Unformatted text preview: ðÃpÞðLÀ1 ðÃpÞÃLðpÞÞ 0 ½LðÃpÞðLÀ1 ðÃpÞÃLðpÞÞ 0 ðk; Þ eiÃpÁx
00 ¼ ½LðÃpÞW ðÃ; pÞ 0 ½LðÃpÞW ðÃ; pÞ 0 ðk; Þ eiÃpÁx :
The little group element is a spatial rotation. For any spatial rotation R , we have
00 0 00 0 "
R 0 R 0 ðk; Þ ¼ R 0 R 0 B ðk; 0 Þ ¼ B R 00 0 " ðk; 00 Þ ¼ ðBÀ1 RBÞ ðk; 0 Þ: 0 Plugging back into the above,
00 Ã 0 Ã 0 ðp; ÞeipÁÃ À1 x 0 00 0 ¼ LðÃpÞ 0 LðÃpÞ 0 W ðÃ; pÞ ðk; 0 ÞeiÃpÁx ¼ W ðÃ; pÞ ðÃp; 0 ÞeiÃpÁx ; where W is the spin 2 representation of the little group in a basis rotated by B, W ¼ BÀ1 RB. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (2.42) (2.43) Kurt Hinterbichler: Theoretical aspects of massive gravity The right side is the identity operator on the space of symmetric tensors.
Solving Eq. (2.43), we ﬁnd
ðP P þ P P Þ
D ; ¼ 2
P P ;
where P þ p p =m2 .
In the interacting quantum theory, internal lines with momentum p will be assigned this propagator, which for large p
behaves as $p2 =m4 . This growth with p means we cannot
apply standard power counting arguments [such as those of
Chapter 12 of Weinberg (1995)] to deduce the renormalizability properties or strong coupling scales of a theory. We see later
how to overcome this difﬁculty by rewriting the theory in a way
in which all propagators go similar to $1=p2 at high energy.
The massive graviton propagator (2.44) can be compared
to the propagator for the case m ¼ 0. For m ¼ 0, the action
Sm¼0 ¼ dD x h E ; h ;
where the kinetic operator is
E ¼ O jm¼0
¼ ðð Þ À Þh À [email protected] @ð Þ Þ
þ @ @ þ @ @ : (2.46) This operator has the symmetries (2.41). Acting on a symmetric tensor Z it reads
; Z ¼ hZ À hZ À [email protected] @ ZÞ þ @ @ Z
þ @ @ Z : (2.47) The m ¼ 0 action has the gauge symmetry (2.2), and the
operator (2.46) is not invertible. Acting with it results in a
tensor which is automatically transverse, and it annihilates
anything which is pure gauge
@ ð; Z Þ ¼ 0; ; ð@ þ @ Þ ¼ 0:
(2.48) To ﬁnd a propagator, we must ﬁx the gauge freedom. We
choose the Lorenz gauge (also called harmonic, or de Donder
@ h À 1@ h ¼ 0:
2 (2.49) We reach this gauge by making a gauge transformation with
chosen to satisfy h ¼ Àð@ h À 1 @ hÞ. This condi2
tion ﬁxes the gauge only up to gauge transformations with
parameter satisfying h ¼ 0. In this gauge, the equations of motion simplify to
hh À 1 hh ¼ 0:
2 (2.50) The solutions to this equation which also satisfy the gauge
condition (2.49) are the Lorenz gauge solutions to the original
equations of motion.
To the Lagrangian of Eq. (2.45) we add the following
gauge ﬁxing term:
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 LGF ¼ Àð@ h À 1@ hÞ2...
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