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Please forgive a string theory novice asking a basic question.

Over at this question Luboš Motl gave an excellent answer, but he made a side comment that I've heard before and really would want to know more about:

Quantum field theory is the class of minimal theories that obey both sets of principles [Ed: SR and QM]; string theory is a bit more general one (and the only other known, aside from QFT, that does solve the constraints consistently).

What are the arguments that string theory is more general than QFT? I get that you can derive many different QFTs as low energy effective theories on different string backgrounds. But from my limited exposures to worldsheet perturbation theory and string field theory I can also see string theory as a very special kind of QFT. I also realize these approaches don't fully characterise string theory, but I don't know enough about their limitations to understand why the full definition of string theory (M theory?) surpasses QFT.

My naive guess would be that no, string theory can't be more general than QFT because surely there are many more QFTs which are asymptotically free/safe than could possibly come from string theory. Think ridiculously large gauge groups $SU(10)^{800}$ etc.. Since these theories don't need a UV completion into something else string theory can't be a more general framework than QFT. Logically you could also have theories which UV complete into something other than string theory (even if no such completion is presently known).

Alternately you could turn this around and say that string theory limits the kind of QFTs you can get in the low energy limit. This is saying that string theory is more predictive than QFT, i.e. less general. I always thought this was the goal the whole time! If it is the other way around and string theory really is more general than QFT, doesn't this mean that string theory is less predictive than, for instance, old school GUT model building?

So is the relationship between string theory and quantum field theory a strict inclusion $\mathrm{QFT} \subset \mathrm{ST}$ or more like a duality/equivalence $\mathrm{QFT} \simeq \mathrm{ST}$, or a more complicated Venn diagram?

Note that I am not asking about AdS/CFT as this only deals with special string backgrounds and their QFT duals. I'm asking about the general relationship between string theory and QFT.

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Right, string theory also brings some constraints that imply that certain low-energy field theories can't emerge as approximations to string theory - they're in the swampland,or they have the wrong unrealized values of parameters, and so on. But string theory still morally covers "every consistent enough" region of QFTs or those that admit unification with symmetry. At any rate, the claim that "string theory is a generalization of QFT" just means that "a" QFT is an approximation to string dynamics, but string dynamics isn't quite QFT. –  Luboš Motl Jul 12 '13 at 7:31
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In some backgrounds and interpretations, like in AdS/CFT, string theory is the same thing as QFT. But the AdS string theory lives in a different "spacetime" than the CFT so string theory isn't a QFT "in the same spacetime". String theory may also be formulated as string field theory - at least perturbatively - in many backgrounds. SFT is like a QFT with infinitely many fields. But from some viewpoint, this map isn't precise. –  Luboš Motl Jul 12 '13 at 7:32
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@LubošMotl Thanks. Looked up swampland - it seems to make sense. I guess that really answers my question, though I'm a little concerned about the "morally covers" hedge. ;) Would I be correct in saying that classifying the "consistent enough" QFTs would amount to mapping the landscape? Also, I would suggest that using the word "generalization" in this way is misleading, perhaps even wrong, since it seems to imply much more than "some QFTs are approximations to strings." –  Michael Brown Jul 12 '13 at 8:04
    
I'm not really concerned about whether string theories and their QFT duals live in the same spacetime. I'm perfectly happy with a more general kind of duality. –  Michael Brown Jul 12 '13 at 8:07
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1 Answer 1

up vote 8 down vote accepted

Notice:

  1. Pertubative string theory is defined to be the asymptotic perturbation series which are obtained by summing correlators/n-point functions of a 2d superconformal field theory of central charge -15 over all genera and moduli of (punctured) Riemann surfaces.

  2. Perturbative quantum field theory is defined to be the asymptotic perturbation series which are obtained by applying the Feynman rules to a local Lagrangian -- which equivalently, by worldline formalism, means: obtained by summing the correlators/n-point functions of 1d field theories (of particles) over all loop orders of Feynman graphs.

So the two are different. But for any perturbation series one can ask if there is a local non-renormlizable Lagrangian such that its Feynman-rules reproduce the given perturbation series at sufficiently low energy. If so, one says this Lagrangian is the effective field theory of the theory defined by the original perturbation series (which, if renormalized, is conversely then a "UV-completion" of the given effective field theory).

Now one can ask which effective quantum field theories arise this way as approximations to string perturbation series. It turns out that only rather special ones do. For instance those that arise all look like anomaly-free Einstein-Yang-Mills-Dirac theory (consistent quantum gravity plus gauge fields plus minimally-coupled fermions). Not like $\phi^4$, not like the Ising model, etc.

(Sometimes these days it is forgotten that QFT is much more general than the gauge theory plus gravity plus fermions that is seen in what is just the standard model. QFT alone has no reason to single out gauge theories coupled to gravity and spinors in the vast space of all possible anomaly-free local Lagrangians.)

On the other hand now, within the restricted area of Einstein-Yang-Mills-Dirac type theories, it currently seems that by choosing suitable worldsheet CFTs one can obtain a large portion of the possible flavors of these theories in the low energy effective approximation. Lots of kinds of gauge groups, lots of kinds of particle content, lots of kinds of couplings. There are still constraints as to which such QFTs are effective QFTs of a string perturbation series, but they are not well understood. (Sometimes people forget what it takes to defined a full 2d CFT. It's more than just conformal invariance and modular invariance, and even that is often just checked in low order in those "landscape" surveys.) In any case, one can come up with heuristic arguments that exclude some Einstein-Yang-Mills-Dirac theories as possible candidates for low energy effective quantum field theories approximating a string perturbation series. The space of them has been given a name (before really being understood, in good tradition...) and that name is, for better or worse, the "Swampland".

For this text with more cross-links, see here:

http://ncatlab.org/nlab/show/string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory

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I like this a lot again, I think I should waste some more time on your nlab site ... :-). Could you write something nice to this question too? It urgently needs a good upvotable answer apart from the very negative and pessimistic ones it only has attracted so far ... :-/ –  Dilaton Jul 13 '13 at 18:07
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