I'll try to provide some new insights.
1) Usually a QFT is derived from a classical field theory via a procedure called quantization. This translates fields into field operators. Classical structures of the Hamiltonian formulation, especially the Poisson algebra, are translated into structures on Hilbert space, especially commutation relations. In a classical, relativistic field theory one can construct generators of the Poincare algebra, i.e.
##P^\mu = (H, P^i), L^i, K^i##
In the QFT context the commutators generate the Poincare algebra, just as the classical Poisson brackets do in the classical field theory context. The Poincare algebra is represented on the Hilbert space, which means that physical states members of irreducable representations of the Poincare algebra, and that there are two Casimir operators
##C^{(1)} = P_\mu P^\mu##
##C^{(2)} = W_\mu W^\mu##
which means that physical states with 4-momentum p have invariant mass
## m^2 = p_\mu p^\mu##
(Wigner classification)
This is nothing else but a translation of relations known from classical field theories into QFT language. So SR cannot be derived from QFT.
SR is a general framework which fixes certain symmetry aspects for both classical and quantum field theories.
2) I don't think that it's correct to say ...
ftr said:
... we cannot apply GR to QFT even so GR is considered to be the generalization of SR.
First we do not want to apply GR to QFT (or formulate QFT on curved but classical spacetime), but we want to apply the above mentioned quantization to GR. I agree that this is still work in progress, but it is not true that this is impossible. It is correct that certain methods (especially perturbation theory and perturbative renormalization) do fail when quantizing GR, but this is not due to a fundamental incompatibility of GR with quantization, but only an indication of the limited applicability of these methods. It is true that we do not have a complete (and testable) theory of quantum gravity (= quantization of GR), but we have a couple of candidate theories passing many consistency tests (Loop Quantum Gravity, Asymptotic Safety approach, Causal Dynamical Triangulation, several Supergravity models, ...)
So my conclusion is that we (or our descendants) will be able to quantize GR.
3) The conclusion in
ftr said:
... many theories of QG try to consider GR as a quantum theory which implies that the origin of SR is indeed in QM system somehow.
is not correct. Yes, many theories (all candidates I have listed above) try to quantize GR. But the conclusion that this quantization results in SR (in some limit) is not correct. SR is already present in classical GR, but only as a local (instead of a global) symmetry. Spacetime in GR need not have any global symmetry at all,
but is always has local Lorentz invariance by construction. This is nothing else but the famous equivalence principle. A free-falling observer cannot detect spacetime curvature locally; to her spacetime always looks flat. So for local experiments a free-falling observer can always use SR calculations.
So the origin of SR in QG is not the Q, it's the G ;-)
And it's the same idea that I already indicated above:
Quantization respects certain classical symmetries, in our case local Lorentz invariance.
