Let's say we have an interacting Quantum Field Theory, such as Quantum Electrodynamics if we want to compute an amplitude such as two electrons scattering off each other, then we take the following steps:

1. We simulate the full interacting theory by making a perturbation about the free theory.

2. Since we're doing a scattering amplitude we can approximate the incoming and outgoing states as free field states

3. We sum up the perturbation terms, using all our renormalization and regularization techniques and we get the amplitude for the process.

Now, I have heard that the perturbation series does not converge for Quantum Electrodynamics and I was wondering what is the reason for this.

Is it because the perturbation series cannot fully represent the full nonperturbative theory, kind of like what happens for the expansion of [tex]\frac{1}{1+x^2}[/tex] for [tex]x > 1[/tex]. This is what I would have guessed.

However I've also read it more so has to do with the interacting Hamiltonian not being a well defined operator (not densely defined) on the Hilbert space of the free field. Which would make sense since the perturbation series is nothing more (in my understanding) than an attempt to simulate evolution in the interacting field Hilbert space due to the full Hamiltonian, by acting repeatedly on the free field Hilbert space with the interacting Hamiltonian [tex]H_{I}[/tex]

I've also heard that full nonperturbative Quantum Electrodynamics has not even been proven to exist. What does this mean?

Also what is the content of Haag's Theorem? I've read it's statement, but I'd like to hear from others in case my understanding is wrong.

Sorry for all the questions, just want to get it right. Fascinating stuff.