M 0 limit in this limit we end up with a massless

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Unformatted text preview: ! A þ @ : (4.17) Z 1 dD xLm¼0 À F F À 2ðh @ @  À h@2 Þ 2 þ h T  : (4.23) We see that this has all 5 degrees of freedom: a scalar-tensor vector theory where the vector is completely decoupled but the scalar is kinetically mixed with the tensor. To see this, we unmix the scalar and tensor, at the expense of the minimal coupling to T  , by a field redefinition. Consider the change h ¼ h0 þ  ;  (4.24) where  is any scalar. This is the linearization of a conformal transformation. The change in the massless spin 2 part is (no integration by parts here) Lm¼0 ðhÞ ¼ Lm¼0 ðh0 Þ þ ðD À 2Þ½@ @ h0 À @ @ h0 þ 1ðD À 1Þ@ @ Š: 2 (4.25) The action (4.15) now becomes S¼ Z 1 dD xLm¼0 À m2 ðh h À h2 Þ 2 12 À m F F À 2m2 ðh @ A À h@ A Þ 2 À 2m2 ðh @ @  À h@2 Þ þ h T  À 2A @ T  þ 2@@T; (4.18) where @@T  @ @ T  and we have integrated by parts in the last term. There are now two gauge symmetries h ¼ @  þ @  ; A ¼ @ Ã; A ¼ À ;  ¼ ÀÃ: and the gauge transformations read (4.19) (4.20) By fixing the gauge  ¼ 0 we recover the Lagrangian (4.15). Suppose we now rescale A ! ð1=mÞA ,  ! ð1=m2 Þ, under which the action becomes S¼ Z 1 1 dD xLm¼0 À m2 ðh h À h2 Þ À F F 2 2   À 2mðh @ A À h@ A Þ À 2ðh @ @  À h@2 Þ þ h T  À 2 2 A @ T  þ 2 @@T; m  m (4.21) and the gauge transformations become h ¼ @  þ @  ; A ¼ @ Ã; A ¼ Àm ;  ¼ ÀmÃ; (4.22) where we have absorbed one factor on m into the gauge parameter Ã. Now take the m ! 0 limit. [If the source is not conserved and the divergences do not go to zero fast enough with m (Fronsdal, 1980), then  and A become strongly coupled to the divergence of the source, so we now assume the source is conserved.] In this limit, the theory now takes the form Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 This is simply the linearization of the effect of a conformal transformation on the Einstein-Hilbert action. By taking  ¼ ½2=ðD À 2ފ in the transformation (4.24), we can arrange to cancel all the off-diagonal h terms in the Lagrangian (4.23), trading them in for a  kinetic term. The Lagrangian (4.23) now takes the form Z 1 DÀ1 @ @  S ¼ dD xLm¼0 ðh0 Þ À F F À 2 2 DÀ2  2 T; (4.26) þ h0 T  þ  DÀ2 h0 ¼ @  þ @  ;  A ¼ @ Ã;  ¼ 0: A ¼ 0; (4.27) (4.28) There are now (for D ¼ 4) manifestly 5 degrees of freedom, two in a canonical massless graviton, two in a canonical massless vector, and one in a canonical massless scalar.6 Note, however, that the coupling of the scalar to the trace of the stress tensor survives the m ¼ 0 limit. We exposed the origin of the vDVZ discontinuity. The extra scalar degree of freedom, since it couples to the trace of the stress tensor, does not affect the bending of light (for which T ¼ 0), but it does affect the Newtonian potential. This extra...
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