http://www.arxiv.org/abs/gr-qc/0409089
        From Gravitons to Gravity: Myths and Reality

reached a different conclusion than me. The author tries to argue that the
usual claims - that diff invariant actions including the Einstein-Hilbert
term can be derived as the only consistent nontrivial (interacting)
extension of the 2-form gauge invariance for a spin 2 tensor h_{ab} - are
incorrect.

Well, I happen to disagree.

There are various potentially awkward details about the paper - he tries
to argue that there can be tadpoles (linear terms in h and it derivatives)
in the expansion of the Einstein-Hilbert action as g=\eta+h, even though
"h" is a deviation from a classical solution - which implies that there
are no linear terms in h. He does not want to throw away total derivatives
in cases in which he certainly should, and so on.

But I think that the main problem that leads him to his incorrect
conclusion is that he does never get the right gauge invariance. If you
look at his equations (2) and (66) - and I think that there are no better
prescriptions for gauge invariance in his paper

        h_{ab} \to h_{ab} + \partial_{a}\xi_{b} + \partial_{b}\xi_{a},

you see that he always considers the invariance under the *linearized*
gauge transformation for the tensor h. Of course, the gauge
transformations generated above form an abelian group, which cannot be
isomorphic to the nonabelian diff invariance, and therefore it's not
surprising that Paddy could not derive the correct GR if he used an
incorrect formula for the gauge transformation throughout his paper.

I believe that his actions - e.g. equations (50)-(54) - must be clearly
gauge-non-invariant under his gauge invariance, and this is probably the
most obvious technical complaint I would raise to invalidate the paper.

The paper explains many times quite explicitly that Paddy is not happy
with the idea to accept the simple fact that the diff invariance *is* the
same thing as the gauge-invariance for the tensor "h", although written in
different variables.

...

However, the issue is what to do with the linearized gauge invariance
above. The gauge transformation above is linear, and therefore
gauge-invariant functions of "h" would have to be gauge-invariant
piece-by-piece in an expansion in "h". I think that is is pretty obvious
that it implies that all gauge-invariant functions of this "h" and its
derivatives would have to be functions of a linear gauge-invariant
expression involving "h", but there is no one.

This prevents the spin 2 tensors with the simple, linear (abelian) gauge
invariance to self-interact. Paddy could still obtain the correct
description of the interactions between matter and h_{ab} - it's because
neglecting the nonlinear terms in the gauge invariance is equivalent to
working with G=0 and allows one to neglect gravitational
self-interactions, while getting the interaction with other fields
correctly.

For spin 2, there is a cure that allows gravitons to self-interact, which
is to deform the equations (2) and (66) and add the nonlinear terms.
These effectively replace the partial derivative by the covariant
derivative with respect to the metric (\eta_{ab}+h_{ab}). It is analogous
to the nonabelian groups used in Yang-Mills theory - they can also be
viewed as a nonlinear deformation of U(1)^{dimension}.

For the symmetric tensor's gauge invariance, there is nothing such as
F_{\mu\nu} for electromagnetism, I believe. For example, F_{\mu\nu} is the
antisymmetrized derivative of A_\mu, and the antisymmetrization is
necessary so that - when combined with a symmetric second derivative - the
gauge variations of F_{\mu\nu} cancel.

However for the symmetric tensor, we would have to antisymmetrize it, too,
to cancel the variation, but an antisymmetrization of a tensor that has a
pair of symmetric indices gives zero. Therefore, there is no nonzero
gauge-invariant "field strength"  computed from h_{ab}, and consequently
there are also no higher-order nonzero gauge-invariant functions of "h"
and its derivatives.

The only way how to get the self-interactions is to modify the gauge
invariance, and add nonlinear terms. This identifies the spin 2 tensor
gauge invariance with diff invariance in different variables, and the
Einstein-Hilbert action (plus possible higher-order terms, such as R^2) is
the only gauge-invariant (under the nonlinear gauge invariance for the
spin 2 field h_{ab}) action one can construct, up to field redefinitions.

For fields with spin above 2, there is nothing even like that, and one
cannot consistently define an interacting higher spin field respecting the
gauge invariance (which is necessary to eliminate its negative norm
polarizations). For massive spin 1, the usual Higgs mechanism is the only
choice, too. The exceptions I know - which are able to make interactions
for higher spin fields etc. - are theories with infinitely many fields -
string theory itself, and that crazy gravitational dual of the linear
vector O(N) sigma models, proposed by some Russian physicists in the past,
that Polyakov studied as a toy-model for AdS/CFT.

Is the description above correct? Are there some other loopholes you know?

Cheers,
Lubos
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