atyy said:
Why does the wrong derivation work (I think I've read that it reproduces the right derivation term by term)?
I've also heard that Fell's theorem explains why the wrong derivation works. Is there any substance to that?
Let's just look at the wrong derivation. I'll use the path integral approach where Haag's theorem becomes Nelson's theorem since it is easier to discuss.
We have the path integral:
$$\int{\mathcal{O}\left(\phi\right)d\nu}$$
We then separate the interacting measure into two components, the free measure ##d\mu## and an exponential ##e^{-S_{I}}## to get:
$$\int{\mathcal{O}\left(\phi\right)e^{-S_{I}}d\mu}$$
If we expand the exponential we then get an asymptotic series:
\begin{align*}
\int{\mathcal{O}\left(\phi\right)d\nu} & \approx \int{\mathcal{O}\left(\phi\right)d\mu} \\
& + \int{\mathcal{O}\left(\phi\right)\left(-S_{I}\right)d\mu} \\
& + \dots
\end{align*}
This asymptotic relation is valid in the continuum, it's simply that in the continuum ##d\nu \neq e^{-S_{I}}d\mu##, i.e. it's not the free measure times a function. That's Nelson's theorem, the path integral version of Haag's theorem. So that part of the derivation doesn't work.
However the derivation holds at every finite lattice spacing and thus the asymptotic relation holds at all lattice spacings as well. You can take the continuum limit on both sides of the relation and show it continues to hold in that limit and thus the perturbative series is valid in the continuum.
So one can consider the usual derivation to be a short hand. Introduce a cutoff, then expand the measure, get the asymptotic relation and take the continuum limit on both sides. You just can't use that expansion method directly in the continuum. If you want to prove the relation directly in the continuum there are other methods but they are much more mathematically involved.
Haag/Nelson's theorem just tells you the free and interacting theory are disjoint in the continuum. It doesn't however change the fact that the terms in the expansion in the coupling constant of the interacting theory's moments can be calculated with the free (Gaussian in path integral) theory.
So expanding the moments:
$$\mathcal{W}\left(x_1,...,x_n,\lambda\right) \rightarrow \sum_{n}\lambda^{n}\mathcal{G}_{n}\left(x_1,...,x_n\right)$$
The ##\mathcal{G}_{n}\left(x_1,...,x_n\right)## functions can be computed from Fock space/the Gaussian measure.
There is one side effect of the fact that they are disjoint that shows up when using the free theory to compute the terms. The need to renormalize the terms.
The perturbative series ends up being only asymptotic of course, not convergent. Though that happens in NRQM as well. In lower dimensions for some theories you can use the Borel transform to sum the series and thus existence of the interacting theory can be proved directly from perturbation theory.
In 4D but also for Yang Mills in lower dimensions there are poles in the Borel plane preventing resummation. The poles mean one has to take a contour around them to obtain the interacting theory, but there are infinite such contours introducing an ambiguity of order ##\mathcal{O}\left(e^{-\frac{1}{\lambda}}\right)##. Some are from instantons and others are from renormalons. Renormalons are finite terms resulting from coupling constant renormalization that cause the perturbative series to have an extra ##n!## growth term that leads to poles in the summed Borel series.
