I know that you can come across non-perturbative effects in QFT, particular when the coupling constant lies outside the radius of convergence of the asympototic perturbation series.
From the perspective of canonical quantization I understand how and why such instantons crop up. Specifically, they give (non-trivial) exact solutions of the equations of motion for the full interacting theory. These may be canonically quantized to produce a quantum instanton.
My question relates to the origins of instantons from the path-integral approach. Weinberg (in Volume II of his QFT text) claims that in perturbation theory
we expand the action $S$ around spacetime independent vacuum values of the fields, keeping the leading quadratic term in the exponential $\exp(iS)$
What does he mean by this? I've definitely never done any expansions around vacuum values to derive perturbation theory for the path integral. Doesn't one usually just split e.g. $\exp(iS_{free})\exp(iS_{int})$ and that immediately gives you the series?
He goes on to say that non-perturbative effects
arise because there are stationary points of the action which are spacetime dependent
I understand this in the context of canonical quantization, but it doesn't make much sense to me with the path-integral. Unless of course one introduces the stationary phase approximation, but I don't know its relation to usual pertubation theory. Hopefully this will be clearer once the previous paragraph makes sense to me!