Can we get full non-perturbative information of interacting system by computing perturbation to all order?

Perturbation theory gives for the solution an asymptotic series in the coupling constant $g$. There are infinitely many functions having the same asymptotic series, since for example adding a function of $e^{-c/g^2}$ vanishing at zero will not change the asymptotic series.

Thus in general, the perturbation series does not give full perturbative information. Every summation procedure needs to make additional assumptions about the solution; it will resum the series correctly when these assumptions are satisfied but in general not otherwise.

In many toy instances one can prove that the assumptions of Watson's Borel summation theorem can be shown to hold; then Borel summation works. But it is known not to work in other cases, e.g., in the (frequent) presence of renormalons.

In 4D relativistic quantum field theory it is not known of any resummation method whether it will work. The most powerful resummation technique, based on resurgent transseries has the most promise.

  1. It's my understanding that taking the sum up to the least term in an asymptotic series gives exponentially good accuracy, but no further. The exponentially small error term can be attributed to the topological sectors in some cases.

  2. When we perform Borel resummation, there is a phenomenon called "resurgence" where these exponential terms come in series which look just like the "vacuum" perturbation series but with a prefactor like $e^{-S_0/g^2}$ where $S_0$ is interpreted as the instanton action. The list of examples of this is growing every day. See this paper for instance (and those referencing it): . Presumably after you resum the series it converges to an exact answer, at least in cases with resurgence. I don't know any theorems though.