"Even if the interacting fields were elements of the same algebra W for all $\lambda$ and had a common set of states, as Dyson noted more than 60 years ago, for a theory
such as the one with interaction Lagrangian $\lambda \varphi^4$, a ground state $\omega_0(\lambda)$ cannot be expected to be analytic in $\lambda$ at $\lambda = 0$, since no ground state can exist when $\lambda<0$. Thus, perturbative expressions for quantities such as the S-matrix should not converge—even if it were the case that notions of “in” and “out” particle states could be defined."