Based on Pohlmeyer’s charges and using techniques from Loop Quantum Gravity, the closed string can even be quantized in any dimension. However, the non-separable Hilbert spaces used in the construction are difficult to handle, so one usually tries to avoid such spaces, if possible.

Another possible approach, the one I am primarily concerned in this post, is perturbative. A framework for such a perturbative treatment was developed in a joint paper, published in CMP, with Dorothea Bahns and Kasia Rejzner. It is a conservative approach, in that it uses well-established techniques of Quantum Field Theory (QFT). Also here it turns out that there are no anomalies, yielding a well-defined effective field theory in any dimension. It can be seen as a toy model for quantum gravity, sharing some its features (diffeomorphism invariance, perturbative non-renormalizability).

The fact that the Nambu-Goto string can be quantized perturbatively in any dimension seems to be largely unknown, even to experts. Let me thus describe this approach and in particular sketch the argument for the absence of anomalies.

So how does one quantize the Nambu-Goto string perturbatively? The point is to consider, well, perturbations; more precisely, perturbations around arbitrary non-degenerate classical solutions of the Nambu-Goto string (in the following also referred to as the background). So let be a classical solution, i.e., a world-sheet (a submanifold) in the target space . A perturbation can be conveniently described by a vector field on , taking values in . For an illustration, consider this Figure:

For a flat target space, the dynamical world-sheet is thus given by the embedding

where is the embedding of in , and is a formal perturbation parameter. Expanding the Nambu-Goto action in yields an action for the perturbation .

Obviously, things are not as simple as that. Due to diffeomorphism or reparametrization invariance, the equations of motion for that one obtains in this way are not hyperbolic. A gauge fixing is necessary. This can be achieved as for ordinary gauge theories, in the BRST formalism.

An infinitesimal reparametrization is given by a vector field on , with values in , cf. the following Figure:

The infinitesimal reparametrization acts on the field as

One promotes to a ghost and introduces the BRST differential defined by

In the quantization of a gauge theory, anomalies may occur, i.e., the quantum theory may no longer be invariant under the symmetry. Such anomalies can be removed unless certain cohomology classes and at positive ghost number are not trivial. Hence, if one can show that these cohomology classes are trivial, then the theory can be quantized consistently.

Now the free ( independent) part of the BRST differential acts as

By the assumption of non-degeneracy of the classical solution, has maximal rank. It follows that all the ghost are in the range of , so and are trivial at positive ghost number (in other gauge theories, derivatives of ghosts appear in , so only these derivatives drop out of the cohomology). Cohomological perturbation theory implies that then also and are trivial. Hence, the theory can be quantized without anomalies.

A gauge fixing still needs to be performed. Suitable choices would be harmonic gauge , with the metric induced by the embedding , or transversal gauge , i.e., setting all longitudinal modes to zero. The latter seems to have been first proposed by Kleinert, and it is particularly suitable for calculations, as it turns out that the longitudinal modes which are set to zero do not occur anyway in the free part of the action. Hence, on the level of the free theory, one can work entirely only with the physical (transversal) modes.

In the end, we thus arrive at a well-defined effective field theory living on the submanifold . This is in general a curved space-time, and the field equations only depend on the geometric data of the background, i.e., its induced metric and the second fundamental form. It is thus natural to quantize using techniques from QFT on curved space-times. In particular, one performs renormalization locally, i.e., for the definition of a local quantity at only the background geometric data at can be used, c.f. also my previous post. Furthermore, Poincare covariance can be ensured.

The last thing to check is that the physical subspace of the representation space, as defined via the BRST charge, is positive definite, i.e., a Hilbert space. In the paper, we checked this for an open string with Dirichlet boundary conditions, but in more recent work, I showed that this is also the case for rotating open and closed strings. These results will be discussed in a future post.

The model has an interesting feature. We can easily think of observables that measure the position of the string in target space. Just pick a test function on the target space, restrict it to the dynamical submanifold , and integrate it. If this observable gives a non-zero value, then the string intersects the support of . The construction is obviously reparametrization invariant, and also independent of the split into background and perturbation . If we now pick a background with a non-trivial compact intersection with the support of and expand in , we obtain a localized observable on the background , despite diffeomorphism invariance. Hence, it is not just diffeomorphism invariance which leads to the absence of local observables in gravity, it is the absence of external structures (the target space in the present case).

You may now legitimately ask: What can we do with the theory in the end? The theory is certainly not a fundamental theory. Apart from being perturbatively non-renormalizable, there is also no preferred classical solution around which to expand, i.e., no vacuum. The closest thing to a classical ground state of the Nambu-Goto string would be an event or a world-line, but these are degenerate (one may also debate on whether one should call these solutions, as these are configurations at which the action is not continuously differentiable).

However, if one fixes a non-zero angular momentum, then there are well-defined classical ground states, i.e., rotating string solutions. Doing perturbation theory around these solutions, one can compute semi-classical corrections to the Regge trajectory, i.e., the relation between energy and angular momentum. These are interesting from a phenomenological point of view, as the Nambu-Goto string can be seen as a model for the vortex line connecting the valence quarks in mesons. Furthermore, one can compare with the corresponding results of the non-perturbative quantization schemes, in order to check whether these have the correct semi-classical limit. This will be discussed in a future post.

]]>This book costs 75Euros (99Dollars) in Springer’s online shop, and you get a quality that is comparable to downloading it from some Russian server and printing it yourself (actually, you might even get better quality this way, as you do not need to scale it down). I thus complained at Springer’s customer office. They were very friendly and proposed to enquire whether the quality of the print can be enhanced. This, however, turned out to be impossible, so they offered me to choose another book (without me having to return the copy of Haag’s book).

In any case, I find it hard to believe that it is impossible for Springer to produce a print of better quality. Perhaps some reader with good connections to Springer can do something about it…

]]>The second notion of locality, the one that I am interested in this post, is relevant in the context of QFT on curved space-times or in the presence of external (background) fields. It can be roughly stated as follows: For the definition of an observable localized at , only the local geometric data at should be relevant. A precise definition in the context of QFT on curved space-times was provided by Hollands and Wald (2001) and Brunetti, Fredenhagen, and Verch (2003), and it can be straightforwardly generalized to gauge background fields.

Now even in the context of free fields, most relevant local observables are quadratic in the fields, i.e., non-linear. Examples are the *stress-energy tensor*,

for a massless, minimally coupled scalar field, and the *current*,

for a Dirac field. Both are relevant for the study of back-reaction effects, the former in the context of QFT on curved space-times, the latter for QFT in the presence of external electromagnetic fields.

In a QFT, the fields are singular, mathematically they have to be treated as distributions. In particular, one can not simply multiply them point-wise. This is easily seen from the fact that the *two-point function* diverges in the limit of coinciding points. So how should one evaluate the above observables in the state ?

The answer one finds in typical QFT textbooks is that, before taking the limit of coinciding points, one should subtract the vacuum two-point function , i.e., for the definition of the expectation value of one should define

Analogously, one proceeds with expressions containing derivatives, such as the stress-energy tensor. This procedure is called *point-splitting*.

The prescription is equivalent to Wick normal ordering and perfectly sensible on Minkowski space (in the absence of any other background field). However, the difficulties begin already when one restricts to sub-regions of Minkowski space, as in the Casimir effect. There, one restricts the fields to space-times with boundary on which one imposes, say, Dirichlet boundary conditions . On this space-time, there is again a vacuum state for the quantum field. However, this vacuum state should \emph{not} be used for point-splitting. This would lead to a vanishing of the energy density in the ground state and thus to the absence of the Casimir effect. The correct result is obtained by using the Minkowski space vacuum two-point function for the point-splitting.

Now this sounds paradoxical: In order to define a local observable in , one has to use the two-point function of Minkowski space. How does the field in know that is embedded in Minkowski space? This seems to grossly violate the locality requirement that only the local data at should enter the definition of the observable .

The situation becomes even more confusing when one tries to define the stress-energy tensor or the current on generic curved space-times or in the presence of generic external potentials. In such a situation there is no preferred vacuum state, as there is no time-translation symmetry that can be used to single out positive energy modes. So how should one then define the above local observables?

In the context of QFT on curved space-times, it was realized in the 1970’s, that physically reasonable states have two-point functions of so-called *Hadamard form*, i.e., they can be locally, for near , written as , where the *Hadamard parametrix* contains all the singularities and is determined solely by the local geometric data, independently of the state . The idea is thus to point split with , i.e., to define . Building on work of Hadamard (1923) on fundamental solutions to the wave equation and later work of DeWitt and Brehme (1960) and DeWitt (1975), this was worked out by Christensen (1976), Adler, Liebermann and Ng (1977) and Wald (1977). This procedure is called *Hadamard point-splitting*.

The above paradox concerning the Casimir effect is now easily resolved by noting that on flat space-time, the Hadamard parametrix (essentially) coincides with the Minkowski vacuum two-point function. A complete understanding of Hadamard point-splitting was achieved with the work of Hollands and Wald, which covers all local field polynomials (not just the stress-energy tensor) and also classifies the (renormalization) ambiguities inherent in the construction.

In the context of Quantum Electrodynamics (QED) in external potentials, the definition of the current is usually achieved by Schwinger’s prescription: By exploiting the (anti-) symmetry of the two-point function under exchange of and , one can remove some of the singularities. However, some divergences remain if the current of the external field does not vanish, and even if it does, the result in general depends on the direction from which the limit is taken. Even worse, in textbooks on the subject, a crucial parallel transport in Schwinger’s expression is omitted, leading to further divergences and gauge dependences.

Motivated by this unsatisfactory state of affairs, Jan Schlemmer and myself reconsidered QED in external potentials in the light of the lessons learned from QFT on curved space-times. Not surprisingly, the current can indeed be satisfactorily defined via Hadamard point-splitting, and it can even be used to perform concrete calculations. However, there was one surprising aspect. When trying to figure out the historical development, we found an article of Dirac (1934), in which he states:

A distribution [two-point function] such as occurs in nature according to the above assumption can be divided naturally into two parts ,

where contains all the singularities and is also completely fixed for any given [background] field, so that any alteration one can make in the distribution of electrons and positrons will correspond to an alteration in but none in .[…]

It therefore appears reasonable to make the assumption that

the electric and current densities corresponding to are those which are physically present, arising from the distribution of electrons and positrons.In this way we can remove the infinities […].

This is, I think, an admirably clear description of the Hadamard point-splitting procedure. This conceptual breakthrough seems to have been forgotten quite rapidly, probably due to the difficulty of doing actual calculations in position space. The method was still used by Uehling (1935), Serber (1935) and partly also by Heisenberg and Euler (1936) in their seminal works on vacuum polarization. Nowadays, one finds historical reviews on QED in external potentials that do not even cite Dirac’s paper. The re-discoverers of Hadamard point-splitting in the 1970’s also seem to have been unaware of Dirac’s work.

It appears that important insights even of eminent researchers can be forgotten quite quickly.

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