The String Coffee Table

Bert Schroer writes:

[begin forwarded email:]

In his discussion contribution Robert Helling refers the localization issue to the realm of interacting strings. But if he wants to defend the conceptual framework of what is called string theory, maybe he should have declared the whole localization issue as irrelevant.

By history as well as invariant content, string theory is a recipe to manufacture a “dual” S-matrix of an unknown off-shell relativistic quantum theory. This is achieved in terms of infinite towers of particles i.e. duality is a particular extreme form of crossing (field theoretic crossing requires a finite number of particles but in its formulation also multiparticle scattering cuts participate, it can never work with one-particle states only). In the field theoretic setting one can show that the inverse scattering has a unique solution (B.S. AOP) i.e. if one assumes that there exists a loccal net for an S with crossing than this net (of course not the individual fields) is uniquely determined (something which has been known by practitioners of the d=1+1 bootstrap-formfactor program for factorizing models for a long time). Hence the right localization concept for the S-matrix is that of the underlying off-shell operator theory and not that od an auxiliary “cooking recipe” which turned out to be useful in its construction. But this is certainly not a viewpoint which suits Pohlmeyer’s setting; he is dealing with an integrable system whose invariant charges should at the end lead to some setting in which the classical string has an intrinsic counterpart.

The very fact that Robert Helling does worry about this issue shows that he does not want string theory to end up in some mysterious S-theory (thus adding another Big Latin Letter). I sympathize with this desire, but how can it be implemented? Dimock’s work does not seem to be liked very much by string theorist. One can of course criticise his result on the basis that he simply ignored the tachyon. But presumably one would also obtain a center of mass localization in the supersymmetric case (somebody should do it!) if one follows his lightfront gauge. I would bet that in the covariant treatment it would come out to be completely delocalized. I agree that the word string is not only useful for the classical N-G Lagrangian, but it also helps to visualize Euclidean cooking recipe’s of combining and splitting tubes. But if there is any connection to real time intrinsic physical properties (the Euclidean framework as an analytic continuation is limited to pointlike fields!!) connected with worldsheets and strings remains a mystery (if there is a connection than the mystery is shifted to the necessity of interpreting it as a S-matrix). If the word “stringy” refers to the appearance of infinite mass towers (are they stable against higher orders?) then there is a certain analogy with the helicity towers in MSY except that the latter are really the internal degrees of freedom of a semiinfinite spacelike string (By the way the important property of thoses strings is that the operators A(x,e) do not commute if only the endpoints are spacelike separated i.e. they are described by a sophisticated amalgamation of Minkowski and de Sitter which is far from a tensor product between the two localizations).

One way out of this dilemma would be to say: let us leave the conceptual high grounds of local quantum physics and confront muddy paths in order to end up eventually with something which is much better than what we had before. I would be willing to do this if somebody could convince me that I will not stay in the blue yonder. But this is a decision which everybody has to make for himself.

The fact that I have a critical distance to strings of course does not mean that I cannot appreciate excellent mathematical physics work on strings as e.g. that of Pohlmeyer and his collaborators. By the way the comment that Rehren’s AdS-CFT framework leads to strings is not quite correct (see his later work with Duetsch on this point). If you start with a standard pointlike field theory on the AdS side you end up with a nonstandard CFT (its Borchersclass of relatively local fields is not denumerable as the Wick monomials for free fields but rather non-denumarable as for the so-called generalized free fields). On the other hand if you start with a standard CFT there will be some sort of Nielsen-Olsen threads or strands i.e. field objects which only depend on 3 instead 4 spacetime parameters.

Why don’t young folks learn a little bit about local quantum physics (the Lagrangian approach only covers a thin crust), after all despite my age I have been able to pick up several things in string theory.

Bert Schroer

[end forwarded email]

Hi Robert -

you wrote:

But one should be careful to generalize that result to the interacting theory.

That sounds plausible. On the other hand, the authors whose results disagree with that by J. Dimock all worked in free string field theory, too.

There are for instance

E. Martinec, Strings and Causality (1993)

and

H. Hata & H. Oda, Causality in Covariant String Field Theory (1996).

In

J. Dimock, Local String Field Theory (2003)

these papers are cited, but it is not discussed why their results might differ from those found by J. Dimock.

My suspicion is that the procedure of second quantization which is used in section 4 of the above paper by Dimock is not equivalent to that which for instance follows by quantizing an action of the form $⟨\psi \mid Q\mid \psi ⟩$. That’s because the only role played by the string oscillations in that section 4 is to determine the mass parameter of a free field. It is therefore maybe not surprising that equation (49) of that paper follows from the assumptions made there. Perhaps it are these assumptions or postulates about the nature or definition of string field theory, which leads to the discrepancy in the results on string field causality.

in the end locality should be with respect to the dynamical metric

Could you help me understand in which sense we can expect to see a dynamical metric in open bosonic string field theory?

As far as I understand one idea is that as the open string tachyon condenses the D25 brane disappears and somehow the open string field theory then describes closed strings.

IIRC Aaron once said something about poles in OSFT which correspond to closed strings. But does that mean that OSFT can capture the gravitational degrees of freedom?

Bert Schroer wrote:

The standard string only exists in $d=10,26$ whereas the P-strings […] exist in any spacetime dimension $d>1+1$.

How do you know this?

I am not aware of any consistent quantization of the Pohlmeyer invariants except for that obtained by expressing them in terms of DDF invariants. If I am missing something, please let me know.

As far as I know the work by Meusburger & Rehren remains conjectural until the ‘quadratic generation hypothesis’ is proven (see section 6 of the above paper). But I cannot judge how likely this is.

So as far as I am aware all we know so far is that the Pohlmeyer program can be solved in the usual critical dimension. (Of course one could still consider noncritical strings in $D<26$ in the usual way, though.)

In the introduction of your paper you say that canonical data and Virasoro constraints of Polyakov strings and Pohlmeyer strings are the same.

What I say is that, in particular, from the canonical coordinates and momenta of the NG action (without any gauge fixed) you can construct both the Pohlmeyer and the DDF invariants.

I didn’t even realize that this point could be controversial until D. Bahns told me that she believed that the construction of DDF invariants required fixing conformal gauge. (See the conclusion of her hep-th/0403108)

Probably this wrong belief is due to the fact that usually the DDF invariants are written down in a CFT context, which is derived from a Polyakov action with conformal gauge fixed. Luckily K.-H. Rehren agrees that no gauge needs to be fixed to construct the DDF invariants.

I furthermore claim that the Pohlmeyer invariants are a proper subset of all DDF invariants, that the (or at least one) quantization of the latter is obvious, and that it trivially induces a consistent quantization of the Pohlmeyer invariants.

I don’t know (and as far as I can see nobody does) if it is possible to perform another quantization of the Pohlmeyer invariants, inequivalent to that of the DDF invariants. But from the perspective that the Pohlmeyer invariants are a subset of the DDF invariants it looks a little unlikely to me.

Bert Schroer wrote:

Dimock’s pointlike localization of canonical strings (math-ph/0308007) is an accepted result within the mathematical physics community (Dimock is a first rate mathematical physicist).

I am no expert on this question, but discussion on sci.physics.strings suggests that Dimock’s version of string field theory is not generally accepted.

I will try to clarify the cause for this disagreement. Maybe there are some assumptions that people don’t agree on.

One may of course point to the incompleteness of the knowledge about Pohlmeyer strings but it does not make a convincing argument to base a hope for d=26,10 on these remaining imperfections unless one envisages a concrete mechanism of which kind of breakdown of the Meusburger-Rehren quadratic generation hypothesis could cause such a weird phenomenon.

I wouldn’t refer to ‘hope’ in this context. (I sure don’t ‘hope’ for any number d of spacetime dimensions. But if it turns out to be 10 I can accept that.)

But all that is known for sure is that in more than 25 years of trying no proof for a consistent quantization of the Pohlmeyer algebra in noncritical d has been found, while it is pretty easy in the critical dimension. Unfortunatly, as I said, I am lacking intuition for the ‘probability’ that the quadratic generation hypothesis is true.

On the other hand, depending on what we want to understand under a ‘consistent quantization of the Pohlmeyer algebra’ it is possible to demonstrate that it is possible for arbitrary d:

The classical Pohlmeyer invariants are a proper subset of the classical DDF invariants. Where does the critical dimension even enter when only the algebra of the DDF invariants is quantized, i.e. as long as we don’t talk about states in a Hilbert space and whether or not there are negative norm and null states, and how many of them?

The answer is of course: The critical dimension appears in a quantum correction to the bracket of the longitudinal DDF observables ${\tilde{A}}_m^{-}$.

Classically these obey the unextended Virasoro algebra, but quantumly they pick up a central term:

(cf. (2.3.98) of GSW).

When proceeding in the sense of Brower’s proof of the no-ghost theorem this correction term is responsible for the fact that the space of states generated by the DDF operators has non-negative norm and a maximum of null states precisely for $d=26$.

But let’s assume (as I think it is the case with Pohlmeyer) that we are not interested in any Hilbert spaces, but just in the algebra itself. Then the only crucial point is if the quantum algebra closes on invariants in the sense that the quantum commutator of any two quantized Pohlmeyer invariants is again an invariant (maybe the sum of a Pohlmeyer invariant with a non-Pohlmeyer invariant).

But trivially, this condition is fulfilled for every value of d when quantizing the Pohlmeyer invariants in the DDF context. The quantum correction for $d\ne 26$ is proportional to the identity operator and hence itself invariant. Hence by using the standard quantization of the DDF invariants for the Pohlmeyer invartiants one can get a quantization of the algebra of the latter for arbitrary d which closes on invariant objects. I.e. no matter in which number of dimensions we work the quantum correction to the algebra of Pohlmeyer invariants will be an invariant. From this perspective $d=26$ is only special in that here the quantized algebra closes not just on DDF invariants but on the smaller set of Pohlemeyer invariants. (Here I am actually ignoring the issues that further anomalies might be hidden in the infinite sums over oscillators involved in the definition of the Pohlmeyer invariants.)

So if just a closed quantum algebra of the Pohlmeyer invariants is sought for, then the proof and construction of its existence for each d is easy when using DDF invariants.

But is this all we want? I did discuss precisely this point with H. Nicolai and K.-H. Rehren in Ulm. Nicolai inquired what Pohlmeyer will do when he has found a consistent quantization of his algebra. Will he want to represent it on some Hilbert space and demand positive norm of physical states and a decoupling of longitudinal exceitations?

If this is what he wants, then so far this is only known how to do in $d=26$.

If this is not what he wants, then he will probably have to explain how he is going to calculate any scattering amplitudes.

Well, this is at least my understanding of the matter. Comments are very welcome.

I also should add that the work of Pohlmeyer and his group on strings which started more than a quarter century ago does not draw its importance and interest from its present fashionable attention in connection with loop gravity.

Agreed. And I never intended to (and believe I never did) claim the oppsite. It was a coincidence that Thomas Thiemann’s paper on the LQG-string made me aware of the Pohlmeyer invariants. Indeed, I think that Thomas Thiemann’s embedding of the Pohlmeyer invariants into the LQG context clarifies none of the above issues, since it all still depends fully on the Meusburger&Rehren work and the quadratic generation hypothesis.

String-localized fields are not introduced because some people were bored by 60 years of pointlike fields and wanted to explore other more interesting looking extended objects

It seems like you are insuating that the opposite is true for (Polyakov-)string theory. This would be a strange claim, I’d think. After all people got interested in strings not because they were bored by points but because strings describe quantum gravity. (If it is the quantum gravity of the experimentally accessible world is another question. But it sure is one theory of quantum gravity - and in fact the only one so far.)

Unfortunately I cannot comment much on the other interesting things that you wrote - but perhaps others here can.

Bert Schroer writes:

[begin forwarded mail]

The resolution of the paradox between Dimock’s and Martinec’s localization is obtained by taking a closer look at the commutators of the string fields. The operators are smeared with with test functions which in addition to the center of mass coordinate depend on all the internal string oscillator variables. But for the commutator to vanish it is enough
that the test finctions are othogonal in the symplectic sense in the center of mass only irrespective of the behavior in the string oscillators (something which vanishes already after integrating over the c. m. x is not undone by integrations over the oscillator variables).

The use of the Schroedinger representation (see appendix of Dimock) in order to convert the content of the Wave equation with the oscillator mass operator into an infinite dimensional hyperbolic differential equation (the way to obtain martinec’s condition) is not wrong, but it may create the incorrect picture of a genuine string localization. To spell out in more detail what is at stake here let us look at the MSY case where we do obtain a genuine string localization. In that case the inner string degrees of freedom which reside in the infinite spin tower are encoded into the geometrical spacelike direction e of a linear string and the the result is a field A(x,e) which is localized (i.e. has vacuum fluctuations) in Minkowski as well as in De Sitter
spacetime. The sophisticated amalgamation between the two spaces (so that the de Sitter point e really plays the correct role in the causal localization of two strings) is the very nontrivial result of the application of modular theory.
Martinec’s localization is a localization in a multidemensional (infinite dimensional) configuration space of Schroedinger wave functions which has nothing to do with the notion of string localization (as opposed to pointlike localization). The result of Dimock is the only genuine localization which remains after canonically quantizing the N-G string. The strings of string theory are not string-localized; the (in my view unfortunate) terminology refers to some euclidean manufactoring recipe.

[end forwarded mail]

Returning to the issue of string-localization raised by Barton I would like to again emphasize that even though there is no disagreement (as proven at the end of Dimock’s paper) between the properly interpreted Martinec’s localization result (derived from a hyperbolic wave equation in which the configuration space contributions from the oscillators contained in the mass operator increase the spacelike part) one should be aware of the fact that the configuration space interpretation of the oscillator degrees of freedom in the setting of an infinite dimensional Schroedinger description leads to the strange result that the causal dependency region of Dimock’s center of mass point is a region inside the forward cone. In fact that region can be pushed more and more inside the forward cone by increasing the contribution of the configuration space coordinates of the oscillators. This leads to an creeping of the local commutativity beyond the spacelike region into the timelike region and hence the whole situation becomes physically inconsistent with any causal localization concept. This again adds to my point that the whole idea that classical localization is transferred by quantization to a corresponding quantum localization is limited to pointlike fields:

classical string localization does not seem to pass to quantum string localization (so the word string only refers to the classical object) and quantum string-localized fields do not have a classical parallelism.

There are too many words these days without any intrinsic quantum meaning (another example: “world sheet”). What is dearly needed is a conceptual cleansing on the level of Bohr, Heisenberg and Wigner but regrettably this art seems to have been lost.

In Urs answer to some of my remarks he mentioned that there exists Brower’s proof leading to the distinction of d=26, 10. Of course I have no quarrel with that proof although the transition from Lie-group relations involving unbounded operators to a unitary representation of the Poincare group is a quite delicate business and requires the check of certain joint domain properties known under the name of (Edward) Nelson criterion which was not not checked by Brower (neither by anybody else as it seems). I don’t want to dwell on this point but I think control is superior to faith. My skeptic attitude comes rather from my physical instinct and I would rather go along with Henning Rehren and think that there is something wrong with the quantization. In any case if the investigation of string-localized fields would have led to such a result in our the MSY paper probably would only have been written after having reached a complete conceptual understanding (and by this I certainly mean much more than the result of a calculation a la Brower)or not at all.

Urs was less than happy about my calling string theory an invention so I should be more explicit. Of course this S-matrix formalism of what is nowadays called string theory goes back to Veneziano’s dual model (Polyakov’s contribution came later). In fact Veneziano’s contribution may be called a theoretical discovery simply because phenomenogists (for reasons which now would be totally obscure) wanted a situation in which the scattering continuum was approximated by an infinite set of particle poles in such a way that there could be a crossing relation between the s t and u channels. By an ingenious engineering with property of Gamma functions Veneziano discovered the dual model which fulfilled the dream of the strong interaction phenomenologists of those days. It is the second event which took place in the middle of the 70s in Paris (Scherk at al.) which people nowadays call the second (or first?) string revolution which I however would prefer to call the “Bartholomew nightmassacre” of the old (Regge pole associated) string theory of strong interactions. This second step was a pure invention, the only observation going into it was that strings contain spin 2 and anything containing spin 2 and is not outright diverging could contain a quantum theory of gravity. In fact the most surprising aspect (given the general belief that going up on the energy scale by a factor 1000 the probability that a theory would still work decreases by say 10%) was that this outrageous step of keeping the mathematical content of a
theory intact over 15 orders of magnitude while only changing the semantics (the semantic functor from strong interaction strings to quantum gravity containing strings). Whoever accomplished acceptance of this new gospel really succeeded to revolutionize particle physics, because from thereon everything goes and one has a free range for the roaming spirit unhindered by experiments or restrictions to old fashioned physical principles.

The somewhat tragic aspect of this development is that a really deep problem, certainly one of the deepest of particle physics, was lost in the course of this conceptual turmoil. I am talking about the S-matrix-bootstrap program. This was of course not just the result of the above events but primarily because their overzealous protagonists wanted to distinguish it from the conceptual setting of QFT and got into a cleansing rage against QFT from which they never recovered. They were so hung up on this point that they did not (or did not want to) see that some time later this program was converted into the bootstrap-formfactor construction and it worked beautifully for the limited class of “factorizing” models for which it was possible to separate the bootstrap S-matrix construction from the problem of constructing matrix elements of off-shell operators. All the desires which led Heisenberg to advocate to place the S-matrix into the center of the stage (namely to have a ultraviolet finite formulation of particle physics) were fulfilled for this limited set of models and even if their S-matrix was purely elastic, they had the full vacuum polarization of field states and led to fields with arbitrary inverse power short distance behavior as one is used to in the general case. This was a beautiful illustration of the message that the so-called ultraviolet problems are not intrinsic problems of the theory but rather are an inevitable consequence of doing calculations in terms of correlation functions of (inexorably singular) pointlike fields which Lagrangian quantization imposes on us. So the problem was how one has to modify the bootstrap formfactor program in order to be able to go beyond these special situation?

Unfortunately the knowledge about these important things are lost and as a result of the string theory propaganda (string theory is superior to QFT in the ultraviolet) the caravan moved on and left the most poignant problems of particle physics behind in the sand.

Some brief comments:

1) There is a difference between saying that a paper is nonsense and saying that it is mathematically correct but flawed on the part of the interpretation. A two-dimensional local theory with a E(2) spectator theory can not be interpreted in any different way because the interpretation of QFT (in constrast to QM) is coming with the specification of the model defined by its correlation functions. I also think by applying a Wigner-Moyal functor to a local Wightman theory you do not create a new physical reality; in fact the statement that the CPT invariance of the Wightman theory continues to play this role in the suggested ‘noncommutative version’ seems to highlight this point. There is no reason whatsoever why a non-local theory should have a TCP theorem as a structural
consequence; rather (as in the case of the Coester-Polyzou “direct particle interaction”) you may impose this invariance and be interested in the restrictions it imposes on the nonlocal interaction.

2) The Pohlmeyer quantization (of the invariant charges) is what was done successfully (I think first by Faddeev) with integrable systems and Pohlmeyer’s main observation was that the classical Nambu-Goto model is an integrable system (for integrable systems you determine the conserved charges before you quantize). This is not utilized in the Polyakov canonical approach (or functional, which is equivalent). It is of course completely correct to say that the kind of formalism which string theory in the sense of the dual model needs as an auxiliary device (to reproduce the dual model) is the Polyakov version and not Pohlmeyer’s. Of course I am aware of the existence of Brower’s proof of the correct Lie commutation relation only in D=10,26 but for the Poincare invariance one has to do more, namely in order to integrate the infinitesimal generators of a would be noncompact group to the finite unitary representers one has to check those rather sophisticated domain properties which appearantly has not been done. Whereas such a requirement would usually be considered as nit-picking, this changes of course if the result of such an investigation takes on high significance.

My sceptical attitude is however not bases on this apparent mathematical gap but rather results from the following remarks.
The Veneziano dual model was of purely phenomenological origin and I wonder whether apart from Veneziano and some of the oldtime Mandelstam representation (+fast converging dispersion relation)-based phenomenologist remember why it was invented (perhaps even
Green-Schwartz-Witten and Polshinski do not remember). Such a fine question as to
restrictions with respect to spacetime dimensions would have bothered none of the phenomenological crowd. Theories which totally change their interpretations in ‘revolutions’ (from forgotten phenomenology to the selection of spacetime dimensions) however create suspicions, but this seems to be my personal problem (it should not be like that after more than 30 years of physical failures and some mathematical successes).