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**Unformatted text preview: **. As shown by Nicolis, Rattazzi, and
Trincherini (2008), the terms (8.19) are the only polynomial
terms in four dimensions with these properties.
The Galileon was ﬁrst discovered in studies of the DGP
brane world model (Dvali, Gabadadze, and Porrati, 2000a),
for which the cubic Galileon L3 was found to describe the
leading interactions of the brane bending mode (Luty, Porrati,
and Rattazzi, 2003; Nicolis and Rattazzi, 2004). The rest of
the Galileons were then discovered by Nicolis, Rattazzi, and
Trincherini (2008), by abstracting the properties of the cubic
term away from DGP. They have some other very interesting
properties, such as a nonrenormalization theorem [see, e.g.,
Sec. VI of Hinterbichler, Trodden, and Wesley (2010)], and a
connection to the Lovelock invariants through brane embedding (de Rham and Tolley, 2010). Because of these unexpected and interesting properties, they have since taken on a
life of their own. They have been generalized in many
directions (Deffayet, Deser, and Esposito-Farese, 2009,
2010; Deffayet, Esposito-Farese, and Vikman, 2009;
Padilla, Safﬁn, and Zhou, 2010; Deffayet et al., 2011;
Goon, Hinterbichler, and Trodden, 2011; Khoury, Lehners,
and Ovrut, 2011) and are the subject of much recent activity
[see, for instance, the many papers citing Nicolis, Rattazzi,
and Trincherini (2008)].
The fact that the equations are second order ensures that,
unlike Eq. (7.10), no extra degrees of freedom propagate. In
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 699 fact, as pointed out by de Rham and Gabadadze (2010a), the
properties (A.17) of the tensors X guarantee that there are
no ghosts in the Lagrangian (8.10) of the decoupling limit
theory.14 By going through a Hamiltonian analysis similar to
that of Sec. II.A, we see that h00 and h0i remain Lagrange
multipliers enforcing ﬁrst class constraints [as they should
since the Lagrangian (8.10) is gauge invariant]. In addition,
the equations of motion remain second order, so the decoupling limit Lagrangian (8.10) is free of the Boulware-Deser
ghost and propagates 3 degrees of freedom around any
background.
Once the 2 degrees of freedom of the vector A are
included, and if there are no ghosts in the vector part or its
interactions, the total number of degrees of freedom goes to 5,
the same as the linear massive graviton. The vector interactions were shown to be ghost free at cubic order by de Rham
and Gabadadze (2010b). de Rham, Gabadadze, and Tolley
(2010) showed that the full theory beyond the decoupling
limit, including all the ﬁelds, is ghost free, up to quartic order
in the ﬁelds. This guarantees that any ghost must carry a mass
scale larger than Ã3 and hence can be consistently excluded
from the quantum theory. Finally, Hassan and Rosen (2011a,
2011c) showed, using the Hamiltonian formalism, that the
full theory, including all modes and to all orders beyond the
decoupling limit, carries 5 degrees of freedom. The Ã3 theory
is therefore free of the Boulwar...

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