X we nd g y x f jy f jy g fy

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Unformatted text preview: ðY ðxÞÞ: (6.8) We now expand Y about the identity, Y ðxÞ ¼ x þ A ðxÞ: (6.9) The quantity G is expanded as G @Y ðxÞ @Y ðxÞ ¼ g ðY ðxÞÞ @x @x @ðx þ A Þ @ðx þ A Þ g ðx þ AÞ ¼ @x @x  ¼ ð þ @ A Þð þ @ A Þ g þ A @ g    1 þ A A @ @ g þ Á Á Á 2 ¼ g þ A @ g þ @ A g  þ @ A g  1 þ A A @ @ g þ @ A @ A g 2 þ @ A A @ g  þ @ A A @ g þ Á Á Á : (6.10) We now look at the infinitesimal transformation properties of g, Y , G, and Y , under infinitesimal general coordinate transformations generated by fðxÞ ¼ x þ ðxÞ. The metric transforms in the usual way, g ¼  @ g þ @  g þ @  g :    (6.11) The transformation law for the A’s comes from the transformation of Y , Y  ð xÞ ! fÀ1 ðY ðxÞÞ % Y  ð xÞ À A ¼ À ðx þ AÞ ¼ À À A @  À 1A A @ @  À Á Á Á : 2 (6.12) The A are the Goldstone bosons that nonlinearly carry the broken diffeomorphism invariance in massive gravity. The combination G , as we noted before, is gauge invariant (6.13) ¨ We now have a recipe for Stuckelberg-ing the general massive gravity action of the form (5.3). We leave the Einstein-Hilbert term alone. In the mass term, we write all the h ’s with lowered indices to get rid of the dependence on the absolute metric, and then we replace all occurrences of h with Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 where indices on A are lowered with the background metric. ¨ This is exactly the Stuckelberg substitution we made in the linear case. In the case where the absolute metric is flat, gð0Þ ¼  ,  we have from Eq. (6.10), H ¼ h þ @ A þ @ A þ @ A @ A þ Á Á Á : (6.16) Here indices on A are lowered with  and the ellipsis are terms quadratic and higher in the fields and containing at least one power of h. This takes into account the full nonlinear gauge transformation. As in the linear case, we usually want to do another ¨ scalar Stuckelberg replacement to introduce a Uð1Þ gauge symmetry, A ! A þ @ : (6.17) Then the expansion for the flat absolute metric takes the form H ¼ h þ @ A þ @ A þ 2@ @  þ @ A @ A þ @ A @ @  þ @ @ @ A þ @ @ @ @  þ Á Á Á ; (6.18) where again the ellipsis are terms quadratic and higher in the fields and containing at least one power of h. The gauge transformation laws are h ¼ @  þ @  þ L h ;  ¼ ÀÃ: Y  ¼ À ðY Þ; H ðxÞ ¼ G ðxÞ À gð0Þ ðxÞ;  (6.15) A ¼ @ Ã À  À A @  À 1A A @ @  À Á Á Á ; 2  ðY ðxÞÞ; G ¼ 0: 691 (6.14) (6.19) ¨ This method of Stuckelberging can be extended to any number of gravitons and general coordinate invariances, as done by Arkani-Hamed, Georgi, and Schwartz (2003) and Arkani-Hamed and Schwartz (2004), in analogy with the gauge theory little Higgs models and dimensional deconstruction (Arkani-Hamed, Cohen, and Georgi, 2001a, 2001b). When multiple gravitons are...
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