2002 appendix g xavier bekaert nicolas boulanger and

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Unformatted text preview: Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . In order to prepare the ground for further discussion, we denote by ‘‘electromagnetic minimal coupling’’ the coupling of a charged particle to the electromagnetic field obtained by replacing the partial derivatives appearing in the Lagrangian describing the free, charged matter field in flat space, by the uð1Þ-covariant derivative, viz. @ ! @ À igð1Þ A . i 2. Equivalence principle: The spin-two case As argued by Weinberg (1964), the equivalence principle can be recovered as the spin-two case of his low-energy theorem. On one side, Lorentz invariance for the emission of a soft massless spin-two particle leads to the conservation P law i gð2Þ p ¼ 0. On the other side, translation invariance i i P implies momentum conservation i p ¼ 0. Therefore, for i ´ generic momenta, Poincare invariance requires all coupling constants to be equal: gð2Þ ¼ gð2Þ ¼: gð2Þ (8 i; j). In other i j words, massless particles of spin two must couple in the same way to all particles at low energies. This result has far-reaching consequences as it resonates with two deep properties of gravity, namely, its uniqueness and its universality. On the one hand, the local theory of a self-interacting massless spin-two particle is essentially9 unique: in the low-energy regime (at most two derivatives in the Lagrangian) it must be described by the EinsteinHilbert action. Therefore, the massless spin-two particle rightfully deserves the name ‘‘graviton.’’10 On the other hand, the gravitational interaction is also universal (Weinberg, 1964): if there exists a single particle that couples minimally to the graviton, then all particles coupled to at least one of them must also couple minimally to the graviton. According to Weinberg himself, this theorem is the expression of the equivalence principle in quantum field theory, so, from now on, it will be referred to as the Weinberg equivalence principle. A proper understanding of this crucial theorem involves, however, some subtleties on the precise meaning of ‘‘minimal coupling.’’ Consider the quadratic Lagrangian Lð0Þ ð’s ; @’s Þ describing a free spin-s matter field denoted by ’s . In general relativity, the equivalence principle can be expressed by the Lorentz minimal coupling prescription, i.e., the assumption that the transformation rules of tensor fields under the ´ Poincare group extend naturally to the diffeomorphism group and the replacement of partial derivatives by Lorentzcovariant ones, viz. @ ! r ¼ @ þ gð2Þ Àlin þ Á Á Á , in the matter sector. It must be observed that this prescription does not apply to the spin-two field itself because the Einstein-Hilbert Lagrangian is not the covariantization of the Fierz-Pauli quadratic Lagrangian Lð0Þ ð’2 ; @’2 Þ. One focuses on cubic couplings Lð1Þ ðh; ’s ; @’s Þ of the type 2-s-s, i.e., linear in the spin-two fi...
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