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**Unformatted text preview: **ally transform tensorially under isometry (implemented quantum
mechanically via similarity transformations). The remaining
carriers are different types of potentials obtained by integrating various curvature Bianchi identities (and which one may
thus think of as representing different ‘‘dual pictures’’ of one
and the same particle); such integrals in general transform
under isometry with inhomogeneous pieces that one can
identify as Abelian gauge transformations.
Thus, in the standard perturbative interaction picture one is
led to the Fronsdal program: the construction of interaction
Hamiltonians starting from Lorentz invariant and hence
gauge-invariant nonlinear Lagrangians built from the aforementioned carriers.
We stress that the Fronsdal program is based on a working
hypothesis: that standard canonical quantization of free ﬁelds
in ordinary spacetime is actually compatible with the presence of higher-spin translations in higher-spin gauge theories.
We proceed in this spirit in the bulk of this paper.
B. The Weinberg low-energy theorem The Weinberg low-energy theorem is essentially a
by-product of dealing with the more general problem of
emissions of soft massless particles. Consider a (nontrivial)
scattering process involving N external particles with (say,
ingoing) momenta pi (i ¼ 1; 2; . . . ; N ) and spin si . The emission of an additional massless particle of integer spin s with
arbitrary soft momentum by the ith external particle is controlled by a cubic vertex of type s-si -si (i.e., between a gauge
boson of spin s and two particles of spin si ) with coupling
constant gðsÞ . The Weinberg low-energy theorem (Weinberg,
i
1964) states that Lorentz invariance of (or, equivalently, the
absence of unphysical degrees of freedom from) the deformed amplitude imposes a conservation law of order
s À 1 on the N external momenta8:
N
X
i¼1 gðsÞ p1 Á Á Á psÀ1 ¼ 0:
i
i
i (1) A. Preamble: The gauge principle and the Fronsdal program The key feature of the ﬁeld-theoretic description of interacting massless particles is the gauge principle: a sensible
perturbation theory requires compatibility between the
interactions and some deformed version of the Abelian gauge
symmetries of the free limit. The necessity of gauge invari7 The S-matrix no-go theorem (Benincasa and Cachazo, 2007) is
not discussed here because it relies on slightly stronger assumptions
than the others; see, e.g., the conclusion of Porrati (2008) for more
comments.
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 1. Charge conservation: The spin-one case Lorentz invariance for the emission of a soft massless spinone particle (such as a photon) leads to the conservation law
P ð1Þ
i gi ¼ 0; thus it requires the conservation of the coupling
constants (such as the electric charges) that characterize the
interactions of these particles at low energies.
8 For pedagogical reviews, see, e.g., Weinberg (1995), Sec. 13.1 or
Blagojevic (2002), Appendix G. Xavier Bekaert,...

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