauge symmetry into a quantum global symmetry, which acts on the Hilbert space instead of reducing it. This is not fatal, because unitarity is not violated.

It is true that the anomaly violates unitarity when D > 26, and that the interacting string is inconsistent also when D < 26. However, this is irrelevant since your paper specifically deals with the free string. You can of course demand that the symmetry remains gauge after quantization, but then you are putting in an extra condition by hand.

### Re: Lessons from the LQG string

OK, you are right. But we want to go to the invariant subspace as the physical Hilbert space (what you call “remains gauge”) and that is just $\left\{0\right\}$ if the central charge does not vanish.

Posted by: Robert on October 18, 2006 3:36 PM | Permalink | Reply to this

### Re: Lessons from the LQG string

That is an exclusive we. However, what matters is not what you or I want, but what nature decides. If the correct quantum theory has a global infinite-dimensional symmetry, there is no way to distinguish it from a gauge theory by looking at the classical limit alone. Classically, you can always write down a nilpotent BRST operator, and thus reduce the theory, both if the original quantum symmetry was global or gauge.

It is also important which version of the gauge algebra you consider. For Laurent and Fourier polynomials, an anomaly is necessary for nonzero charge (L_0 != 0 implies all L_m != 0 because [L_m, L_-m] = 2m L_0), whereas for compact support and ordinary polynomials (L_m with m <= -1), there exists no anomaly. It is easy to see that this is true for all kinds of gauge symmetries - conformal, diffeomorphisms, or Yang-Mills does not matter. Since we know that nonzero charge exists, and string theory tells us that Laurent polynomials are admissible, I believe that gauge anomalies are inevitable.

But none of this has any bearing on LQG, of course.

### Re: Lessons from the LQG string

10 19 06

Robert thanks so much for putting out a prose version. I really like the interchange between you and Thomas. I am quite curious to see your thoughts on that new paper claiming that there is unitary equivalence between the Pohlmeyer rep and the fundamental rep cited on Christine’s site. Accordingly, there is a new use of formalism that wasn’t employed previously.

I will read it this weekend but am so darned curious as to what results this will yield.

I have come to look at LQG and stringiness and other theories as a sort of curiosity but the question is always whether or not the exotic curiosity can yield something sensible. Let us see…

Posted by: Mahndisa on October 19, 2006 8:08 PM | Permalink | Reply to this

### Re: Lessons from the LQG string

I think it’s definitely not a unitary equivalence. This would not work between a separable space and one that is not separable. My understanding is that they have some approximation procedure, but I hesitate so say something specific, yet.

Posted by: Robert on October 19, 2006 8:28 PM | Permalink | Reply to this

### Polymer vs. Pohlmeyer

new paper claiming that there is unitary equivalence between the Pohlmeyer rep and the fundamental rep

Careful with the spelling here. There is a Pohlmeyer algebra (named after German physicist Klaus Pohlmeyer, advisor of K.-H. Rehren) involved in some work that carries “LQG” in its title, and there is what people call polymer representations (named after polymers) that plays a role in these papers.

These are two very different entities.

Posted by: urs on October 20, 2006 2:38 PM | Permalink | Reply to this

### Re: Lessons from the LQG string

10 19 06

Robert thanks so much for putting out a prose version. I really like the interchange between you and Thomas. I am quite curious to see your thoughts on that new paper claiming that there is unitary equivalence between the Pohlmeyer rep and the fundamental rep cited on Christine’s site. Accordingly, there is a new use of formalism that wasn’t employed previously.

I will read it this weekend but am so darned curious as to what results this will yield.

I have come to look at LQG and stringiness and other theories as a sort of curiosity but the question is always whether or not the exotic curiosity can yield something sensible. Let us see…

Posted by: Mahndisa on October 19, 2006 8:10 PM | Permalink | Reply to this

### Re: Lessons from the LQG string

Nice paper, I can’t see anything wrong with your calculations (for that matter your previous LQG paper either), so thumbs up, I think the situation is now fairly clear.

I can see the next line of attack though, namely to look for nonperturbative renormalization group improvements that either spoil things further for them or alternatively to justify some of their construction.

Posted by: Haelfix on October 19, 2006 11:18 PM | Permalink | Reply to this

### Re: Lessons from the LQG string

I should say the imposing of an IR regulator for the 2 coupled harmonic oscillator makes this even harder to believe

Posted by: Haelfix on October 19, 2006 11:27 PM | Permalink | Reply to this
Read the post The Role of Rigour
Weblog: Musings
Excerpt: Don't trust strangers bearing Theorems.
Tracked: October 21, 2006 5:08 PM

### Re: Lessons from the LQG string

10 21 06
Urs:
Thanks very much for this clarification. I think I called the Pohlmeyer rep due to reading Thiemann’s paper and then wondered for a long while if there was something lost in translation or a typo occured. Thanks a bunch for helping me out:)

As to your comment Robert, yes the authors of the Corichi (spelling?) paper do employ an approximation procedure for representing p and p^2, as we all know that they are not well defined in the Polymer representation.

I think the same issue pops up from before; how do we know which deformed Hamiltonian to choose? How can we uniquely determine the functional form of p for these non standard reps in which p is not well defined? I don’t feel comfortable with the results of the paper. It sort of reminds me of the shadow states paper, which I think was a bit arbitrary in some ways. Although I still think that quantizing spacetime makes sense…

Thanks for the responses. I anxiously await your analyses of this paper, because you all are on a high enough level to adequately critisize it, while I am simply a curious person:)

Posted by: Mahndisa on October 22, 2006 12:12 AM | Permalink | Reply to this

### Stone - von Neumann

All quantizations of a system with finitely many dofs are unitarily equivalent. If polymer quantization of the harmonic oscillator is not unitarily equivalent to Fock quantization, it violates the Stone-von Neumann theorem, right?

### Re: Stone - von Neumann

Not right. SvN assumes the continuity of the representation of the Weyl operators that the polymer state does not have. Thus it evades SvN. As I said, the polymer Hilbert space is not separable thus it is genuinely different from Fock.

Posted by: Robert on October 25, 2006 9:10 AM | Permalink | Reply to this

### Re: Stone - von Neumann

Sorry, I did not express myself well. Anyway, SvN and LOST are incompatible, and hence the axioms of both cannot hold true in nature.

### Re: Stone - von Neumann

I would not say incompatible. SvN says “if continious then it’s Fock (up to unitary equivalence)” and LOST says “if invariant, then it’s polymer”. Where is the tension?

Posted by: Robert on October 25, 2006 9:28 AM | Permalink | Reply to this

### Re: Lessons from the LQG string

t dimensionality is the essence inthis lqg.but in the light of Ramanujam’s elliptic function “the term vanishing” indicates a loop quantum.intentionally or unintentionally Ramanujam has laid a foumdation stone for lqg string.the famous 26 dimensions are studdeed with so many theories evolved out since Einstein
and until this Stone von Newman.can i get the intrinsic concepts get clarified and enlightened?
expecting replies
E.Paramasivan
(

Posted by: E.paramasivan on June 10, 2007 8:44 PM | Permalink | Reply to this

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