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In Quantum Field Theory, the calculation of scattering amplitudes relies on interaction terms like $\lambda \phi^4$ or $\psi ' \gamma ^{\mu} \psi$. These are products of field operators calculated at a spacetime point. Derivation of most interaction terms employs local gauge symmetries, which is also an idea tied to field theories.

String theory supposedly calculates the same interaction amplitudes, but without employing fields. How does it do it?

I know about String Theory's replacement of the Feynman propagator. It replaces it with the path integral of the world-sheet action. But you need more than the propagator to calculate interaction amplitudes, i.e. you need terms like $\lambda \phi^4$. The propagator alone isn't enough.

Qmechanic
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Ryder Rude
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2 Answers2

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  1. As Connor Behan's answer correctly states, string theory not only has an integral over the worldsheet but also a sum over topologies of worldsheets.

  2. It is a fact that the amplitudes computed by this sum over worldsheets have, in the low-energy limit, the same shape as the scattering amplitudes of certain SUGRA QFTs. However, string theory is not a quantum field theory and is defined by this sum over worldsheets - there is as of yet no known underlying "non-perturbative" theory to which this sum over worldsheets could be seen as a perturbative approximation (in analogy to how the sum over Feynman diagrams as a perturbative approximation to the underlying non-perturbative dynamics in QFT.

    So any questions that rely on drawing an analogy to the non-perturbative (i.e. non-Feynman diagram) part of QFT are simply not questions string theory, as it currently stands, can answer. There is no time evolution, no non-perturbative S-matrix - there is just the perturbative prescription for the sum over worldsheets. As an analogy: This is the same as if we defined QFT by writing down the Feynman diagram prescriptions for its scattering amplitudes instead of deriving them as a perturbative approximation to some underlying dynamics.

  3. Most gauge theories in particular and interactions in general don't directly arise from the perturbative string scattering amplitude in 10d - they come from specific properties of the compactification manifold when that theory is broken down to 4d (see also this answer of mine), and this is not a particularly "stringy" phenomenon. The concept of dimensional reduction that produces a plethora of different and differently interacting theories from a few 10d SUGRA theories is firmly part of "normal" QFT, even though string theorists of course have a particular interest in it.

ACuriousMind
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The path integral you mention includes an integral over possible manifolds for the worldsheet. This might appear innocuous at first because the worldsheet metric in the Polyakov action is always $\eta_{\alpha\beta}$ (since we gauge fixed the Nambu-Goto action). But this only determines the local structure of a manifold. There is still a sum over topologies left to do. You can read about this in David Tong's notes, specifically chapter 6.

Focusing on just the lowest topology would essentially be ignoring "instanton contributions". This is safe to do in Yang-Mills theory which has a whole series worth of more dominant interactions. But doing the same thing in string theory would just give us something trivial. Of course the situation becomes more complicated when you have compactifications and fluxes which introduce QFT-like interactions in addition to the dependence on the topology.

Connor Behan
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