### 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 $\mid m\u27e9$ and $\mid m\prime \u27e9$ goes like

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

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

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

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
## 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.