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**Unformatted text preview: **E. Quantum corrections in the Ã3 theory
IX. Brane worlds and the Resonance Graviton
A. The DGP action
B. Linear expansion
C. Resonance gravitons
X. Conclusions and Future Directions
Appendix: Total Derivative Combinations 698
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707 I. INTRODUCTION Our goal is to explore what happens when one tries to give
the graviton a mass. This is a modiﬁcation of gravity, so we
ﬁrst discuss what gravity is and what it means to modify it.
A. General relativity is massless spin 2 General relativity (GR) (Einstein, 1916) is by now widely
accepted as the correct theory of gravity at low energies or
large distances. The discovery of GR was in many ways ahead
of its time. It was a leap of insight, from the equivalence
principle and general coordinate invariance, to a fully nonlinear theory governing the dynamics of spacetime itself. It
provided a solution, one more elaborate than necessary, to the
problem of reconciling the insights of special relativity with
the nonrelativistic action at a distance of Newtonian gravity.
Had it not been for Einstein’s intuition and years of hard
work, general relativity would likely have been discovered
anyway, but its discovery may have had to wait several more
decades, until developments in ﬁeld theory in the 1940s and
1950s primed the culture. But in this hypothetical world
without Einstein, the path of discovery would likely have
been very different and in many ways more logical.
Ó 2012 American Physical Society 672 Kurt Hinterbichler: Theoretical aspects of massive gravity This logical path starts with the approach to ﬁeld theory
espoused in the ﬁrst volume of Weinberg’s ﬁeld theory
text (Weinberg, 1995). Degrees of freedom in ﬂat fourdimensional spacetime are particles, classiﬁed by their spin.
These degrees of freedom are carried by ﬁelds. If we wish to
describe long-range macroscopic forces, only bosonic ﬁelds
will do, since fermionic ﬁelds cannot build up classical
coherent states. By the spin statistics theorem, these bosonic
ﬁelds must be of integer spin s ¼ 0, 1, 2, 3, etc. A ﬁeld c ,
which carries a particle of mass m, will satisfy the KleinGordon equation ðh À m2 Þ c ¼ 0, whose solution a distance
r from a localized source is similar to $rÀ1 eÀmr . Long-range
forces, those without exponential suppression, must therefore
be described by massless ﬁelds m ¼ 0.
Massless particles are characterized by how they transform
under rotations transverse to their direction of motion. The
transformation rule for bosons is characterized by an integer
h ! 0, which we call the helicity. For h ¼ 0, such massless
particles can be carried most simply by a scalar ﬁeld . For a
scalar ﬁeld, any sort of interaction terms consistent with
Lorentz invariance can be added, and so there are a plethora
of possible self-consistent interacting theories of spin 0
particles.
For helicities s ! 1, the ﬁeld must carry a gauge symmetry
if we are to write interactions with manifest L...

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