# Intuitive sketch of the correspondence of a string theory to its limiting quantum field theory

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I'm looking for an intuitive sketch of how one shows the correspondence of string theory to a certain QFT. My best guess is that one calculates the scattering amplitudes in the string theory as a series in some parameter (string length?) and shows that the leading order term is equal to the scattering amplitudes in the corresponding QFT.

If this is the case then my hope is that someone can elaborate and perhaps point me to some references. If I'm off base then my hope is that I can get a sketch and not be bogged down in heavy math (at this stage).

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recategorized Apr 20, 2014
If you only want an intuitive sketch, this 600-character comment is more than enough. Histories in string theory look like Riemann surfaces. The long-distance limit inevitably makes all the tubs in the diagram much thinner than they're long - because one pays with energy for the spatial circumference of the cross section (i.e. length of the string). So then one has Feynman rules involving the lowest vibration states of strings - and they look like pointlike particles and have discrete spectrum - and they interact by some vertices (given by the tube junctions), so we get Feynman rules of QFTs.

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Hi thanks for this. If you don't mind perhaps you could elaborate (pedagogical references?). In particular why do you mean by long-distance limit? Is this equivalent to taking the string length to 0 (in the same way that the classical limit is found by taking $\hbar$ to 0)?

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If this is how string theory corresponds to QFT, then is it a fair assessment to say that the notion of a quantum field is only useful in that in some limit, where one can ignore the spatial extent of a string, the theory makes consistent predictions? That is to say we shouldn't ascribe "reality" to quantum fields in the same way that we presumably do to strings?

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This correspondence is what Lubos said, but he glosses over the main technical difficulty: you need to make sure that a single string diagram, which includes all contributions of a given loop order, reproduces combinatorially many Feynman diagrams with the correct weight in the field theory, infinite string-tension, limit. Joel Scherk showed that this limit works properly in 1969 or so, using world-sheet duality, but to read the paper, back then world-sheet duality was called Dolen Horn Schmidt duality, and the infinite tension limit was called the zero slope limit, meaning zero Regge slope, infinitesimal $J/m^2$, so higher $J$ resonances are at infinite mass. World-sheet duality tells you that the exchange of all the Regge particles in any one channel, say the s-channel, reproduces a t-channel sum and a u-channel sum also, so when taking a zero slope limit, each channel separately turns into a particle propagation which adds up, while each effective string nonlocal blobby vertex turns into the appropriate local field theory vertex in the limit, with the appropriate combinatorial factor (because in each channel, the string propagation tube is separately reducing to a line). The method of reducing to a field theory is not so laborious as you say, otherwise we would never be able to actually do it, you can figure out what the field theory is just from the spectrum of massless particles and the effective vertices in the field theory limit, without going through the full scattering amplitude at all loops. It is usually made easy because we know the supersymmetry, so we know the low-energy limit just by symmetry. A proper answer should review Scherk's paper, I didn't do that, but that's the argument.
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