What goes wrong with strongly coupled theories?

Question

Let $\lambda$ be the coupling constant of a quantum field theory. It is said that

1. Perturbation theory is only valid when the theory is weakly coupled ($\lambda \ll 1$).
2. In most cases, the series of Feynman diagrams is divergent.

I would like to know the reasoning behind the above two statements.

Since the exponential function has an infinite radius of convergence one would think that by truncating sufficiently far in the series we can obtain a good approximation to the theory. Is the statement (1) better phrased as "the first few terms in the sequence are a good approximation only when the theory is weakly coupled"?

Similarly, why should the series of Feynman diagrams diverge if they are obtained from a convergent power series (of the exponential functional)?

Answer 1

Yes, the problem with strong coupling is typically that you will need many terms to get a decent approximation. However, it is also true that the perturbative series often does not converge. Hence, if you need way too many terms, perturbation theory will just fail, because the series will eventually start to diverge.

The most famous example is probably the $\lambda \phi^4$ theory. For $\lambda < 0$, the theory is unstable (the potential does not have a global minimum), while this is false for $\lambda > 0$. Hence, the theory is not analytic at $\lambda = 0$ due to this stability problem. Hence, perturbation theory will eventually diverge.

Another way of understanding this is noticing that adding loops to Feynman diagrams increases the number of diagrams very fast. You get way too many diagrams for many loops and the series end up growing too fast.

While the series for the exponential function is convergent, calculating physical quantities from it is non-trivial and involves other mathematical manipulations. These manipulations end up spoiling the convergence properties.