RevModPhys.84.987

2 s s ie linear in the spin two eld h and quadratic in

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Unformatted text preview: eld h and quadratic in the spin-s field ’s . The symmetric tensor of rank two  :¼ Lð1Þ =h is bilinear in the spin-s field. For 9 See, e.g., Boulanger et al. (2001) for a precise statement of the very general hypotheses and references therein for previous literature on this issue. 10 A thorough discussion on the observability of the graviton is presented by Boughn and Rothman (2006) and Rothman and Boughn (2006). Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 991 consistency with the linearized diffeomorphisms  h ¼ @  þ @  , the cubic coupling Lð1Þ to a massless spintwo field h must arise through a bilinear conserved current of rank two, i.e., @  % 0, where the weak equality denotes the equality up to terms that vanish on the solutions of the free equations of motion for ’s . For s ¼ 2, the cubic self-coupling of type 2-2-2 coming in the Einstein-Hilbert Lagrangian gives rise to a conserved tensor  which is equivalent to the Noether energy-momentum tensor T  for the Fierz-Pauli Lagrangian. For s Þ 2, the cubic 2-s-s coupling Lð1Þ comes from the Lorentz minimal coupling prescription applied to the free Lagrangian Lð0Þ if and only if  is equal (possibly on shell and modulo an ‘‘improvement’’) to the Noether energy-momentum tensor T  for Lð0Þ . It is this precise condition on  (for any spin) that should be understood as minimal coupling in the Weinberg equivalence principle. 3. Higher-order conservation laws: The higher-spin cases Lorentz invariance for the emission of soft massless higher (s ! 3) spin particles leads to conservation laws of higher (s À 1 ! 2) order, i.e., for sums of products of momenta. For generic momenta, Eq. (1) has no solution when s À 1 > 1, therefore all coupling constants must be equal to zero: gðsÞ ¼ 0 for any i when s > 2. In other words, as stressed i by Weinberg in his book (Weinberg, 1995), p. 538: ‘‘massless higher-spin particles may exist, but they cannot have couplings that survive in the limit of low energy’’ (that is, they cannot mediate long-range interactions). Moreover, strictly speaking the Weinberg low-energy theorems concern only s-s0 -s0 couplings. Nevertheless, notice the existence of a simple solution for Eq. (1) corresponding to so-called trivial scattering, i.e., elastic scattering such that the outgoing particle states are permutations of the incoming ones, as in the case of free or possibly integrable field theories. For example, if we denote the ingoing momenta by ka (a ¼ 1; 2; . . . ; n) and the outgoing ones by ‘a , then the higher-order conservation laws P P ðs Þ  1 sÀ1 ¼ ðÀ1ÞsÀ1 a gðsÞ ‘1 Á Á Á ‘sÀ1 of order a aa a ga ka Á Á Á ka s À 1 > 1 imply that the outgoing momenta can only be permutations of the incoming ones, and that gðsÞ ¼ gðsÞ for a all a if s is even, while gðsÞ ¼ a gðsÞ with ða Þ2 ¼ 1 for all a if a s is odd. C. Coleman-Mandula theorem and its av...
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