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**Unformatted text preview: **eld h and quadratic
in the spin-s ﬁeld ’s . The symmetric tensor of rank two
Â :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., @ Â % 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 Â 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
Â is equal (possibly on shell and modulo an ‘‘improvement’’) to the Noether energy-momentum tensor T for
Lð0Þ . It is this precise condition on Â (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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