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This question came to mind while reading about Weinberg's folk theorem that any quantum theory that is Poincare covariant and satisfies cluster decomposition will look like a quantum field theory at low energies, the caveat being that at high energies, string theory stands as a counterexample. In fact, Lubos Motl claimed here: Any resources for string theory using algebraic quantum field theory? that the axiomatic formalism of AQFT is simply incapable of describing string theory. This is surprising to me, because the Haag Kastler axioms seem to merely encode the very minimal conditions of locality, causality, and covariance. These are physical principles that are seemingly universally applicable in physics. Therefore, I would like to know if there are particular aspects of this set of axioms which a hypothetical nonperturbative string theory would struggle to satisfy.

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String theory is not an (algebraic) QFT because it doesn't take place on a fixed spacetime. The Haag-Kastler axioms are of the form "given a spacetime manifold $X$, then...", but string theory doesn't (have to) work that way.

See this answer of mine for a longer discussion of the ontology of spacetime in string theory. The upshot is that string theory "generates" the spacetime as the target space of the non-linear $\sigma$-model representation of a conformal field theory, and if you deform this conformal field theory, you deform the spacetime. This type of topology change/dynamical spacetime is not part of any of the usual axiomatizations of QFT because it does not happen in QFT - if this were a normal QFT, then the base space of the non-linear $\sigma$-model would be spacetime, not the target space.

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  • $\begingroup$ Within a fixed string theory, does this deformation of spacetime occur on its own? $\endgroup$
    – Prox
    Aug 13, 2022 at 1:43
  • $\begingroup$ @Prox That's a bit like asking whether within a "fixed QFT", the parameters can change - you can't really talk about a single fixed theory (neither in QFT nor in string theory) because you have things like renormalization flows that always mean you have to consider a class of theories related to your "fixed" theory to really understand what it does. $\endgroup$
    – ACuriousMind
    Aug 17, 2022 at 9:57
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I finally found the basis of a satisying answer in the following article by Bert Schroer. https://arxiv.org/abs/1612.00003.

" Since Haag’s LQP comprises all models which fulfill the causal localization principles in a (positivity obeying!) Hilbert space setting and ST falls according its protagonists into this category, the obvious question is whether the objects of ST describe, as string theorists claim, string-localized objects in the sense of causal localization in Minkowski space. If localization in ST really means what the terminology suggests, two string operators should commute if the strings are spacelike separated (the quantum version of Einstein-causality); there is no other physical meaning which one can attribute to quantum strings localized in spacetime.

Freed from a quantization parallelism to classical physics, the LQP formulation is synonymous with a realization of causal localization principles in the context of quantum theory which means in particular that string-local operators are defined as objects in spacetime which are causally localized i.e. two string operators commute if they are relatively spacelike separated.

Causal localization is inexorably connected to vacuum polarization and the strength of the vacuum polarization clouds depend on the tightness of localization. This affect in particular the short distance scale dimensions of pl fields. If the alleged ”stringy” objects of ST bear any relation to spacetime strings they must be related to the sl fields of LQP even if they had been constructed in a different way from that of sl fields. The main point of contention is whether the objects of ST are really string-local in any with relativistic causality compatible sense.

In order to understand that string theorists use the terminology ”string” for something which bears no relation with localized quantum objects in spacetime it is helpful to look at what they are doing and understand why they think they are addressing propertie of quantum localization. Before addressing the quantization of the Nambu-Goto action or constructing their 10 dimensional ”superstring model” from the action of a particular 10 component supersym- metric $d = 1 + 1$ conformal current model string theorists it is helpful to take a critical look at their view of the quantum theoretical counterpart of particle world lines

The model is defined in terms of the relativistic action $\sqrt{-ds^2}$ but the resulting covariant classical world line has no quantized counterpart since particle operators $\vec{q}(t)$ only exist in (nonrelativistic) quantum mechanics (the nonintrinsic ”Born localization”) and the quantum theoretical description of a single relativistic particle uses Wigner representation theory. From the latter one can construct free fields which and the point-local free fields can be reformulated in terms of a relativistic action. There is simply no access to wave functions of relativistic particles in terms a quantization of actions describing relativistic world lines and hence this construction turns out to be a squib load.

The theory which describes relativistic particles is Wigner’s construction of unitary representations of the Poincar´e group which cannot be accessed by quantization of classical actions; in fact his 1939 unitary representation theory was the first successful intrinsic quantum construction of a relativistic particle theory. As we know nowadays this theory already containes the germ of causal localization10 in the form of modular localization of positive energy states which is closely related to the causal localization of fields.

Only on this level of causal localization of fields can one make contact with the quantization of pl fields (section 3). The more generic and important co- variant sl fields cannot be accessed in this way (section4). They are objects which are pure quantum in that the umbilical cord of an alleged quantization parallelism has been cut. This is our main motivation for giving much space to causal localization in an article dedicated to the memory of the protagonist of LQP which places causally locaiized operator algebras into the center stage.

This leaves the question of what remains of ST if it is not a theory of quantum strings in spacetime. An authoritative answer from somebody who has spend a good part of his professional life to understand the physical content of the Nambu-Goto action is that it describes an infinite set of conserved charges as one finds in $d = 1 + 1$ integrable QFTs. But different from integrable $d=1+1$ QFT there is no trace of anyspacetime localization in N-G models

The fusion and splitting of world sheets as a description of spacetime strings in analogy to the interpretation of perturbative Feynman graphs as coalescing and splitting of point-like particles represents an attempt of string theorists to create localized interactions in terms of classical metaphors. On the other hand the fact that this is based on misunderstandings of quantum causal localization does not invalidate the mathematical use of such constructions as an inspiration for interesting topological, algebraic and geometric constructions. "

I have additionally found a very old series of discussions between some string theorists and Schroer about the subject here: https://golem.ph.utexas.edu/string/archives/000338.html (Thanks to @Slereah for sharing this with me)

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  • $\begingroup$ Schroer is not to be relied upon in a discussion of string theory, e.g. in his paper he cites Rehren (reference 7) as an authority on AdS/CFT... He also talks about "string-localized field theory", as an authentic way to get string-like objects within QFT. I have no idea if it is formally valid or not, but I would point out that mainstream QFT already has observables associated with extended object, the famous Wilson loops, and now also "surface operators"... $\endgroup$ Jul 21, 2023 at 0:39
  • $\begingroup$ As for the way in which string theory is and is not like QFT, @ACuriousMind is correct to say that it differs in not having a fixed background space. On the other hand, it does often have a fixed asymptotic boundary similar to QFTs, e.g. string theory in flat space has an S-matrix, and scattering of strings in AdS space is similarly formulated in terms of the boundary. $\endgroup$ Jul 21, 2023 at 0:42

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