This is an attempt to ask separately about aspects of my previous question, which was closed as too broad. Note that I strongly prefer results that are or can be made mathematically completely rigorous. This question is essentially the same as proof of radius of convergence of perturbation series in quantum electrodynamics zero (2)
Related questions include How can an asymptotic expansion give an extremely accurate predication, as in QED? and Asymptoticity of Pertubative Expansion of QFT, which has a nice reference to some lecture notes on the connection of perturbative and non-perturbative aspects of QFT.
As far as I know, the free quantum theories for both the Dirac field as well as the Photon field on Minkowski space are well defined, that is: One has a Hilbert space (the fermionic and bosonic Fock spaces respectively) and a self-adjoint (in the precise mathematical sense of the term) Hamiltonian operator for the free evolution. In addition one has a unitary representation of the full (disconnected) Poincaré group (correct me if I'm wrong, precision is a big issue for me here). This allows one to treat the free theories just as nonrelativistic quantum theories. In some inertial frame of reference one can specify an initial state, specifying a field configuration. One would then be able to calculate the state at a later time (in that frame), or as it would be seen from a different reference frame, without the need for any perturbation theory. I think one can also meaningfully answer questions as "what is the probability of detecting exactly one photon within a given spacetime region". One would have to work in "position space" but it seems like a meaningfull question. In principle using suitable numerics and arbitrary amounts of computation time, all such well-defined problems can be given arbitrarily precise answers?!
In the interacting theory (full QED, the two above free theories + minimal coupling), this does not seem to be the case. One does not have a Hilbert space and it goes without saying that there are no well-defined operators to speak of. The theory is "defined perturbatively", where one (after renormalization) only (Correct me, as I assume this claim is wrong) makes predictions for scattering cross sections and decay rates in the form of formal power series (where each term is finite after renormalization) in something somehow related to the fine structure constant. For this I would like to quote part of one answer from (2):
[...] EDIT: Many think the perturbation parameter in QED is α∝$e^2$ , but it is not the case. To obtain a meaningful calculation result, one has not only to perform renormalization, but also fulfill soft diagram summation. The latter is equivalent to taking into account α in another initial approximation. In other words, the meaningful perturbation expansion is different from the Dyson's one and may converge. The subtlety is not immediately visible, but it is implied that each term of Dysons' expansion is different from zero, and it is not so! For example, a Compton scattering amplitude calculated in the first Born approximation (Klein-Nishina) represents an elastic process that never happens in reality and in the theory because soft photon radiation. Only inclusive cross sections are different from zero in QED. The Klein-Nishina formula is multiplied in QED by an elastic form-factor whose first term of the Taylor series is unity, but the rest, after summation, gives zero, like $e^{−x}=1−x+x^2/2−...$ when $x\to \infty$.
My question is:
In the final state of full QED (renormalized etc.), has it been shown rigorously (Dyson's argument is not rigorous!) that the perturbation expansion for physical quantities (for example the magnetic moment of the electron) does not converge? Surely this has been tried. Is it an open problem in 2017? The above answer suggests that the expansion is not in powers of the fine structure constant, which contradicts many other sources. What is the expansion parameter really?
