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I was reading a paper on Coleman–Mandula theorem and Ward Identity [The Coleman-Mandula Theorem by Sascha Leonhardt]1, where I saw it says that-

Let a higher spin current $\hat{B}_{\mu\nu}$ is non commuting with Poincare generator $\hat{p}_\mu$; $[\hat{B}_{\mu\nu},\hat{p}_\mu]\neq 0$. From this we mean $\hat{B}_{\mu\nu}$ is neither a generator of Poincare symmetry nor a linear combination of four momentum operator $\hat{p}_\mu$. (From here, we have a feeling it generates internal symmetry). So, from its symmetry and tracelessness we can say that, $$ \left<p\right|\hat{B}_{\mu\nu}\left|k\right>\propto p_\mu p_\nu-\frac{1}{4}\eta_{\mu\nu}p^2$$

Now I can not understand the following things,

  1. What is the significance of this term- higher spin current? [I saw somewhere on the internet that higher spin current corresponds to free theory, but how?]
  2. From the relation $[\hat{B}_{\mu\nu},\hat{p}_\mu]\neq 0$, how can we identify that $\hat{B}_{\mu\nu}$ is neither a generator of Poincare symmetry nor a linear combination of four momentum operator $\hat{p}_\mu$?
  3. From the symmetry and tracelessness How can we say that, $$ \left<p\right|\hat{B}_{\mu\nu}\left|k\right>\propto p_\mu p_\nu-\frac{1}{4}\eta_{\mu\nu}p^2$$
  4. If $ \hat{B}_{\mu\nu}\propto \hat{p}_\mu \hat{p}_\nu-\frac{1}{4}\eta_{\mu\nu}\hat{p}^2$, then how can the ward identity, that is $\hat{p}^\mu \hat{p}^\nu \hat{B}_{\mu\nu}=0$, be proved?
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    $\begingroup$ When you cite things, please give author, title, and ideally a link to the source instead of referring to it nebulously as "a paper". Also, if the quote is not verbatim but you are paraphrasing it, please do not use the blockquote formatting. $\endgroup$ – ACuriousMind May 7 at 14:48

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