RevModPhys.84.987

# Weinberg witten theorem weinberg and witten 1980

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Unformatted text preview: theories where the graviton is a bound state of particles with spin one or lower. Its proof Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . 1006 involves S-matrix manipulations which will be discussed in more detail in the next section on its reﬁned version. If one assumes locality, then it becomes surprisingly easy to prove the Lagrangian version of the Weinberg-Witten theorem. Let ½s denote the integer part of the spin s. Lemma: Any local polynomial which is at least quadratic in a spin-s massless ﬁeld, nontrivial on shell and gauge invariant, must contain at least 2½s derivatives. Proof: Corollary 1 of Bekaert and Boulanger (2005) states that, on shell, any local polynomial which is gauge invariant may depend on the gauge ﬁelds only through the Weyl-like tensors. The latter tensors contain ½s derivatives, thus the lemma follows. j A straightforward corollary of this lemma is a version of the Weinberg-Witten theorem. Weinberg-Witten theorem (Lagrangian formulation): (i) Any perturbatively local theory containing a charge current J  which is nontrivial, Lorentz covariant, and gauge invariant, forbids massless particles of spin s > 1=2. (ii) Any perturbatively local theory containing a Lorentzcovariant and gauge-invariant energy-momentum tensor T  forbids massless particles of spin s > 3=2. Proof: In the free limit, any Noether current in a perturbatively local theory must be a quadratic local polynomial. For massless ﬁelds of spin s > 1=2, the lemma implies that this polynomial must contain at least two derivatives (or four derivatives if s > 3=2). However, the charge current contains one derivative and the energy-momentum tensor contains two derivatives. j The lower bound s > 3=2 of this version is slightly weaker than the lower bound s > 1 of the original Weinberg-Witten theorem (Weinberg and Witten, 1980). Anyway the case s ¼ 3=2 is low spin and thereby is not a main concern of this paper. 2. Reﬁnement of Weinberg-Witten theorem Porrati (2008) takes gauge invariance into account in order to still use Weinberg-Witten’s argument but in a context where the stress-energy tensor need not be gauge invariant (or Lorentz covariant, which is the same in a secondquantized setting) any more. In the original work (Weinberg and Witten, 1980) a particular matrix element was considered: elastic scattering of a spin-s massless particle off a single soft graviton. The initial and ﬁnal polarizations of the spin-s particle are identical, say þ s, its initial momentum is p and its ﬁnal momentum is p þ q. The graviton is off shell with momentum q. The matrix element is hþs; p þ qjT j þ s; pi: (B1) In the soft limit q ! 0 the matrix element is completely determined by the equivalence principle, as recalled above when reviewing Weinberg’s low-energy theorem. Using the relativistic normalization for one-particle states hpjp0 i ¼ 2p0 ð2Þ3 3 ðp À p0 Þ, we...
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