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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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