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Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Rather than unitary gauge, we can instead ﬁx a Lorentzlike gauge for the action (4.3),
@ A þ m ¼ 0: (4.9) This gauge ﬁxes the gauge freedom up to a residual gauge
parameter satisfying ðh À m2 ÞÃ ¼ 0. We can add the gauge
ﬁxing term
SGF ¼ Z 1
dD x À ð@ A þ mÞ2 :
2 (4.10) As in the massless case, quantum mechanically this term
results from the Fadeev-Popov gauge ﬁxing procedure.
Adding the gauge ﬁxing term diagonalizes the Lagrangian,
S þ SGF ¼ Z 1
1
dD x A ðh À m2 ÞA þ ðh À m2 Þ
2
2
1
þ A J À @ J ;
(4.11)
m and the propagators for A and are, respectively,
Ài
;
p2 þ m 2 Ài
;
p2 þ m 2 (4.12) which are similar to $1=p2 at high momenta. Thus we have
managed to restore the good high energy behavior of the
propagators.
It is possible to ﬁnd the gauge invariant mode functions for
A and , which can then be compared to the unitary gauge
mode functions of the massive photon. In the massless limit,
there is a direct correspondence; is gauge invariant and
becomes the longitudinal photon, A has the usual Maxwell
gauge symmetry and its gauge invariant transverse modes are
exactly the transverse modes of the massive photon.
¨
B. Graviton Stuckelberg and origin of the vDVZ discontinuity Now consider massive gravity,
S¼ Z (4.7) and the gauge symmetry is 683 1
dD xLm¼0 À m2 ðh h À h2 Þ þ h T ;
2
(4.13) where Lm¼0 is the Lagrangian of the massless graviton. We
want to preserve the gauge symmetry h ¼ @ þ @
¨
present in the m ¼ 0 case, so we introduce a Stuckelberg ﬁeld
A patterned after the gauge symmetry,
h ! h þ @ A þ @ A : (4.14) The Lm¼0 term remains invariant because it is gauge invariant and Eq. (4.14) looks like a gauge transformation, so all
that changes is the mass term,
S¼ Z 1
1
dD xLm¼0 À m2 ðh h À h2 Þ À m2 F F
2
2
2
À 2m ðh @ A À h@ A Þ þ h T
À 2A @ T ; (4.15) where we have integrated by parts in the last term, and where
F @ A À @ A . Kurt Hinterbichler: Theoretical aspects of massive gravity 684 There is now a gauge symmetry
h ¼ @ þ @ ; A ¼ À ; S¼
(4.16) and ﬁxing the gauge A ¼ 0 recovers the original massive
gravity action (as in the vector case, this is a gauge condition
for which it is permissible to substitute back into the action,
because the potentially lost A equation is implied by the
divergence of the h equation). At this point, we might
consider scaling A ! mÀ1 A to normalize the vector kinetic term, and then take the m ! 0 limit. In this limit, we
end up with a massless graviton and a massless photon, for a
total of 4 degrees of freedom (in four dimensions). So at this
point, m ! 0 is still not a smooth limit, since we would be
losing one of the original 5 degrees of freedom.
We go one step further and introduce a scalar gauge
¨
symmetry, by introducing another Stuckelberg ﬁeld ,
A...

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