Dorothea Bahns: The Invariant Charges of the Nambu-Goto String

[marcus]

Dorothea Bahns, a former student of K. Pohlmeyer, is a theoretical physicist at Uni Freiberg. Her 1999 thesis, done under Pohlmeyer, was cited with special acknowledgment by Thiemann in his recent "Loop-String" paper, lengthily discussed here at PF. Bahns has recently assembled the results obtained in her thesis in condensed form and posted the preprint as
http://arxiv.org/hep-th/0403108
Her results now seem to be potentially of some interest. Thanks to Urs (together with Thomas Thiemann) for calling attention to Bahns research.

-------exerpt---------
The invariant charges of the Nambu-Goto String and Canonical Quantization

Dorothea Bahns
(Fakultaet fuer Mathematik und Physik der Universitaet Freiburg)

Abstract
It is shown that the algebra of diffeomorphism-invariant charges of the Nambu-Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space.

1 Introduction
The action of the Nambu-Goto string is a generalization of the reparametrization-invariant action of the relativistic particle in d-dimensional Minkowski space, where instead of a point-particle, a one-dimensional extended object (a string) is considered. Correspondingly, the solutions of the equations of motion are surfaces swept out by the string in spacetime (called world-sheets) which are extremal with respect to the Minkowski metric.

The parametrization of these surfaces is not fixed by the equations of motion, and hence, a change of the parametrization corresponds to a symmetry transformation which does not change the physical state of the system. Therefore, the Nambu-Goto string is a system with gauge group given by the diffeomorphisms of a surface. As such, it provides an interesting model to study the fundamental problem of quantizing a system with gauge freedom given by the diffeomorphism group.

For closed strings, the world-sheet is tube-shaped. It was shown especially in this case, that the Nambu-Goto string can be treated as an integrable system and that its integrals of motion can be constructed from a suitably defined monodromy [2]. These integrals of motion are functionals on the world-sheet which are invariant under arbitrary reparametrizations (gauge transformations) and as such are observable quantities. They form a graded Poisson algebra [3, 4], the Poisson algebra of invariant charges, and were shown to be complete in the sense that, up to translations in the direction of its total energy-momentum vector, the string can be reconstructed from the knowledge of the invariant charges, together with the infinitesimal generators of boosts [5]. In this scheme, the constraints which are present in the system enter as a condition on the representation of the algebra, and – together with conditions regarding Hermiticity and positivity of the energy – distinguish its physically meaningful representations.

The algebra of invariant charges provides the starting point of the algebraic quantization of the Nambu-Goto string [2]. This scheme is based on the idea that the correspondence principle should be applied to physically meaningful quantities only, which in a theory with gauge freedom means that it is applicable only to gauge-invariant observables. In this spirit, the graded Poisson algebra of invariant charges of the Nambu-Goto string is quantized by application of the correspondence principle, replacing the Poisson brackets by commutators and allowing for particular (observable) quantum corrections which are restricted by demanding structural similarity of the classical and the quantum algebra.

So far, it does not seem at all likely that in this scheme an obstruction regarding the dimension d of the underlying Minkowski space should appear (other than d > 2). In contrast to this, the canonical quantization of the Nambu-Goto string is consistent only
in certain critical dimensions. Here, the correspondence principle is assumed to hold for the Fourier modes of some particular parametrization, i.e. for quantities which are not observable. It leads to the well-known construction of Fock space which contains the
physically relevant states as a subspace.

In this paper, which is an exposition of results gained some years ago [1], it is shown that canonical quantization does not yield a representation of the algebra of invariant charges. After a short exposition of known results regarding the algebraic approach to
the quantization of the Nambu-Goto string [3, 4, 6] in the the following two sections, the fourth section contains an investigation of the canonical quantization and its application to the algebra of invariant charges. It is shown that unobservable anomalies arise in the defining relations of the algebra in 3+1 dimensions. In section 5 it is then shown that the problem cannot be cured by adjusting the dimension of the underlying Minkowski space.
------end quote----

 

[Urs]

Marcus quoted D. Bahns' thesis:

In this paper, which is an exposition of results gained some years ago [1], it is shown that canonical quantization does not yield a representation of the algebra of invariant charges.

Depending on what this is supposed to mean, I think that the claim is problematic. I do believe that there is a canonical quantization of the Pohlmeyer invariants which does yield a rep of an algebra of invariant charges and which furthermore reproduces the results obtained by the standard way of quantization.

See here for further discussion and http://www-stud.uni-essen.de/~sb0264/p5.pdf for the technical details.

I am somewhat puzzled because I had a private discussion with D. Bahns about the DDF invariants and I thought I had made it clear that her claim that these require fixing a worldsheet gauge is not maintainable. This is not a subtle issue. The DDF invariants, like the Pohlmeyer invariants, can be written down in the theory of the Nambu-Goto string, and this does not even have an auxiliary worldhseet metric with respect to which one could even speak of fixing a conformal gauge. Shrug.

As soon as I am back from Ulm I should see that I get http://www-stud.uni-essen.de/~sb0264/p5.pdf published.

 

[marcus]

Urs, thanks for your reply. And for including (as "shrug" and "the above notes") links to your comment about Bahns paper on SPR,

http://groups.google.de/groups?selm=4053318b$1@news.sentex.net

and your own draft paper

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

You are certain to be having a great time at the Ulm Fruelingstagung and to be energetically pursuing the Elders with many questions. Probably several of us, myself included, wish we could be there. Please report to us at PF your discussions with the "LQGists" at the conference including hopefully maybe Ashtekar.

 

[ranyart]

Marcus one reason I had posted here:https://www.physicsforums.com/showthread.php?postid=160606#post160606

the Dorothea Bahns paper was the obvious linking to TT 'amazing paper'.

This seems to have stalled the whole thread you created, I appologize for this.

From Monday 14th March I will disappear from the PF and web-forums until my computer gets back online, its been an absolute pleasure in seeing the internet being used for relaying some great intuitive debates.

 

[marcus]

Originally posted by ranyart
Marcus one reason I had posted here:https://www.physicsforums.com/showthread.php?postid=160425#post160425

the Dorothea Bahns paper was the obvious linking to TT 'amazing paper'.

This seems to have stalled the whole thread you created, I appologize for this.

From Monday 14th March I will disappear from the PF and web-forums altogether, its been an absolute pleasure in seeing the internet being used for relaying some great intuitive debates.

I see! You noticed Dorothea's paper and posted a link to it already
on 9 March in this post:

https://www.physicsforums.com/showthread.php?postid=160425#post160425

I did not catch that at the time.

You often find interesting things and share them. I am sorry to
here you are leaving PF as of Monday.
I am curious why and wish you would write me a PM.
BTW I don't understand your worry about stalling that thread
you did not stall it at all that I know of and wouldn't have
mattered had you. Hope your absence is only temporary. Be well.

 

[Urs]

K.-H. Rehren now agrees with my assertion that, as opposed to what it says in Bahn's thesis, the DDF invariants do NOT require fixing any worldsheet gauge. For more details of definitely the very latest from the field of Pohlmeyer invarians see here :-)

 

[marcus]

Originally posted by Urs
... For more details of definitely the very latest from the field of Pohlmeyer invarians see here :-)

continuing this pleasant vicarious experience of the spring conference at Ulm:

----quote from Urs CT post---
Wednesday

Ulm is nice little town at the foots of two mountains. One of these mountains carries the name ‘Einstein’ and enjoys sainthood. The other is called Eselsberg (‘donkey mountain’) and can actually be reached by mortals. On top of the Eselsberg there is the university and other scientific and industrial institutions, the total of which is called Wissenschaftsstadt (‘science city’).

There is a bus which takes me from my hotel in Ulm to this acropolis of science. Today I was late (again) for the first lecture. But I was lucky. K.-H. Rehren was late, too, and on the same bus.

He greeted me with the words that he had looked at my notes regarding Pohlmeyer invariants, which I had shown him the day before, and that he found some of the steps problematic. On our way to the lecture hall I tried to briefly sketch the resolution, but couldn’t quite convince him in terms of words.

During the first talks we both scribbled lots of algebra on scratch paper and as coffee break arrived we were able to agree that there is in fact no problem but that at one point notation and at another point an argument must and can be improved.

I very much enjoyed this constructive communication. While sipping our coffee we could even agree that the construction of classical DDF invariants for string does not have anything to do with fixing conformal worldsheet gauge, as opposed to what has been claimed recently.

That was great. K.-H. Rehren even demonstrated that the relation between Pohlmeyer invariants and DDF invariants holds on a larger part of phase space than I was originally able to show. There is only a subset of measure 0 on phase space where the equality between Pohlmeyer invariants and suitable polynomials of DDF invariants is technically problematic. (But I think by being careful we can even deal with that subset.)

After coffee break one of the highlights of the symposium was due, namely H. Nicolai’s talk on cosmological billards and the bold conjectures associated with them, which have been put forward by Damour, Henneaux and Nicolai, as I have mentioned before This is all very intriguing and maybe I’ll find the time to say more about it.

At lunch I was still discussing technical aspects of classical string invariants with Rehren, when Nicolai joined us and asked if we were making progress. I said that we made much progress in mutual understanding. Rehren still wants to go through the calculations again in private before signing my claims about Pohlmeyer invariants and DDF invariants, and I can only appreciate that.

Hermann Nicolai asked if this wouldn’t show that there must be things like critical dimension etc. in Pohlmeyer’s approach, too, whereas Pohlmeyer et al. argue that their approach works, if it works (so far a consistent quantization of the algebra of Pohlmeyer invariants has not been found), in any number of dimensions. But K.-H. Rehren pointed to the argument that he had presented before here at the String Coffee Table and according to which it is not clear yet if by relaxing some of the usual requirements on the reps of the Virasoro algebra one could avoid certain consequences.

When Rehren and Nicolai left for the afternoon talk sessions I noted that at the table next to me somebody was carrying a badge which indicated that his name was ‘Christoph Schiller’. I said: ‘Hey, I think I know you from the discussion forum sci.physics.research! You must be the one concerned with how general relativity has to do with a maximal force.’

And I was right. He briefly explained the idea to me, and it didn’t sound that crazy at all. It is essentially a modification of the old argument by Jacobson

T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state (1995)

who showed that you can derive GR from certain assumptions about thermodynamics. Schiller apparently noted that at the beginning of Jacobson’s derivation one may replace the thermodynamic assumptions by the assumption that there is a maximum value of the force between any two extended bodies. Here it is important that these are not taken as pointlike. He says that when you put an extended test particle above the horizon of a black hole, the distance being equal to the test particle’s Schwarzschild radius, it will experience precisely that maximal force.

I haven’t checked any of this yet, but if right the software engineer Schiller may have added a curious observation to physics, which, he says, might be of value in teaching GR.

It is such a great weather outside that I’ll stop at this point and see if I can produce some endorphine by exposing my myself to the sunshine.

---end quote---

 

[Urs]

Now Bert Schroer has joined the discussion on Pohlmeyer invariants:

http://golem.ph.utexas.edu/string/archives/000338.html#c000881