# Perturbation theory and size of the perturbation

In quantum field theory, we usually perturb the free field by a little bit. What would be so bad about using a large perturbation to the free field?

• Then the exact solution would be very different - quantitatively and qualitatively - from the initial approximation. You cannot imagine how different it would be. Commented Dec 29, 2021 at 9:07

1. In general, perturbation theory schematically gives you results for quantities of interest $$q$$ of the form $$q = q_0 + \sum_n\lambda^nq_n$$, where $$\lambda$$ is a parameter meaningfully related to the size of your perturbation. The idea is that for small $$\lambda$$, it then suffices to compute this expression up to low values of $$n$$ to get "most" of $$q$$. When $$\lambda$$ is large, each $$q_{n+1}$$ becomes more relevant than its predecessor $$q_n$$, making your perturbative approach useless as you have to compute an ever more relevant infinity of terms.
2. Specifically in QFT, most perturbative series in $$\lambda$$ are asymptotic series whose number of "useful" terms is roughly given by $$\lambda^{-1}$$ - meaning that at large $$\lambda$$, there are no useful terms at all. See this question and its linked questions for more on the asymptotic nature of the perturbation series of QFT.
• A remark: for perturbatively well defined theories like $\phi^4$ in 2 and 3 spacetime dimensions, the series diverges but is nevertheless Borel summable. In other words, the nonperturbative information is contained in the perturbation series. It's just that successive summing over terms, that one usually does, is the wrong way to extract that information. Commented Dec 29, 2021 at 23:05