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Perhabs my perspective of string theory is wrong, but I have no idea how the following works: A string representing a single fermion (like an incoming electron) splits into an other fermionic (outgoing electron) and an outgoing bosonic (photon-like). The photon means a bosonic mode , but it only appears on an outgoing leg of the "graph". According to my interpretation it must violate a rule. Could someone tell me where my logic is faulty?

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The problem is that not every Feynman diagram of strings reduces to a Feynman diagram in the effective field theory arising from the interacting theory of strings.

The correct logic is that the "internal" dynamics of the strings, and the interactions among them, induce the spectrum and quantum dynamics of the background fields. But it's not true that you can represent every single EFT excitation, say an electron, by a vibrating string ignoring all the other states of the string.

The key problem with the phrase "a string representing a single fermionic excitation" is that it's not generally possible to take an arbitrary set of EFT degrees of freedom and represent that product state as a "vibrating string" and nothing more. If a string vibrates at the level $n$ of its spectrum, all the states at that level must be present in the EFT limit, you cannot integrate them from the EFT (without the addition of other mechanisms).

What string theorist really does is the following. Take a background, infer the spectrum of background quantum fields by quantizing the classical theory of strings on the given target space (when possible). After integrating the string modes above its ground states, an (effective) quantum field theory is produced, and string theory exactly behaves as ordinary quantum field theory. Indeed, that's the beauty of string theory Why string theory is quantum mechanics on steroids.

In summary, the correct stringy-way to think about QFT degrees of freedom is the QFT-formalism in itself, because string theory exactly behaves as an ordinary QFT when it deals with QFT degrees of freedom.

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    $\begingroup$ Thanks for the answer, it gave me a new perspective of string theory. But the point of the question is about the perturbative string theory in a naive approach. So ingoing electron and photon are two strings. Is it possible to loose the bosonic mode if the output is only an electron-string? $\endgroup$ – RobertSzili Sep 14 '20 at 18:24
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    $\begingroup$ Ramiro: I take your first parahraph as an answer for my question. Because according my view a real electron cannot be represented by a single string mode. I needed some time to process :) $\endgroup$ – RobertSzili Sep 20 '20 at 13:26

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