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I understand how we get particle spins in string theory. It comes from the spin representation of the internal degrees of freedom of the the string excitations.

But how do we impose Bosonic/Fermonioc statistics in string theory? In Quantum Field theory, there is the spin-statistics theorem which constraints integer spin particles to be Bosons and Half-integer spins to be fermions. How do we get particle statistics in string theory

Put another way, are particle statistics constrained in string theory like they are in QFT due to spin-statistics theorem?

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To get fermions in string theory, you need to quantize the superstring (as opposed to the bosonic string). The superstring has worldsheet fermions that are the superpartners of the bosonic spacetime coordinates. The supersymmetry algebra naturally involves anti-commutation relations for the fermionic degrees of freedom. The fact that the low energy spectrum of the superstring involves anti-commuting fermionic fields comes from that fact.

From an effective field theory point of view, you could also say that the low energy effective theory of string theory has to generate anti-commuting fermions, if string theory is a consistent UV theory. So in that sense, the low energy limit does constrain string theory to obey the spin-statistics theorem.

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  • $\begingroup$ Thanks for the answer. Does the quantisation of the superstring enforce the spin statistics theorem or is the spin statistics theorem enforced by experiments? $\endgroup$
    – Ryder Rude
    Commented Oct 17 at 13:04
  • $\begingroup$ @RyderRude I would say that question is comparing apples and oranges, in the sense that you are asking about two different levels of explanation. On a purely mathematical/theoretical level, the spin-statistics theorem in QFT is a result that if you follow the rules of bosonic quantization and apply it to a fermion field (ie using commutation relations), then you get an inconsistent quantum theory, since you get negative norm states and you can't define probabilities. Therefore within the framework of QFT, you need to use anti-commutation relations. We believe string theory reduces to (...) $\endgroup$
    – Andrew
    Commented Oct 17 at 14:20
  • $\begingroup$ (...) a consistent QFT in the low energy limit, so it will also obey the spin-statistics theorem. The way that happens from first principles if you quantize a string, is the worldsheet supersymmetry I described. However, physics is an empirical science. We could absolutely imagine a world where identical spin-1/2 particles obey Bose-Einstein statistics. Since QFT can't describe that world, then we'd conclude that QFT was not a correct theory of Nature. You wouldn't be able to derive that conclusion logically sitting in your armchair, you'd actually have to do experiments and see what happens. $\endgroup$
    – Andrew
    Commented Oct 17 at 14:23
  • $\begingroup$ All experiments we've done (not involving gravity) are consistent with the Standard Model of particle physics, which is a QFT, so we have good reasons to expect reality to be explained by a QFT, and therefore to expect that the spin-statistics theorem to hold in Nature (and, indeed, it does hold in every experiment we've done, and has even been used to infer properties of particles before we've discovered them). But, I just want to make the point that you cannot ask whether a result follows from a theoretical derivation or an experimental result, since those are two different questions. $\endgroup$
    – Andrew
    Commented Oct 17 at 14:25
  • $\begingroup$ The reason I asked this question was that I decided that my credence in string theory would increase if it "predicted" (post-dicted) particle statistics, like QFT does. This is why I asked if it follows mathematically or experimentally in string theory. $\endgroup$
    – Ryder Rude
    Commented Oct 17 at 14:33

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