Bound states in QED

[Jamister]

TL;DR Summary

bound states are problematic in QED and QFT in general.

Weinberg writes in his book on QFT Vol1 that bound states in QED are problematic because perturbation theory breaks down. consider the case of hydrogen atom, electron+proton. Weinberg explains this case and I copy from the book:

https://www.physicsforums.com/attachments/247655
what is time ordered diagrams of old fashion perturbation theory?
I don't understand his explanation. What is this factor $[q^2/ m_e]^{-1}$ ?
In Addition,Weinberg claims the momentum space integration is $q^3$ but I think it is necessary to the integral to know its contribution.
but aside those technical question, how is that perturbation theory breaks down, while we know from QM that for a coulomb interaction we have a perfect ground state for the hydrogen atom, and the coulomb interaction is a direct result of QED.

 

[vanhees71]

Well, just read on in Weinberg's book.

To get the hydrogen spectrum you start perturbation theory from another split of the Hamiltonian into $H_0$ and $H_I$, using the Coulomb gauge, which is more convenient than the covariant gauges in this case.

You can also see this from usual Feynman diagrams. The bound state energies of a proton and an electron is formally given by the poles of the the scattering amplitude for ep->ep. To get poles you have to resum an infinite number of diagrams (at least "ladder diagrams"). This approach is nicely discussed in Landau&Lifshitz, vol. 4.