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**Unformatted text preview: **k; Þ ¼ B ðk; 0 Þ; where B is any unitary matrix.
^
Given a particular spatial direction, with unit vector ki ,
there is an SOðd À 1Þ subgroup of the little group SOðdÞ
^
which leaves ki invariant, and the symmetric traceless
representation (rep) of SOðdÞ breaks up into three reps of
SOðd À 1Þ, a scalar, a vector, and a symmetric traceless
tensor. The scalar mode is called the longitudinal graviton
and has spatial components
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
d
ij
^
^i kj À 1 ij :
(2.20)
k
L ¼
dÀ1
d
^
After a large boost in the ki direction, it goes similar to
L $ p2 =m2 . As we see later, in the massless limit, or large
boost limit, this mode is carried by a scalar ﬁeld, which
generally becomes strongly coupled once interactions are
taken into account. The vector modes have spatial components
pﬃﬃﬃ
^
(2.21)
ij ¼ 2kði jÞ ;
V;k k ^
and after a large boost in the ki direction, they go similar to
L $ p=m. In the massless limit, these modes are carried by a
vector ﬁeld, which decouples from conserved sources. The
remaining linearly independent modes are symmetric traceless
^
tensors with no components in the ki directions, and they form
the symmetric traceless mode of SOðd À 1Þ. They are invari^
ant under a boost in the ki direction, and in the massless limit,
they are carried by a massless graviton. In the massless limit,
we therefore expect that the extra degrees of freedom of the
massive graviton organize themselves into a massless vector
and a massless scalar. We see later explicitly how this comes
about at the Lagrangian level.
Upon boosting to p, the polarization tensors satisfy the
following properties: they are transverse to p and traceless,
p ðp; Þ ¼ 0; ðp; Þ ¼ 0; We choose the standard boost to be
1
ð À 1Þpi pj ;
jpj2
pi qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 À 1;
Li 0 ðpÞ ¼ L0 i ðpÞ ¼
jpj
Li j ðpÞ ¼ ij þ (2.14)
L0 0 ðpÞ ¼ ; where
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ p0 =m ¼ jpj2 þ m2 =m
is the usual relativistic . See Chapter 2 of Weinberg (1995) for
discussions of this standard boost and general representation theory
´
of the Poincare group.
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (2.22) and they satisfy orthogonality and completeness relations
ðp; ÞÃ ðp; 0 Þ ¼ 0 ;
X
(2.23) 1
ðp; ÞÃ ðp; Þ ¼ ðP P þ P P Þ
2
À 3 (2.19) 1
P P ;
DÀ1 (2.24) where P þ p p =m2 . The right-hand side of the
completeness relation (2.24) is the projector onto the symmetric and transverse traceless subspace of tensors, i.e., the
identity on this space. We also have the following symmetry
properties in p, which can be deduced from the form of the
standard boost (2.14):
ij ðÀp; Þ ¼ ij ðp; Þ;
0i ðÀp; Þ ¼ À0i ðp; Þ;
00 ðÀp; Þ ¼ 00 ðp; Þ: i; j ¼ 1; 2; . . . ; d; (2.25) i ¼ 1; 2; . . . ; d; (2.26)
(2.27) Kurt Hinterbichler: Theoretical a...

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