## March 29, 2006

### String Localization, Once Again

#### Posted by Urs Schreiber

In

Bert Schroer
String theory and the crisis in particle physics
physics/0603112

the main technical argument, apart from several sociological arguments ($\to$), is a certain subtlety in the commutators of string fields.

Bert Schroer interprets the nature of these commutators as saying that from the intrinsic point of view of quantum string field theory strings are pointlike, and this he regards as a fatal flaw, as far as I understand.

We had discussed this at great length a while ago with Bert Schroer himself here on the Coffee Table ($\to$).

The relevant literature ($\to$) is

E. Martinec
Strings and Causality
hep-th/9311129

as well as

H. Hata, H. Oda
Causality in Covariant String Field Theory
hep-th/9608128

and in particular

J. Dimock
Local String Field Theory
math-ph/0308007 .

At least superficially, there appears to be a certain disagreement between these results. According to the last paper, two free, bosonic string fields have vanishing commutator if the center of mass of the two strings described by the two fields are spacelike seperated.

The first two papers, however, seem to come to the conclusion that the excitations of the string (its spatial extension) also enters the string field commutator. In our last discussion, this was confirmed by Barton Zwiebach ($\to$).

I must admit that I have not taken the time to look at this closely enough to sort this out properly. I had asked J. Dimock about this ($\to$), who replied that there is no contradiction.

One obvious but easily overlooked fact at least seems to be important to note:

What is usually called the center of mass of the string is not in any way an intrinsic, invariant quantity.

What is usually called the “center of mass” is really the “center of coordinate density”, if you like. It depends on the parameterization of the string and hence on whether or not any gauges have been fixed.

Of course, when the string is quantized in the usual way, there is a natural and obvious choice of “center of mass mode” which does play the role that one would expect naively. But in particular if one is interested in “intrinsic” properties of quantum systems, one should probably be careful with interpreting any condition that involves the center of mass of two strings too literally.

In any case, even if we agree that two free string fields commute precisely when their respective “centers-of-mass” are spacelike seperated, I don’t quite see why this is supposed to be problematic.

I think the question is this: are we facing technical inconsistencies or just inconsistencies with our intuitive expectations of what technical results should look like?

Posted at March 29, 2006 7:48 AM UTC

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### Re: String Localization, Once Again

Is this just another quantum gravity issue with regards to good observables.

I mean do even *classical* strings on a curved manifold, have spacelike vanishing commutators that don’t depend on gauge choices in general?

I wouldn’t expect them too.

Posted by: Haelfix on March 29, 2006 12:15 PM | Permalink | Reply to this

### Re: String Localization, Once Again

Is this just another quantum gravity issue with regards to good observables.

I don’t think so. For starters, we can and should restrict attention to open string field theory (not much of a choice, really). So there is no gravity complicating issues here.

I mean do even classical strings on a curved manifold, have spacelike vanishing commutators

Not sure what you mean by the commutator of a classical string.

What makes this question of string localizability hard is that it requires quantum string field theory, i.e. genuinely second quantized strings. Most of the literature on string field theory is only concerned with the classical string field action. But doing the full thing quantumly should lead to commutation relations for string fields, analogous to those for ordinary quantum fields.

I am not aware of any detailed understanding of quantum string field theory. But maybe I am just being ignorant.

Posted by: urs on March 29, 2006 12:32 PM | Permalink | Reply to this

### Re: String Localization, Once Again

I should have said one loop truncated gravity with say the Nambu Goto action.

But yea I understand the current technical limitations with quantum string field theory.

Posted by: Haelfix on March 29, 2006 1:23 PM | Permalink | Reply to this

### Re: String Localization, Once Again

I should have said one loop truncated gravity with say the Nambu Goto action.

Sorry, just to make sure, are you talking about worldsheet gravity?

Posted by: urs on March 29, 2006 1:29 PM | Permalink | Reply to this

### Commutators

In QFT, non-gauge-invariant operators need not (and, frequently, do not) commute outside of the lightcone. The commutators of gauge-invariant operators, however, must vanish outside of the lightcone. This is required for micro-locality.

Long before you were born, our forefathers realized that the requirements of unitarity, Lorentz-invariance, and micro-locality could be codified in terms of the analytic properties of the S-matrix.

Veneziano constructed his famous S-matrix as an example satisfying these required analytic properties. Ergo it, and all the string-theoretic constructions which followed from it, satisfy the requirements of micro-locality.

(It is worth noting the recent paper of Adams et al, which I talked about on my blog, which makes very explicit the connection between the analytic properties of the S-matrix and the absence of superluminal signaling.)

Studying commutators of non-gauge-invariant operators (like the string field in string field theory) sheds no light whatsoever on the issue of micro-locality in string theory.

Schroer, presumably, knows this, too. But, then, his paper would be much shorter if he cut out all the obviously wrong stuff.

Posted by: Jacques Distler on March 29, 2006 3:05 PM | Permalink | PGP Sig | Reply to this

### Re: Commutators

It is hard to say for me what precisely Bert Schroer is concerned about. It seems that he argues precisely the other way around. Like “if we have micro-locality, there are hence no intrinsically extended objects in the theory”, where intrinsic is supposed to mean something like “as measured by some genuine quantum observable” or the like. He seems to be considering his “string-localized fields” as a counterexample.

In any case, as I said, my point was that the fact that some string fields do or do not commute in dependence of their center-of-mass position is not a relevant question, as you confirm.

Posted by: urs on March 29, 2006 3:33 PM | Permalink | Reply to this

### Re: String Localization, Once Again

In my paper I do not say anywhere that string theory has a fatal flaw as a result of localization properties. I only lament a bit that for a newcomer the terminology “string” may be misunderstood to mean that the objects the theory deals with are localized along strings in Minkowski spacetime in the same (quantum) sense as standard quantum fields are localized at a point. To be sure, there are objects (string-localized fields) which only exist in a string-localized form, but they have nothing to do with string theory (they are mentioned with references); that they have infinite helicity towers instead of mass towers seems to be purely coincidental.
I also want to point out that I have nothing against string theory per se, it is the social concommitant phenomenon of string theory which constitutes the backbone of may essay (this is also why I posted it in physics/ it only got to hep-th/ by crossing). I find it conceivable that this sociological phenomenon could have happened with another subject (if famous people especially Nobel prizes give their blessing), infact I allude that it may be manifestation of the Zeitgeist; actually towards the end of my paper I mention such a case (although that one is cooking on a much smaller flame). I am completely in agreement with Fredenhagen, Rehren and Seiler that one only can know what something is supposed to be after one obtained an intrinsic understanding (it does not have to be complete, just a hint). I go beyond these three authors by asking the question that given these shortcomings, what is the reason for its popularity; I seriously think there is something to be understood here. Although I use in my article profound (I think) scientific arguments, I am unable to do this in a scientific style as FRS and since I want to be honest to myself I call it a scientific polemic (in the tradition of Jost) and posted it on physics and society. The FRS paper is really a scientific review whereas my is a scientific polemic (in which I only became personal on one occasion where I did not see any rational way of arguing). The catalyst for writing it at all was of course Susskin’s manifesto and I am astonished that this has not led to any discussion within the string community.
Hoping that this may remove some of the misunderstandings

Bert Schroer

Posted by: Bert Schroer on March 29, 2006 6:42 PM | Permalink | Reply to this

### Huh?

In my paper I do not say anywhere that string theory has a fatal flaw as a result of localization properties. I only lament a bit that for a newcomer the terminology “string” may be misunderstood to mean that the objects the theory deals with are localized along strings in Minkowski spacetime in the same (quantum) sense as standard quantum fields are localized at a point. To be sure, there are objects (string-localized fields) which only exist in a string-localized form, but they have nothing to do with string theory (they are mentioned with references); that they have infinite helicity towers instead of mass towers seems to be purely coincidental.

Even after reading your explanation, I am not sure I know what the heck you are talking about.

So let me try to elicit a clarification. For definiteness, let us focus on Witten’s covariant open string field theory. The string field is a functional

$\Phi \left[{X}^{\mu }\left(\sigma \right),b\left(\sigma \right),c\left(\sigma \right)\right]$

Is that an object which is “localized along strings in Minkowski spacetime in the same (quantum) sense as standard quantum fields are localized at a point”?

(Presumably, since it depends not just on ${X}^{\mu }\left(\sigma \right)$, but also on the ghosts, $b\left(\sigma \right),c\left(\sigma \right)$, perhaps we should say “localized along strings in some superspace which includes $b,c$ as odd coordinates.”)

If not, why not?

If yes, do you have some complaint about the implementation of micro-locality in string theory, based on the fact that the string field is localized along some 1D extended object, rather than at a point?

If you do have such a complaint, please take note of the gauge-invariance

$\delta \Phi =Q\Lambda +\Phi *\Lambda +\Lambda *\Phi$

Commutators of gauge-variant operators need not (and, typically, do not) vanish outside the lightcone.

Posted by: Jacques Distler on March 30, 2006 12:54 AM | Permalink | PGP Sig | Reply to this

### Re: Huh?

[…] vanish outside the lightcone.

But the light cone of what? Of a point? (The string’s center of mass, for instance?) Of the entire string (being the union of the light cones over all of its points)?

I believe Bert Schroer wants to regard some object as being “intrinsically” a “string field” of some sort only if its commutators vanish at most outside the light cone(s) of an (two) entire string(s).

Posted by: urs on March 30, 2006 8:11 AM | Permalink | Reply to this

### Re: Huh?

I believe Bert Schroer wants to regard some object as being “intrinsically” a “string field” of some sort only if its commutators vanish at most outside the light cone(s) of an (two) entire string(s).

Which is a false expectation, because the string field is not gauge-invariant.

All of these calculations, as Haelfix notes, are horribly gauge-dependent. I don’t think there’s any meaningful content you can extract from them.

A gauge-invariant statement is the one I made above, about analyticity properties of the S-matrix. If Schroer has a gauge-invariant question to raise, it would be fun to discuss it.

Otherwise, there is an infinite amount of time that can be wasted, computing the commutator of two string fields in different gauges and attempting to compare the results …

(N.B.: in response to Haelfix’s first comment: this is not a problem of quantum gravity, per-se. It’s a problem that occurs in any gauge theory, including open string field theory, which is where we are discussing it).

Posted by: Jacques Distler on March 30, 2006 3:56 PM | Permalink | PGP Sig | Reply to this

### Re: String Localization, Once Again

Urs, yes worldsheet gravity. But anyway Jacques more or less answered what I was struggling to ask. Note its already unclear in Minkowski space, nevermind dealing with something else.

I just read Dimocks paper, and its fairly clear I think. In light cone gauge there is no problem with finding good local observables (equation 49) with respect to the *string lightcone* (page 13), but in the covariant case its not clear (even with various additional positive representation constraints).

So I’ll just reiterate what Urs already said in his opening, it seems all of this are artifacts of what gauge you pick rather than some deep intrinsic understanding.

Posted by: Haelfix on March 30, 2006 9:48 AM | Permalink | Reply to this

### Re: String Localization, Once Again

In light cone gauge there is no problem with finding good local observables (equation 49) with respect to the string lightcone (page 13), but in the covariant case its not clear

And it makes sense when compared with the formula given by Hata and Oda ($\to$), which states that a sufficient condition for the covariant string fields at strings $X$ and $\stackrel{˜}{X}$ to commute is that the difference

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

satisfies

(2)$\left(\delta {x}_{0}{\right)}^{2}+\sum _{n}\left(\delta {x}_{n}{\right)}^{2}>0\phantom{\rule{thinmathspace}{0ex}}.$

If we naively apply light cone gauge to this formula the contribution ${\sum }_{n}\left(\delta {x}_{n}{\right)}^{2}$ becomes non-negative and we indeed get that spacelike CM seperation is sufficient.

Posted by: urs on March 30, 2006 1:38 PM | Permalink | Reply to this

### Re: String Localization, Once Again

(I sent these comments several hours ago. If they were not lost and still arrive, please remove this copy.)
One more attempt to get some light into the exasperating topic of localization in string theory, although I only raised this issue in connection with the opening mantra of popular string theory talks and with a bit of confusion this terminology creates with newcomers (the chosen terminology is never a matter of life and death concerning the object you want to define, it may however have some bearing on its intrinsic characterization, the big open problem in string theory).
Dimock’s work was done on the bosonic string and he had to cut out the tachyon (i.e. an object with a notorious relation with respect to localization) and hence it would be nice to do it with the supersymmetric string (e.g. start from the nice presentation of Griorge hep-th/0506100 which is very much in the spirit of Dimock’s) to be absolutely on the safe side.
Looking in somewhat more details of Dimock’s result there seems to be a fine point: depending on the internal state of the string the vanishing of the commutator extends into the timelike region up to an invariant “mass” whose position depends on the internal state i.e. the object is “super-localized” in the pointlike sense. This may serve as a reminder that one is dealing with an object with a rich supply of internal degrees of freedom but it is even further away from a string in spacetime. Again I have no problem with this, to the contrary I find it very interesting since it must be a special object of AQFT since it is Einstein-causal and fulfills the spectrum condition (and everything in AQFT which looks like an elephant is really an elephant); the interesting problem would then be to find it in that huge set which comprises local quantum physics. There is of course (as Distler points out, I think) the possibility that the equivalence between the lightfront- and the covariant quantization breaks down, but this in itself would be a very startling issue.
I continue to be surprized that the issue of Susskind’s manifesto (without this I would not have written my article) has not received any commentary in the community.
Bert

Posted by: Bert Schroer on March 30, 2006 12:34 PM | Permalink | Reply to this

(I sent these comments several hours ago. If they were not lost and still arrive, please remove this copy.)

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Posted by: urs on March 30, 2006 1:04 PM | Permalink | Reply to this

### Re: String Localization, Once Again

Here is a remark I forgot to add.
There is a very simple and concrete illustration of what I (and also Fredenhagen, Rehren and Seiler) mean by “intrinsic”. Consider the Nambu-Goto string (just because of its simplicity). The quantization (e.g. take the lightcone gauge) gives a local field which is really a collection of infinitely many irreducible representations of the Poincare group (the mass tower) where the strength of the different components is governed by the internal state of the string. The question about intrinsicness is: what is the physical principle which selects the tuning of the relative strength of the components i.e. what characterizes this particular “tower” of particles (which you insist should be placed together) from any other more general way of doing this? (The extrinsic answer is of course it happens to come out in the canonical quantization of the N-G string).
Contrast this with the helicity tower of the family of massless infinite helicity Wigner representation. Tis representation is irreducible so the above problem of highly reducible representations does not enter. In this case the string-localization is a consequence of the more basic positive energy Poincare representation requirement (http://xxx.lanl.gov/abs/math-ph/0511042).
In fact this huge family is besides the (m,s) and finite helicity representations (0,h) the third big class of positive energy representation (which Weinberg in his book has dismissed as unphysical without giving any theoretical reason why), it depends on a continuous “Euclidean” mass parameter of the little group. Already Wigner noticed that these objects have very weird thermal properties (see remarks and references in: http://xxx.lanl.gov/abs/math-ph/0402043). They also do not allow a Noether type energy momentum tensor. But who knows, these may be just the carriers of dark matter/energies. It would be very strange if mother nature does not make use out of this third big positive energy representation class (unless one still finds a reason why this third class is “pathological”). Nature seems to follow principles and not recipes and formulas, and quantum nature does not need the helping hand of a classical description; Wigner’s theory may be very limited but it is genuinely “intrinsic”.
Finally I have a question (in particular to Urs Schreiber, since he seems to have experience with conceptual and mathematical questions). Venezianos dual S-matrix model leads to a crossing symmetric one-particle tower (where all the particle poles enter with the correct sign, so that there are no ghosts) description which is well-defined in any spacetime dimension (??). It is the insistence to read the Veneziano construction as a canonically quantized N-G string which filters out the dimensions?? If the answer is yes, then my question would be: why does one do this (see above)?

Posted by: Bert Schroer on March 31, 2006 6:48 AM | Permalink | Reply to this

### Veneziano

Venezianos dual S-matrix model leads to a crossing symmetric one-particle tower (where all the particle poles enter with the correct sign, so that there are no ghosts) description which is well-defined in any spacetime dimension (??). It is the insistence to read the Veneziano construction as a canonically quantized N-G string which filters out the dimensions??

Veneziano’s S-matrix was for just $2\to 2$ scattering of the lowest members of the tower (“tachyons”). By itself, examining the residues of the poles in Veneziano’s amplitude is not sufficient to ensure that all states in the theory have positive norm.

Indeed, as Lovelace first noticed, they do not, unless ${m}^{2}$ of the lowest member is chosen correctly, and we are in the critical dimension.

To see this, however, requires some procedure for generalizing Veneziano’s S-matrix, either to arbitrary n-point tachyon S-matrix elements, or to S-matrix elements for other members of the tower.

You might object, at this point, that perhaps some other generalization (other than string theory) of Veneziano’s S-matrix is possible.

If so, you are invited to try to write one down …

Posted by: Jacques Distler on March 31, 2006 8:51 AM | Permalink | PGP Sig | Reply to this

### Re: Veneziano

This is really very interesting. Presumably Veneziano could have played his gamma/beta function game with a 2–> 2 particle in the supersymmetric context whose Lovelace extension would have then led to the tachyon-free superstring. In this case one deals with ordinary particles, and whatever extension to n—>m particles one takes (in case it is nonunique) the multiparticle S-matrix would have to fulfill the cluster properties. It would be interesting to know whether this holds for the superstring (it would be amazing if Lovelace also checked it in a nontrivial case because this seems to me quite demanding since it apparently mixes different geni amplitudes).

Posted by: Bert Schroer on March 31, 2006 11:36 PM | Permalink | Reply to this

### The super case

Presumably Veneziano could have played his gamma/beta function game with a 2→2 particle in the supersymmetric context…

Much harder. In the super case, the scalar (the tachyon) is projected out, and the lowest-mass member of the tower is a gauge-boson. The $2\to 2$ gauge boson S-matrix element has a much more complicated structure because, with 4 polarization vectors to play with, in addition to the momenta, there are 16 independent Lorentz-invariant kinematical variables, instead of the 2 that appear in the scalar case.

Correspondingly, guessing the form of the 4-gauge boson S-matrix is much more difficult. A ferry ride would not have sufficed.

Posted by: Jacques Distler on April 1, 2006 1:30 AM | Permalink | PGP Sig | Reply to this

### Re: The super case

I accept that this is technically quite demanding but do you have serious doubt that a Veneziano amplitude could have been obtained this way? One could of course extract from the knowledge of the superstring such an amplitude and see whether their is a chance to get it within the Veneziano rational.

Posted by: Bert Schroer on April 1, 2006 11:37 AM | Permalink | Reply to this

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