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I've read from Figueroa-O'Farrill's The Theory of Induced Representations in Field Theory and this answer that massless particles of helicity $h$ must be associated with fields transforming under the $\left(m,n\right)$ Lorentz representation where $n-m=h$.

In the case of the $h = \pm 1$ photon, we sum the $\left(1,0\right)$ and $\left(0,1\right)$ representations, but gauge symmetry allows us to define and use a potential transforming under $\left(\tfrac{1}{2},\tfrac{1}{2}\right)$ (with two independent components).

Is there a similar line of reasoning to show why the $h = \pm 2$ graviton is associated with a $\left(1,1\right)$ potential (with two independent components)?

tomdodd4598
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    see this answer by JamalS for how to count the d.o.f. of the metric perturbation and arrive at 2 in 4d after subtracting gauge d.o.f. – ACuriousMind Jan 10 '23 at 15:43
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    Never done it explicitly, but it is perfectly possible to generalize the spin one analysis. The detailed spin one case is worked out in Weinberg's The Quantum Theory of FIelds, section 5.9. If you try to embed the creation/annihilation operators of helicity $\pm 1$ massless particles into a vector field you get something that transforms as a four-vector up to a gauge transformation. Weinberg then explains that the analysis can be generalized to spin two. He gives some details on a more general case at the end of the section. – Gold Jan 16 '23 at 17:56
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    yes, the story for the graviton is basically the same as for the photons, just twice. It's also quite well known since the work of Weinberg in the mid 60's on this. See e.g. page L6/10 of this lectures for an account https://drive.google.com/file/d/1twimpDVituHS33tqJt3jEZdbqEFk_UGM/view – TwoBs Jan 17 '23 at 21:28
  • @TwoBs Please consider summarizing your link in a standalone answer. Links to Google Dive leak personal information for both the sharer and the sharee; furthermore such private links are especially susceptible to link rot. – rob Jan 17 '23 at 21:54

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