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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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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. – LuboŇ° Motl Jan 26 '12 at 5:58
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)? – Kyle Jan 27 '12 at 2:08
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? – Kyle Jan 27 '12 at 2:18

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