Poincare Symmetry in QFT Given that spacetime is not affine Minkowskispace, it does of course not possess Poincare symmetry. It is still sensible to speak of rotations and translations (parallel transport), but instead of
$$[P_\mu, P_\nu] = 0$$
translations along a small parallelogram will differ by the curvature. I have not thought carefully about rotations and translations, but basically you could look at the induced connection on the frame bundle, to figure out what happens.
This is all to say that spacetime has obviously not exact Poincare symmetry, although the corrections are ordinarily very small. Most QFT textbooks seem to ignore this. Of course it is possible to formulate lagrangians of the standard theories in curved space and develop perturbation theory, too. But since there is no translation invariance, one can not invoke fourier transform.
My questions are: 


*

*Why is it save to ignore that there is no exact poincare symmetry? Especially the rampant use of fourier transforms bothers me, since they do require exact translation invariance. 

*How does one treat energy momentum conservation? Presumably one has to (at least) demonstrate that the covariant derivative of the energy momentum tensor is zero.


Any references that discuss those issues in more detail are of course appreciated.
 A: When done properly, none of the problems exists and some of your assumptions are invalid. First, concerning the two questions,


*

*in topologically trivial but arbitrarily curved spacetimes, the Poincaré symmetry holds in the sense that it is a small subgroup of the infinite-dimensional group of all diffeomorphisms; general relativity and all of its extensions are invariant under arbitrary coordinate transformations which – for spacetimes of normal topology – include the Lorentz transformations as well.

*on the contrary, the energy-momentum conservation law doesn't hold locally in general relativity. There's no coordinate-system-independent, nonzero definition of the stress-energy tensor in general relativity whose divergence using partial derivatives would vanish; this is related to the breaking of the translational symmetries by general backgrounds because these translational symmetries are needed, by Noether's theorem, for the conservation laws to exist. In GR in asymptotically flat or otherwise translationally symmetric backgrounds, one may still define a conserved total ADM mass and other things.
I would also like to correct some other statements:


*

*the existence of a Fourier transform doesn't depend on the Lorentz symmetry. In $\exp(ip\cdot x)$, the inner product is really an action of a linear form on a vector (they live in mutually dual spaces) so one doesn't need an inner product on either of the two spaces.

*general relativity, especially in the presence of fermions etc., is often formulated using tetrads/vierbeins/vielbeins (they're really needed for the fermions). Then the total gauge symmetry group (under which the physical states must be invariant) includes not only the diffeomorphisms but also local Lorentz transformations.

*quantum gravity expanded around the Minkowski space does exactly preserve the global Lorentz symmetry as well, despite the fact that coherent states of gravitons are able to add curvature into the spacetime and turn it into a Lorentz-symmetry-violating geometry. That's because these gravitons may still be treated as spin-2 fields that exist in the pre-existing background.

*the gauge symmetries have to annihilate physical states; there may be exceptions for symmetry generators that move the asymptotic region of the spacetime (at infinity); the physical states may carry charges under these generators but these generators must still be isometries of the background (and the corresponding superselection sector of the Hilbert space).
What's true about Nestoklon's comment is that one faces problems when he tries to quantize Einstein's equations including all loop corrections; the theory is non-renormalizable. But these problems don't get imprinted to none of the hypothetical problems you have sketched as long as one is satisfied with the one-loop approximation, for example.
A: It is safe to ignore curvature at the length scales of particle physics, as in the relevant region of space-time one can approximate the manifold very well by its tangent space, which is a flat Minkowski space with Poincare symmetry.
For the same reason, engineers do not use general relativity but work with special relativirty (or even Newton's laws). Carrying the extra conceptual burden would only complicate things without any benefit.
But if curvature is important, one can use a curved version of the Fourier transform ((see, e.g., work by Fonarev, gr-qc/9309005).
