Full form of the Pauli-Fierz action

Question

In Deser's paper on the fully interacting version of the Pauli Fierz theory, he does a rather simple method of treating the Pauli Fierz equation without going with infinite sums, just by treating the metric and the connection separately, hiding the infinite sum in the dynamics linking the (inverse) metric to the connection. But in this paper, he writes the Hilbert action as

\begin{equation} S = \int d^4x\ g^{\mu\nu} R_{\mu\nu} \end{equation}

which I suppose will derive something of the form

\begin{equation} S = \int d^4x\ (\eta^{\mu\nu} + h^{\mu\nu}) R_{\mu\nu} (\Gamma) \end{equation}

But his definition of the Hilbert action lacks the determinant of the metric to fix the integral volume in curved space, which according to Feynman's lecture in gravitation does intervene in the action, of the form

\begin{equation} \sqrt{-g} = 1 + h^\beta_\beta + \mathcal{O}(h^2) \end{equation}

But while he does offer some perturbative expansion of the Hilbert action in terms of the spin 2 field of the Pauli Fierz theory, he does not really give its full expression as an infinite sum.

Is Deser's action correct? I have seen this document implying that the determinant might be absorbed in the field we use for $g^{\mu\nu}$ (as it is just slides it's rather sparse on details), but even then that would not work in general since we would still need it for the matter term of the Lagrangian. Also what is actually the infinite sum formulation? I have yet to see it actually written down properly and not cut off rather quickly.

This post imported from StackExchange Physics at 2015-11-02 20:31 (UTC), posted by SE-user Slereah

Answer 1

There has been some substantial increase in the knowledge of massive gravity (and the non-linear completion) in the past few (maybe less than 10) years. Its worth checking out this paper on arxiv; written by Claudia de Rham who was one of the discoverers of the new non-linear massive gravity.