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Unformatted text preview: e ultraviolet but also in the
Pertinent to this statement is the generalized WeinbergWitten theorem. The assumptions are that (i) the Lorentz
minimal coupling term is always present, and (ii) the theory
extends to all energies without encountering any infrared or
ultraviolet catastrophe. To reiterate, the reﬁned analysis relies
crucially via assumption (i) on Weinberg’s formulation of the
equivalence principle,18 which one may view as a low-energy
constraint on the theory. The result is that massless higherspin particles cannot couple with the universal graviton or
anything that the latter couples to. In other words, if such
massless higher-spin theories in ﬂat background exist in the
mathematical sense, they cannot be engineered to the lowenergy physics that takes place in our Universe.
For instance, one can have a theory with two phases: A
symmetric phase at high energy where higher-spin particles
are massless and the Newton constant vanishes for all
particles, and a broken phase, where higher-spin particles
get a mass and the Newton constant is nonzero. This is an
intriguing possibility; moreover it probably occurs in AdS4
(Girardello, Porrati, and Zaffaroni, 2003); see the discussion
in Sec. IV.D. Nothing forbids the existence of an a priori very
warm universe where such exotic theories are relevant. After
cooling and symmetry breakdown these may then yield an
effective matter-coupled gravity theory in which the graviton
is that ﬁeld that couples to everything in always the same way,
with a single coupling constant introduced, namely, Newton’s
The assumptions (i) and (ii) are indeed vulnerable to the
possibility of phase transitions. This will be discussed in
Sec. IV.D. Looking to the limits of the experimental as well
as theoretical tests of the Lorentz minimal coupling, there is
no reason a priori as to why the speciﬁc mechanism by which
diffeomorphism invariance is implemented in Einstein’s
gravity should work at scales that are very small or very
large. This suggests that the Lorentz minimal coupling can be rehabilitated within theories with infrared as well as ultraviolet cutoffs.
D. Flat background As already stressed, the strict deﬁnition of massless particle and S matrix requires a ﬂat spacetime. Passing to a
slightly curved de Sitter or anti–de Sitter spacetime with
cosmological constant Ã, one sometimes considers the existence of gauge symmetries as the criterion19 of masslessness.
Since there is no genuine S matrix in AdS, a subtle and
fruitful way out is that the S-matrix theorems do not apply
any more when the cosmological constant Ã is nonvanishing;
instead one resorts to a holographic dual conformal ﬁeld
theory. This way out has been exploited successfully by the
Lebedev school and has given rise to cubic vertices and full
nonlinear equations of motion.
E. Finite dimensionality of spacetime Finally, in light of the recent progress made in amplitude
calculations in ordinary relativistic quantu...
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