Of their derivatives actually transform tensorially

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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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