Perturbation theory in large order

AbstractFor many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x24+ϵx44−E)y=0, y(±∞)=0,the perturbation coefficients An in the expansion for the ground-state energyE(ground state) ∼ ∑n=0∞Anϵnsimplify dramatically as n → ∞:An∼−(6π3)12Гn+12).We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λϕ4 model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron