RevModPhys.84.671

As follows 00 0 0 p eip 1 x 00

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Unformatted text preview: ðÃpÞðLÀ1 ðÃpÞÃLðpÞފ 0 ½LðÃpÞðLÀ1 ðÃpÞÃLðpÞފ 0 ðk; Þ  eiÃpÁx 00 ¼ ½LðÃpÞW ðÃ; pފ 0 ½LðÃpÞW ðÃ; pފ 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 à 0 à 0   ðp; ÞeipÁà 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  Ài 1 ðP  P  þ P  P  Þ D ; ¼ 2 22 p þm  1 P P ; À (2.44) DÀ1 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 becomes Z 1 (2.45) Sm¼0 ¼ dD x h E ; h ; 2 where the kinetic operator is E  ¼ O jm¼0 ¼ ðð Þ À   Þh À 2@ð @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 2@ð @ 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 gauge), @ 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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