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I am quite familiar with the proof of Weinberg Witten (WW) theorem. One major result which follow from WW is that the graviton cannot be a composite particle. I have 2 questions here:

  1. How do we tell (in the theory) whether something is a composite particle or not? It should be an irreducible reps of the Poincare group, right? The confusion I have here is that an electron is a Dirac fermion, but the Dirac reps is a reducible one. There is something amiss here. Essentially, the question here is that how to distinguish between elementary and composite particles, in the theory?

  2. How to use the Weinberg-Witten theorem to show that the graviton cannot be a composite particle? Can anyone explain this a bit intuitively.

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    $\begingroup$ Regarding #1: Page 461 in Weinberg's own QFT book (v1, near eq 10.7.23) says, "a 'composite' particle may be understood to be one whose field does not appear in the Lagrangian." In the context, a particle is defined via a pole in a time-ordered vev. Regarding reducibility: The Dirac field operator is reducible with respect to the conn'd part of the Lorentz group (L/R parts), but an electron is a combo of both of those irreps because they're coupled via the mass term, which might as well be an "interaction" term as far as the relationship between field-rep'ns and particle-rep'ns is concerned. $\endgroup$
    – Chiral Anomaly
    Commented Dec 7, 2018 at 19:02

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The Weinberg-Witten theorem says that massless particles with spin $j>1$ can't carry Lorentz covariant (/gauge invariant) stress-energy, and massless particles with spin $j>1/2$ cannot carry current that's Lorentz invariant (/gauge invariant) see https://en.wikipedia.org/wiki/Weinberg-Witten_theorem

Composite gravitons made from "stuff" will violate this because the "stuff" will carry stress-energy (and often charge). Obviously that feels a bit hand-wavy - there are many ways around the theorem, as mentioned in the Wiki page.

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