The only thing that changes is the mass term and the

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Unformatted text preview: the unitary gauge (a gauge condition for which it is permissible to substitute back into the action, because the potentially lost  equation is implied Kurt Hinterbichler: Theoretical aspects of massive gravity by the divergence of the A equation), we recover the original massive Lagrangian (4.1), which means Eqs. (4.1) and (4.3) are equivalent theories. They both describe the 3 degrees of freedom of a massive spin 1 in D ¼ 4. The new Lagrangian (4.3) does the job using more fields and gauge symmetry. ¨ The Stuckelberg trick is a terrific illustration of the fact that gauge symmetry is a complete sham. It represents nothing more than a redundancy of description. We can take any theory and make it a gauge theory by introducing redundant variables. Conversely, given any gauge theory, we can always eliminate the gauge symmetry by eliminating the redundant degrees of freedom. The catch is that removing the redundancy is not always a smart thing to do. For example, in Maxwell electromagnetism it is impossible to remove the redundancy and at the same time preserve manifest Lorentz invariance and locality. Of course, electromagnetism with gauge redundancy removed is still electromagnetism, so it is still Lorentz invariant and local, it is just not manifestly so. ¨ With the Stuckelberg trick presented here, on the other hand, we are adding and removing extra gauge symmetry in a rather simple way, which does not mess with the manifest Lorentz invariance and locality. We see from Eq. (4.3) that  has a kinetic term, in addition to cross terms. Rescaling  ! mÀ1  in order to normalize the kinetic term, we have S¼ Z 1 1 dD x À F F À m2 A A À mA @  4 2 1 1 À @ @  þ A J  À @ J ; (4.5) 2 m and the gauge symmetry reads A ¼ @ Ã;  ¼ ÀmÃ: (4.6) Consider now the m ! 0 limit. Note that if the current is not conserved [or its divergence does not go to zero with at least a power of m (Fronsdal, 1980)], then the scalar becomes strongly coupled to the divergence of the source and the limit does not exist. Assuming a conserved source, the Lagrangian becomes in the limit L ¼ À1F F À 1@ @  þ A J ; 4 2 A ¼ @ Ã;  ¼ 0: (4.8) It is now clear that the number of degrees of freedom is preserved in the limit. For D ¼ 4 two of the 3 degrees of freedom go into the massless vector, and one goes into the scalar. In the limit the vector decouples from the scalar, and we are left with a massless gauge vector interacting with the source, as well as a completely decoupled free scalar. This m ! 0 limit is a different limit than the nonsmooth limit we would have obtained by taking m ! 0 straight away in Eq. (4.1). We have scaled  ! mÀ1  in order to canonically normalize the scalar kinetic term, so we are actually using a new scalar new ¼ mold which does not scale with m, so the smooth limit we are taking is to scale the old scalar degree of freedom up as we scale m down, in such a way that the new scalar degree of freedom rema...
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