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lim hþs; p þ qjT j þ s; pi ¼ p p : q! 0 (B2) This is tantamount to saying that, at low energy, the only
possible coupling between gravity and everything else is done
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 via the minimal coupling procedure, bringing no more than
two derivatives (or one if the spin is half-integer) in the
interaction. More precisely, among all possible interaction
terms there must always be that coming from minimal coupling @ ! @ þ ÀðhÞ, with the nonvanishing coefﬁcient
related to Newton’s constant.
Since q is spacelike (off-shell soft graviton), one goes in
the frame in which q ¼ ð0; ÀqÞ, p ¼ ðjqj=2; q=2Þ, and
p þ q ¼ ðjqj=2; Àq=2Þ (the massless spin-s particle is
on shell), and deduces that a rotation RðÞ by an angle
around the q direction acts on the one-particle states
as RðÞjp; þsi ¼ expðÆisÞjp; þsi, RðÞjp þ q; þsi ¼
expðÇisÞjp þ q; þsi since RðÞ is a rotation of around
p but of À around p þ q ¼ Àp. Decomposing T under
space rotations in terms of spherical tensors as the complex
spin zero tensor T0;0 plus the real components fT1;m g1 ¼À1 and
m
fT2;m g2 ¼À2 , one can write the following relation:
m
eÆ2is hþs; p þ qjTj;m j þ s; pi
¼ hþs; p þ qjRy Tj;m Rj þ s; pi
¼ eim hþs; p þ qjTj;m j þ s; pi (B3) which admits, for s > 1, the only solution hþs; p þ qjT j þ
s; pi ¼ 0. If T is a tensor under Lorentz transformations
then this implies that hþs; p þ qjT j þ s; pi ¼ 0 in all
frames, in contradiction with the equivalence principle (B2).
This seems to kill gravity itself, but of course in that case as
usually happens in gauge theories, T is not a Lorentz tensor
(which is the same as saying that T is not gauge invariant).
One can deﬁne matrix elements for T that transform as
Lorentz tensors only at the price of introducing nonphysical,
pure-gauge states. This is what Porrati (2008) did in order to
accommodate the Weinberg-Witten argument to gauge theories for spin-s ﬁelds, s > 1 and prove that massless higherspin particles cannot exist around a ﬂat background if their
tensor T appearing in hþs; p þ qjT j þ s; pi complies
with the equivalence principle (B2).
Denoting by v all one-particle spin-s states, whether or not
spurious (pure gauge), the matrix element under consideration is denoted hv0 ; p þ qjT jv; pi. The method used by
Porrati (2008) in order to derive the S matrix is to perform
the standard perturbative expansion of the effective action
(where g ¼ þ h )
A¼ 1 Z 4 pﬃﬃﬃﬃﬃﬃﬃﬃ
1 Z d4 q ~Ã
h ðqÞ
d x ÀgR þ
16G
2 ð2Þ4
Â ðhv0 ; p þ qjT jv; pi þ T Þ þ Oðh2 Þ: (B4) The linear interaction terms include the matrix element and
another effective tensor T which summarizes the effect of
any other matter ﬁeld but that we omit from now on without
loss of generality. To linear order, Einstein’s equations
become
L h ðqÞ ¼ 16...

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