### 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]**

## 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.