My question involves an analogy I have to point out. Consider the Lagrangian density for the a complex scalar field: \begin{equation} \mathcal{L}=\frac{1}{2}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{\dagger}\phi \end{equation} which is invariant under the global gauge transformation $$\phi \rightarrow e^{i\alpha}\phi.$$ The divergence-less 4-current is $j=(\rho,\underline{j})$ and the associated conserved quantity is the electric charge. This illustrates a global $U(1)$ symmetry. To see the analogy we now look at spacetime translations $x \rightarrow x + a$ which give $$\phi(x) \rightarrow \phi(x+a) = e^{a^{\mu}\partial_{\mu}}\phi.$$ The Lagrangian is no longer invariant but the action: \begin{equation} S = \int \mathcal{L} d^{4}x \end{equation} is invariant due to the translation invariance of the Lebesgue measure. The Noether current is the stress-energy tensor $T^{\mu\nu}$ and the conserved quantity is the 4-momentum. Due to the role played by $T^{\mu\nu}$ in the Einstein field equations we can say that the 4-momentum plays the role of gravitational charge.
Promoting the global U(1) symmetry to a local one couples $\phi$ to the electromagnetic field. This involves modifying the Lagrangian by introducing a covariant derivative and adding the Lagrangian for the EM field. My question is: does promoting the global translation invariance of S to a local one give the correct coupling of $\phi$ to the graviton field? An action invariant under general coordinate transformations is: \begin{equation} S = \int \mathcal{L} \sqrt{- g}d^{4}x \end{equation} where $\mathcal{L}$ should also be modified appropriately and $g_{\mu\nu}$ is the metric. I rephrase and extent my question:
(1) Do general coordinate transformations include local translations $x \rightarrow x + a(x)$ or are they only $SO(1,3)$ gauge-like transformations?
(2) In view of the above analogy the gauge group for gravity is the translation group in Minkowski spacetime. The group is not compact. Shouldn't the group be compact to guarantee a positive-definite kinetic term for gravitons?
(3) The translation group is also abelian. Isn't this inconsistent with the fact that gravitons self-interact?
(4) In essence what I am saying is that the classical theory of gravity is likely incomplete. Are there any approaches to gravity that modify the underlying spacetime such that the isometry group of the new space admits a compact version of the translation group of Minkowski spacetime? Maybe the universe after formation was compact but due to expansion the compact nature is now hidden.
(5) Related to (4). Shouldn't elementary particles correspond to irreducible representations of the isometry group of the universe? If the universe is Minkowski then the isometry group is the Poincare group. Have other possibilities in this regard been explored?
(6) In c = h = 1 units Newton's constant G has mass dimension -2. The fine structure constant is given by: $$\alpha = \frac{e^{2}}{4 \pi \epsilon_{0}}$$ In gravitoelectromagnetism we have the correspondence $G \leftrightarrow \cfrac{1}{4\pi \epsilon_{0}}$. As the momentum $p$ plays the role of gravitational charge, shouldn't the coupling constant to gravity be given by $$g \sim p^{2} G = m^{2} G$$ where $m$ is mass of the particle in question? $g$ is then dimensionless. Does this imply that the covariant derivative to couple matter to gravity is given by something like $D_{\mu} = \partial_{\mu} - p^{\nu} A_{\mu\nu}$? where $A_{\mu\nu}$ is some tensor potential related to $g_{\mu\nu}$.
(7) The above ideas might imply something about spin 0 bosons and their mass renormalization. Any thoughts? Could the $\Lambda^{2}$ dependence of $m^{2}$ be just an indication that gravity was ignored?
