## March 30, 2004

### Notions of string-localization

#### Posted by Urs Schreiber

Yesterday I was contacted by Bert Schroer.

He asked if in the context of my recent hep-th/0403260 I could see any way to get an

intrinsic understanding of ‘string’ or ‘worldsheet’ as a somehow localized object in target space and which concepts would make that visible .

He said that the context of this question is the recent discovery by himself and collaborators, reported in

Jens Mund, Bert Schroer, Jakob Yngvason, String-localized quantum fields from Wigner representations (2004)

of what is called string-localized fields. In a certain way these fields describe semi-infinite ‘strings’ and have the crucial property, that their commutator vanishes iff the respective ‘string rays’ are strictly spacelike seperated. This in a sense generalizes the phenomenon of commuting spacelike fields of point particle theories and apparently also provides representations of the Poincaé group for vanishing mass and infinite spin.

In the above paper it is pointed out that this notion of ‘string-localization’ is quite different from the properties of string fields as they appear in string field theory. There, instead, the commutator vanishes when the center of mass of two strings is spacelike seperated, irrespective of the extension of these strings.

I answered that I didn’t see how the first quantized theory of the Nambu-Goto/Polykaov string could shed any light on these issues, but that the context of the constructions in Bert Schroer’s paper reminded me of tensionless strings, where there are massless states of arbitrary spin in the spectrum.

Today I received an email where Bert Schroer disagrees with these assessments and points out several other aspects of the question. I find the above concept of string-localization and its disagreement with string field theory interesting, but will probably not be able to make many further sensible contributions to these questions. Therefore, with Bert Schroer’s kind permission, I will reproduce his latest email here in the hope that maybe others can make further comments.

Bert Schroer writes:

[begin forwarded mail]

Dear Urs,

your answer to my question for any intrinsic quantum meaning of the
word “string” was that one cannot expect this in first quantization and you
referred as an illustration to the classical relativistic particle Lagrangian
involving the line element in Minkowski spacetime.

First let me say that as far as relativistic particles are concerned that
description is “artistic” (i.e. has no mathematical meaning nor conceptual
status). It has been introduced by string theorists (see Polshinski) as a
support for the artistic link between Lagrangians with operator structures
which then define the true conceptual/mathematical start (in case of P-charges
only if one can show at the end that there exists an (anomalyfree)
representation theory). It has never played a role in relativistic particle
physics which starts with the 1939 Wigner representation theory of irr. positiv
energy representations of the Poincare group.

This brings me to the second point: I do not agree with your statement that
localization has no place in first quantization. The Wigner theory admits two
localizations concepts, the so-called Newton-Wigner localization and the
spatial version of the modular localization (Fassarella-Schroer J.Phys. A:
Math. Gen.35 (2002) 9123). The former (the adaptated Born-Localization of QM)
is only asymptotically covariant and local and plays an absolutely crucial role
in the formulation of scattering theory where only the asymptotic behaviour
matters. The modular localization on the other hand is local and covariant
throughout but the expectation values to which it leads have no propability
interpretation in terms of quantum mechanical projectors (as e.g. obtained from
The N-W position operator), see again the previous reference and its connection
with the old Hegerfeldt error. It is this localization which preempts the
localization of QFT as well as the spin&statistics connection (see Mund’s
papers quoted in M-S-Y math-ph/0402043).

However the localization concept of first quantized QM is not as much intrinsic
as in QFT (especially the net formulation with its spacetime indexing) since
one cannot make general phase space canonical transformation without wrecking
localization i.e. one has to identify the physical q’s from the outside by hand
in order to compute the correct scattering matrix.

In fact it is one of the great achievments (Haag’s contribution) that thanks to
causal locality of observables the localization of charge carrying objects is
totally intrinsic. Given the structure of the observable algebras one can
compute the statistics of charge-carrying fields and at the end demysrify
completely the origin of inner symmetries (the D-R theory). Another result is
that in order to describe the most general nets of algebras one never has to go
beyond semiinfinite stringlike generators (the spatial localization version for
states is discussed in Brunetti-Guido-Longo see math/ph last year). Our (MSY)
zero mass infinite helicity states is a field theoretic version of BGL. You
probably saw that our construction requires a (more sophisticated than string
theory) description since one has to amalgamate Minkowski spacetime with De
Sitter (in one dimension less). String-localization is despite zero mass
incompatible with conformal symmetry (the Casimir invariant kappa breaks
conformality); there is no relation whatsoever to “tensionless strings”. These
string-localized fields A(x,e) also exist in the free massive case but there
they are not needed, the standard pointlike free fields generate the same
spaces. Their use may be interesting for implementing stringlike interactions.
My paradoxical that string-localized fields have no Lagrangian quantization and
that those objects obtained from quantization of string Lagrangians loose their
classical string properties upon quantization only sounds paradoxical to you
because you have internalized the tight connection between pointlike classical
and quantum fields to such a degree that you apply it outside pointlike
localization. But this historically so important connection ends with pointlike
localization. Hence if you baptize objects by a Lagrangian name you should not
be surprised that this name has no physical meaning on the quantum side. In
retrospect it is an instant of underserved look the Jordan’s idee of Quantelung
der Wellenfelder” worked and that the development of pointlike QFT did not have
to wait for Wigner’s intrinsic approach to particles.

Bert Schroer.

One word concerning my remark that string localization does not matter for
Polshinski etc “string theory. That theory is an S-matrix theory and the string
aspect entered through a completely formal observation: the cooking recipe for
that crossing (but in the sense of infinite particle towers and not in the QFT
sense) becomes easier communicable if one thinks about that Lagrangian, it is
completely void of physical meaning and perhaps it would have been better and
less confusing if one would have used the Big Latin Letters from the very
beginning. Of course this does in no way diminuish the interest in the
Pohlmeyer charges as a model for an integrable representation theory even if
one does not yet know their physical use.

Bert Schroer

[end forwarded mail]

Posted at March 30, 2004 9:31 AM UTC

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### Re: Notions of string-localization

I never understood modular theory so I cannot say anything on that but let me mention one thing: I agree that target space localization of strings should better be addressed in string field theory rather than from a world sheet perspective but of course that quite hard. There are these results by Dimock that at least in the free case the infinite tower of string states appears to be localized at the string centre of mass. But one should be careful to generalize that result to the interacting theory.

What makes this even more difficult is that in the end locality should be with respect to the dynamical metric and not wrt the background target metric. But this should of course only be asked of gauge invariant observables and not direktly for the fields you get by linearly expanding the field equation (gauge dependent fields typically are not local, ie do not commute at spacelike separation). So this question is really infinitely hard to address.

One other thing that comes to my mind: In Henning Rehren’s version of AdS/CFT, at least one of the theories (in the bulk or on the boundary) cannot be generated by pointlike fields. This is because the intersection of all wedges on the boundary that contain a diamond in the bulk vanishes. Thus there have to be some operators that are localized like strings or wilson lines.

Posted by: Robert on March 30, 2004 11:49 AM | Permalink | Reply to this

### Re: Notions of string-localization

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.

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]

Posted by: Urs Schreiber on March 30, 2004 4:16 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

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?

Posted by: Urs Schreiber on April 1, 2004 12:04 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

Bert Schroer writes:

[begin forwarded mail]

Dear Urs,

in some sense you are trying to deemphasize the difference between Pohlmeyer strings and those used for the construction of the S-matrix of GSW-Polshinski S-Matrix (Polyakov strings if written in the functional form). But are these objects not fundamentally different (despite the fact that they both claim to be in some way related to the Nambu-Goto string by different quantizations)?

The standard string only exists in d=10,26 whereas the P-strings (as well as our string-localized fields) exist in any spacetime dimension d>1+1. The prediction of spacetime dimensionality which string theorist claim as their greatest success (see a recent paper of Deser) is absent in Pohlmeyer’s approach. String theorist were even able to convince experimentalists to look for the small curled up dimensions; Pohlmeyer would never propose such a weird idea. In the introduction of your paper you say that canonical data and Virasoro constraints of Polyakov strings and Pohlmeyer strings are the same. Is this consistent with their significant difference?

Bert Schroer

[end forwarded mail]

Posted by: Urs Schreiber on March 30, 2004 1:03 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

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.

Posted by: Urs Schreiber on March 30, 2004 7:06 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

Barton Zwiebach writes on sci.physics.strings:

[begin forwarded post]

My knowledge on the matter indicates that the quoted statement by Urs is not believed to be correct:

Apparently in string field theory string fields commute if the center of mass of the strings is spacelike seperated, irrespective of the oscillation and spatial extension of the string.

I have not done work on this, but the last reference I know is from Hata and Oda (hep-th/9608128), who cites earlier work and claims that one gets a vanishing commutator for open string fields $\Phi \left(X\right)$ and $\Phi \left(\stackrel{˜}{X}\right)$, when

(1)$\int d\sigma \left(X\left(\sigma \right)-\stackrel{˜}{X}\left(\sigma \right){\right)}^{2}>0.$

It does not suffice that the CM’s be spacelike separated to satisfy this inequality. If the strings are fully spacelike separated (any two points, one on each string, are spacelike separated), the inequality is satisfied.

(2)$\left(X-\stackrel{˜}{X}\right)\left(\sigma \right)=\delta {x}_{0}+\sum _{n}\delta {x}_{n}\mathrm{cos}\left(n\sigma \right)$

The inequality is then roughly

(3)$\left(\delta {x}_{0}{\right)}^{2}+\sum _{n}\left(\delta {x}_{n}{\right)}^{2}>0$

which shows that spacelike separated CM’s $\left(\delta {x}_{0}{\right)}^{2}>0$ does not suffice.

Much of the causality questions have been studied in the light-cone gauge. I am not clear about the role of reparameterization invariance when statements are made about covariant closed string fields.

Best, Barton.

[end forwarded post]

Posted by: Urs Schreiber on March 30, 2004 6:29 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

Bert Schroer writes:

[begin forwarded mail]

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). One can criticize this result on physical grounds because the tachyon (which could worsen the localization, but it cannot convert a point into a string) has been omitted. The adaptation of this lightfront gauge calculation for the superstring would remove any residual doubt. Dimock could not perform the calculation in the covariant gauge and one has the impression that the outcome could be a complete delocalization.

Concerning the d=10,26 issue it should be pointed out that even if one takes the canonical framework there is one loophole because the calculation establishes the correct commutation relations of the Lie algebra of the Poincare group but the transition to the unitary representation of the group is a delicate buisyness (see Rehren’s remarks) if the group is noncompact (check of Nelson domain properties) and I could not find a reference where this was done.

To Urs: 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 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. Rather it was always known and appreciated by the mathematical physics community. Let me add some remarks about some fundamental conceptual differences between “string-localized fields” and “string theory” (standard or a la Pohlmeyer).

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 (operators with extended localization exist of course in every standard theory). Rather they were imposed on physicists by the already existing causality principles: in certain cases they require objects with noncompact localization in order to fully unfold.

Take the example of the Wigner 1939 zero mass infinite helicity representation. The field theoretic realization of the infinite helicity tower within the AQFT setting just requires the semiinfinite spacelike extension; it is the sharpest localization by which you can generate those Wigner states. These objects A(x,e) have vacuum fluctuations in Minkowski- as well as in De Sitter spacetime. It is a pity that Weinberg in his famous book delegates the Wigner theory to a service role for obtaining additional arguments in favour of the Lagrangian quantization approach rather than following its intrinsic logic. This of course explains why he missed to understand the infinite helicity tower. He dismisses that interesting family of representations by saying that “nature does not make use of it” (as if this would be the obligation of a theorist!) but of course whether something is physically reasonable or notcan only be decided after knowing its content. In d=1+2 there is another mechanism which requires string-localized fields: the spacelike infinity (a 2-dim De Sitter spacetime) is not simply connected and a value of spin different from (half)integer “activates” the (Bargman) covering group of SO(2,1) as well as the spacelike region at infinity and lead to braid group statistics. This time the “plektons” (anyons if abelian) do not carry any string degree of freedom (like the helicity tower of before); in this respect they are similar to Bosons/Fermions.

By the way every QFT living on covering space (this includes the timelike covering in conformal QFT) is automatically interacting (each subwedge localized operator applied to the vacuum creates in addition to one particle states a vacuum cloud, i.e. there is nothing resembling
free fields). So any nonperturbative constructive approach to higher dimensional field theory should start from this observation of a natural built in interaction which does not have to rely on extraneous concepts as Lagrangian quantization. Anyons are special illustration of so-called Buchholz-Fredenhagen strings but in higher dimendion there is presently no good intrinsic understanding of the mechanism which gets them going (the analog of the covering aspects). One can also imagine closed string localized objects arising in AQFT but their raison d’etre would be quite different (operator 2-cohomology through the possible violation of Haag
duality for spacetime tori localized algebras) Whereas string-localization is a discovery that the principles allow
more general realizations than the very narrow Lagrangian setting, string theory is an invention which starts with Veneziano’s nice use of properties of Gamma functions. It would be interesting to shed a new light on the history (because many string theorists do not know their own history) but this would require a separate Forum.

I think most people in this forum will agree with me that particle physics is in a deep crisis (actually the most serious crisis since I entered professional life in the 60’s). Just look at the situation in this forum, I (and of course even more Henning Rehren) understand (but not always agree with) what the majority is doing. But most of the participants (see the remarks of Robert Helling) have no access any more to what I (and some other people) am doing.

Bert Schroer

[end forwarded mail]

Posted by: Urs Schreiber on March 31, 2004 3:37 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

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 ${\stackrel{˜}{A}}_{m}^{-}$.

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

(1)$\left[{\stackrel{˜}{A}}_{m}^{-},{\stackrel{˜}{A}}_{n}^{-}\right]=\left(m-n\right){\stackrel{˜}{A}}_{m+n}^{-}+\frac{26-d}{12}{m}^{3}{\delta }_{m,-n}$

(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.

Posted by: Urs Schreiber on March 31, 2004 4:43 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

I have had some more discussion with Bert Schroer and Jon Dimock in private.

Jon Dimock says that his result is fully compatible with those of Martinec and others, which were mentioned by Barton Zwiebach. He says that spacelike separation of the centers of mass is sufficient but not necessary for two string fields to commute and that that’s what he demonstrated.

This is however in contradiction to what Barton Zwiebach said here.
(?)

Posted by: Urs on April 5, 2004 10:01 PM | Permalink | Reply to this

### Re: Notions of string-localization

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]

Posted by: Urs Schreiber on April 6, 2004 8:10 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

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.

Posted by: Bert Schroer on April 21, 2004 4:09 PM | Permalink | Reply to this

### Re: Notions of string-localization

Regarding the localization/causality issue I think that two point are important:

Whenever we are working in terms of Lorentzian worldsheets, as we certainly can at least for non-interacting strings, then proper causality is obvious, since the 2D CFT is perfectly causal in a physically totally satisfying way.

As soon as interactions are switched on the true issue of physical causality becomes much more subtle than is apparent in the papers that we have discussed here, since, as for instance Robert Helling has pointed out above, the causal structure of the background will fluctuate, too. Lacking a proper understanding of closed string field theory thus unfortunately prevents any true settling of this question.

But I do agree that you have a point that the result obtained in open bosonic string field theory for a flat background causal structure concerning the causal dependence of string fields looks physically problematic. I’d attribute this to the inability of open string field theory to correctly account for fluctuiations of the metric.

Concerning the quantization of the string:

I am pretty surprised to see that you argue that the standard quantization might be flawed. In fact, I’d consider the fact that the Pohlmeyer program can be solved (at least I think it can) by mapping it to the usual quantization reassuring.

But I am glad that the Pohlmeyer/DDF discussion is considered by some to be relevant to the general question of what the ‘alternative quantization’ of LQG might actually be. (On the other hand, I don’t quite get it. Thomas Thiemann’s LQG string would have one controversial step less if it used DDF instead of Pohlmeyer invariants, which wouldn’t affect anything else in the paper.)

But of course everybody should have his own prejudices concerning theoretical physics. If some people feel that the standard quantization of the string can be circumvented it would be nice to clarify the issue. For instance it would be nice to have some theorem on the question whether the Pohlmeyer program admits more than one solution, or something like that. Unfortunately I am not aware of anything in this direction. All I know is that the one solution known (in my opinon), namely the standard quantization, is consistently checked from many different point of views, like path integral and/or BRST quantization, etc. Seems to me that any attempt at an ‘alternative’ quantization to the path integral quantization would be on very shaky physical grounds.

Posted by: Urs Schreiber on April 28, 2004 6:21 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

Whenever we are working in terms of Lorentzian worldsheets, as we certainly can at least for non-interacting strings, then proper causality is obvious, since the 2D CFT is perfectly causal in a physically totally satisfying way.

The causality at issue here is not that of a chiral conformal field theory but that in target space, which makes even noninteracting strings problematic. I agree that interactions can only make things worse, but what means “worse” if the starting situation is already bad. In addition, non of this conformal stuff is there any more once one implemented the requirement $c=0$.

I am pretty surprised to see that you argue that the standard quantization might be flawed. In fact, I’d consider the fact that the Pohlmeyer program can be solved (at least I think it can) by mapping it to the usual quantization reassuring.

I am not saying that it is flawed. It is simply not the quantization of the Nambu-Goto string, but it is the right auxiliary formalism which reproduces the dual model S-matrix which defines the content of string theory. The Pohlmeyer approach to the Nambu-Goto Lagrangian is an intrinsic way to attribute a quantum system to that Lagrangian since it follows the well established way of placing the integrability at the center of quantization. All experience with quantization points to its uniqueness i.e. there exists only one quantum system to a classical Lagrangian which deserves this name. I would think that the physical content of Pohlmeyer charges has nothing to do with string theory. If “doing something” to the Nambu-Goto Lagrangian helps to set up a nice recipe for the construction of an S-matrix (of the dual model), then this serves its purpose. If your DDF operators, which form a subset of operators in the unphysical Hilbert space in the canonical quantization are able to reproduce the full S-matrix of string theory, well this is fine; you do not want a relation to the Pohlmeyer charges because everything indicates that their (not yet understood) quantization has nothing to do with string theory (a knowledge about the representation theory of the P-charges would make some people including Thomas Thiemann very happy). What you want is a Lagrangian wrapping or some other setting which leads to the rules of splitting and combining euclidean tubes which seems to be the recipe of doing interacting string theory.

But of course everybody should have his own prejudices concerning theoretical physics.

It shouln’t be like this, but unfortunately things have developed in this direction and string theory has its fare share. The hasty naming of things without having properly understood their content (there is unfortunately no patent office to check on the content in theoretical physics) has led to an ever increasing gap between name and content. As a result of this, one ends up with contradictory sounding but nevertheless correct statements like: string theory deals with objects which are not string-localized in the quantum sense.

Posted by: Bert Schroer on April 29, 2004 1:19 PM | Permalink | Reply to this

### Re: Notions of string-localization

There is some related discussion over at sci.physics.research, which I’ll branch off to the Coffee Table in order to make contact with the discussion with Bert Schroer:

When reading Bert Schroer’s latest epos, hep-th/0405105,

I don’t understand everything that Schroer discusses in this paper, but some things I do understand, in particular the issues discussed on the first half of p. 19, and I’d like to comment on these. Some of these arguments have already been exchanged at

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

which Bert Schroer cites as [48], but aybe it is worthwhile to repeat some of the points in the context of s.p.r. (I’ll cc this to Bert Schroer):

Bert Schroer argues that the Pohlmeyer approach to string quantization has advantages over the standard approach in that

there is no mysterious and unphysical distinction (resulting from the non-intrinsic canonical quantization) of d=10,26 rather they exist as Poincaré invariant theory in each dimension $>3$.

(hep-th/0405105, p. 19)

But as far as I am aware this claim is not known to be true. As Bert Schroer acknowledges himself a couple of lines above, the task of of demonstrating the truth of this statement

has been almost completed

(hep-th/0405105, p. 19)

and only almost, I must emphasize, namely up to the so-called ‘quadratic generation hypothesis’. I am not saying that the converse can be proved, but as it stands the claim that there is a quantization of the Nambu-Goto action which does not exhibit the critical dimension is in fact a speculation. And not everybody agrees on how likely it is that this speculation can be proven.

In fact, as I have said before, I claim that there is one known solution of the Pohlmeyer program, as indicated in hep-th/0403260, and that this leads to the just the usual quantization and in particular the usual critical dimension. In the light of this the circumstantial evidence that there is no alternative quantization of the string seems to me to be rather stronger than that (namely which?) pointing in the other direction.

Furthermore, Bert Schroer argues that the fact that open string field theory in light cone gauge exhibits a notion of string localization which seems to violate a certain intuition. Namely for two string fields to commute it is apparently sufficient for the “center-of-mass” of the respective strings to be spacelike seperated. He argues that this shows that there is no ‘intrinsic’, as he calls it, notion of string (with spatial extension) in standard string theory.

Several comments to this point have been made at the String Coffee Table linked above, but maybe one point has not been emphasized, namely that open string field theory is indeed non-local, as can be seen explicitly by looking at the OSFT action in component form (for the first couple of levels). This is given for instance as equation (2.46) in the review hep-th/0102085, where it is crucial to note the appearance of the nonlocal tilded fields defined in equation (2.44).

I learned about two references that might be of interest: hep-th/0305093 and hep-th/0402212. Not unexpectedly, Schroer seems to think that these papers are nonsense, the former more than the latter.

I haven’t read all that, but I would like to know what is concidered nonsense and for which reasons.

Posted by: Urs Schreiber on May 18, 2004 4:11 PM | Permalink | PGP Sig | Reply to this

### Re: Notions of string-localization

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).

Posted by: Bert Schroer on May 18, 2004 4:15 PM | Permalink | Reply to this
Read the post String Localization, Once Again
Weblog: The String Coffee Table
Excerpt: Recap of string localization.
Tracked: March 29, 2006 8:28 AM

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