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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.

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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.

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  • $\begingroup$ I had been thinking that the worldsheet path integral directly corresponds to probabilities in string theory. But it seems like the worldsheet integral plays the exact same role as the Feynman propagator in QFT. One still needs Feynman rules, along with the propagator, to calculate experimental probabilities. It seems like string theory just borrowed Feynman rules from QFT and made a replacement for the propagator $\endgroup$
    – Ryder Rude
    Commented Sep 1, 2022 at 13:20
  • $\begingroup$ Maybe we should expect a weird field theory to be underlying String theory, if its foundations are based on Feynman rules $\endgroup$
    – Ryder Rude
    Commented Sep 1, 2022 at 13:31
  • $\begingroup$ If you need compactification to produce gauge interactions, then what interactions does the original genus expansion describe? Are we saying that gauge interaction are an approximation of the fundamental interactions described in the genus expansion? $\endgroup$
    – Ryder Rude
    Commented Sep 1, 2022 at 13:49
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    $\begingroup$ @RyderRude Note that I said "most" - of course the 10d SUGRA theories contain interactions. But again, string theory doesn't have anything underlying the sum over worldsheets - this sum doesn't "describe" anything, it is the definition of how the string amplitudes are computed. You can interpret the different worldsheets as descriptions of strings splitting and merging, but that's even more just a story than the idea that the lines in Feynman diagrams correspond to particles. $\endgroup$
    – ACuriousMind ♦
    Commented Sep 1, 2022 at 13:57
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    $\begingroup$ @RyderRude The 10d/11d SUGRA QFTs are not renormalizable and hence not viable candidates for a fundamental theory at all scales, cf. e.g physics.stackexchange.com/q/566661/50583. $\endgroup$
    – ACuriousMind ♦
    Commented Sep 1, 2022 at 14:30
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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.

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  • $\begingroup$ So, summing the propagator over topologies is enough to account for field interaction terms? Is there a proof of this? Also, the worldline path integral isn't unitary. If the worldsheet path integral directly gives experimental probabilities, then it must be unitary. If it's unitary, is there a Schroedinger equation corresponding to it? $\endgroup$
    – Ryder Rude
    Commented Sep 1, 2022 at 12:22
  • $\begingroup$ Is it true that string theory is formulated directly in terms of Feynman rules? This is different from QFT in which Feynman rules are not taken to be the definition of the theory, but are rather derived from the field Lagrangian. In David Tong's notes, he uses the worldsheet integral along with Feynman rules (like introducing vertex operators) $\endgroup$
    – Ryder Rude
    Commented Sep 1, 2022 at 12:59
  • $\begingroup$ One of the main drivers of interest in string theory is that general covariance emerges from it. This leads to the problem of time so unitarity is always going to be more subtle than it is for a QFT in flat space. Nevertheless, it has an S-matrix which you can read about in Tong. $\endgroup$
    – Connor Behan
    Commented Sep 1, 2022 at 13:04
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    $\begingroup$ The extent to which you can see field-theory like behaviour in string theory is the topic of chapter 7 on low energy effective actions. But at a general energy scale, there's no proof that the genus expansion yields the same predictions as some "less stringy" field theory and indeed no reason to expect that it would. You are right that we should ultimately want a perturbative expansion like this to be a calculation tool rather than something that the theory itself forces on us. This is the problem of defining non-perturbative string theory which is still open. $\endgroup$
    – Connor Behan
    Commented Sep 1, 2022 at 13:13

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