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 acknowledgement 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(adsbygoogle = window.adsbygoogle || []).push({});

http://arxiv.org/hep-th/0403108 [Broken]

Her results now seem to be potentially of some interest. Thanks to Urs (together with Thomas Thiemann) for calling attention to Bahns research.

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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.

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