Consider the free massless spin-$n/2$ field in a general curved space-time ($M$, $g$): \begin{equation} \nabla^{AA'}\phi_{\underbrace{AB\cdots L}_n} =0\end{equation}If $\phi_{AB\cdots L}$ has charge $e$, then we have the algebraic consistency condition : $$(n-2)\phi_{ABM(C\cdots K}{\Psi_{L)}}^{ABM}=-ie\chi^{AB}\phi_{ABC\cdots L}$$where $\Psi_{ABCD}$ is the gravitational spinor. For $n=0,1,2$ and $e\chi_{AB}=0$, the above consistency condition is vacuous and it follows that a local expression for stress-energy tensor $T_{ab}$ exists such that:

(i) $T_{ab}=T_{(ab)}$

(ii) $\nabla^aT_{ab}=0$

For $n>2$, the consistency condition is very restrictive and no local expression for $T_{ab}$ exists which satisfy conditions (i) and (ii). It is still possible to define non-local $T_{ab}$ which involves integrals of spinor field $\phi_{AB\cdots L}$ rather than derivatives. However, such non-local expression for $T_{ab}$ is not suitable since Einstein's field equations involves only local values. So, does higher spinor fields satisfy Einstein's field equations? Or do we need to modify Einstein's field equations to include non-local $T_{ab}$?

Note: I'm looking for possible extension or modification of EFE or analysis within the context of classical space time geometry which is background independent .

  • $\begingroup$ What do you mean by "stress-energy tensor" here, it's quite possible that the issue stems from confusing the stress-energy tensor with the Belinfante-Rosenfeld tensor $\endgroup$
    – Slereah
    Dec 27, 2021 at 17:58
  • $\begingroup$ @Slereah The original text calls this as "Energy-momentum tensor" only. I was referring to the discussions in section 5.8 of Spinors and Space-time Volume-I. The expression for $T_{ab}$ was symmetric for massive Dirac field , so I guess the method used to compute this Energy momentum tensor must be equivalent to Belinfante-Rosenfeld modification. $\endgroup$
    – KP99
    Dec 27, 2021 at 18:18
  • $\begingroup$ There is Weinberg–Witten theorem and its generalization by Porrati that forbid such stress-energy around Minkowski background. A loophole is AdS background that has Vasiliev theory, with an infinite sequence of massless higher spin fields. $\endgroup$
    – A.V.S.
    Dec 27, 2021 at 20:12
  • $\begingroup$ @A.V.S. Thank you. I'll read about Vasiliev theory, since Wikipedia states that it is background independent. In general, how does higher spinor field in Vasiliev theory couples to gravity? $\endgroup$
    – KP99
    Dec 28, 2021 at 12:47
  • 1
    $\begingroup$ I am not an expert on Vasiliev theory, but my impression is that Riemannian geometry is not a good fit for higher spins gravity since HS gauge transformations also modify the metric, so the notion of local spacetime event is gauge dependent. See e.g. this lecture. $\endgroup$
    – A.V.S.
    Dec 29, 2021 at 18:06


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