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**Unformatted text preview: **orentz symmetry and locality. For helicity 1, if we choose a vector ﬁeld A
to carry the particle, its action is ﬁxed to be the Maxwell
action, so even without Maxwell, we could have discovered
electromagnetism via these arguments. If we now ask for
consistent self-interactions of such massless particles, we are
led to the problem of deforming the action (and possibly the
form of the gauge transformations), in such a way that the
linear form of the gauge transformations is preserved. These
requirements are enough to lead us essentially uniquely to the
non-Abelian gauge theories, two of which describe the strong
and weak forces (Henneaux, 1998).
Moving on to helicity 2, the required gauge symmetry is
linearized general coordinate invariance. Asking for consistent self-interactions leads essentially uniquely to GR and full
general coordinate invariance (Gupta, 1954; Kraichnan,
1955; Weinberg, 1965; Deser, 1970; Boulware and Deser,
1975; Fang and Fronsdal, 1979; Wald, 1986) [see also
Chapter 13 of Weinberg (1995), which shows how helicity 2
implies the equivalence principle]. For helicity ! 3, the story
ends, because there are no self-interactions that can be written
(Berends, Burgers, and van Dam, 1984) [see also Chapter 13
of Weinberg (1995), which shows that the scattering amplitudes for helicity ! 3 particles vanish].
This path is straightforward, starting from the principles of
special relativity (Lorentz invariance) to the classiﬁcation of
particles and ﬁelds that describe them, and ﬁnally to their
possible interactions. The path Einstein followed, on the other
hand, is a leap of insight and has logical gaps; the equivalence
principle and general coordinate invariance, although they
suggest GR, do not lead uniquely to GR.
General coordinate invariance is a gauge symmetry, and
gauge symmetries are redundancies of description, not fundamental properties. In any system with gauge symmetry, one
can always ﬁx the gauge and eliminate the gauge symmetry,
without breaking the physical global symmetries (such as
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Lorentz invariance) or changing the physics of the system
in any way. One often hears that gauge symmetry is fundamental, in electromagnetism, for example, but the more
correct statement is that gauge symmetry in electromagnetism is necessary only if one demands the convenience of
linearly realized Lorentz symmetry and locality. Fixing a
gauge will not change the physics, but the price paid is that
the Lorentz symmetries and locality are not manifest.
On the other hand, starting from a system without gauge
invariance, it is always possible to introduce gauge symmetry
by putting in redundant variables. Often this can be very
useful for studying a system and can elucidate properties
which are otherwise difﬁcult to see. This is the essence of
¨
the Stuckelberg trick, which we make use of extensively in
our study of massive gravity. In fact, as we will see, this trick
can be used to make...

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