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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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